{"paper_meta":{"paper_id":"arxiv:0706.3459","title":"0706.3459","authors":[],"year":null,"published":null,"updated":null,"venue_or_source":"arxiv-papers-shard","primary_category":"cs","secondary_categories":[],"doi":null,"license":"unknown","source_type":"pdf+shard","source_url":null,"pdf_url":null},"abstract":{"raw":"","cleaned":"","offsets":[]},"full_paper_text":{"raw_ordered_text":"arXiv:0706.3459v1 [cs.CC] 23 Jun 2007\nNP by means of lifts and shadows\nG ́abor Kun and Jaroslav Neˇsetˇril\nDepartment of Mathematics, University of Memphis\n373 Dunn Hall, Memphis, TN 38152;\nDepartment of Applied Mathematics (KAM) and\nInstitute of Theoretical Computer Science (ITI),\nCharles University, Malostransk ́e n ́am 22, Praha;\nE-mail: kungabor@cs.elte.hu\nE-mail: nesetril@kam.mff.cuni.cz\n1\nAbstract. We show that every NP problem is polynomially equivalent\nto a simple combinatorial problem: the membership problem for a special\nclass of digraphs. These classes are defined by means of shadows (pro-\njections) and by finitely many forbidden colored (lifted) subgraphs. Our\ncharacterization is motivated by the analysis of syntactical subclasses\nwith the full computational power of NP, which were first studied by\nFeder and Vardi.\nOur approach applies to many combinatorial problems and it induces\nthe characterization of coloring problems (CSP) defined by means of\nshadows. This turns out to be related to homomorphism dualities. We\nprove that a class of digraphs (relational structures) defined by finitely\nmany forbidden colored subgraphs (i.e. lifted substructures) is a CSP\nclass if and only if all the the forbidden structures are homomorphically\nequivalent to trees. We show a surprising richness of coloring problems\nwhen restricted to most frequent graph classes. Using results of Neˇsetˇril\nand Ossona de Mendez for bounded expansion classes (which include\nbounded degree and proper minor closed classes) we prove that the re-\nstriction of every class defined as the shadow of finitely many colored\nsubgraphs equals to the restriction of a coloring (CSP) class.\nKey words: digraph, homomorphism, duality, NP, Constraint Satisfac-\ntion Problem\n1\nIntroduction, Background and Previous Work\nThink of 3-colorability of a graph G. This is a well known hard (and a canonical\nNP-complete) problem. From the combinatorial point of view there is a stan-\ndard way how to approach this problem (and monotone properties in general):\n1 Part of this work was supported by ITI and DIMATIA of Charles University Prague\nunder grant 1M0021620808, by OTKA Grant no. T043671, NK 67867, by NKTH\n(National Office for Research and Technology, Hungary), AEOLUS and also by Isaac\nNewton Institute (INI) Cambridge.\n\n2\nKun, Neˇsetˇril\ninvestigate minimal graphs without this property, denote by F the language of\nall such critical graphs and define the set Forb(F) of all structures which do not\n“contain” any F ∈F. Then the language Forb(F) coincides with the language\nof 3-colorable graphs. Unfortunately, in the most cases the set F is infinite. How-\never the properties characterized by a finite set F are very interesting if we allow\nlifts and shadows.\nLet us briefly illustrate this by our example of 3-colorability. Instead of a\ngraph G = (V, E) we consider the graph G together with three unary relations\nC1, C2, C3 (i.e. colors of vertices) which cover the vertex set V ; this structure will\nbe denoted by G′ and called a lift of G (thus G′ has one binary and three unary\nrelations). There are 3 forbidden substructures: For each i = 1, 2, 3 the single edge\ngraph K2 together with cover Ci = {1, 2} and Cj = ∅for j ̸= i form structure\nF′\ni (where the signature of F′\ni contains one binary and three unary relations).\nThe language of all 3-colorable graphs is just the language Φ(Forb(F′\n1, F′\n2, F′\n3)),\nwhere Φ is the forgetful functor which transforms G′ to G. We call G the shadow\nof G′.\nClearly this situation can be generalized and one of the main results of this\npaper is Theorem 3 which states that every NP problem is polynomially equiv-\nalent to the membership problem for a class Φ(Forb(F′)). Here F′ is a finite set\nof (vertex pair)-colored digraphs, Forb(F′) is the class of all lifted graphs G′ for\nwhich there is no homomorphism F ′ −→G′ for an F ′ ∈F′. Thus Forb(F′) is\nthe class of all graphs G′ with forbidden homomorphisms from F′. (See Section\n2 for definitions.) Theorems 4 and 5 provide similar results for forbidden colored\nsubgraphs and for forbidden induced subgraphs (in both cases vertex colorings\nsuffice).\nWe should add one more remark. We of course do not only claim that every\nproblem in NP can be polynomially reduced to a problem in any of these classes.\nThis would only mean that each of these classes contains an NP-complete prob-\nlem. What we claim is that these classes have the computational power of the\nwhole NP class. More precisely, to each language L in NP there exists a language\nM in any of these three classes such that M is polynomially equivalent to L, i.e.\nthere exist polynomial reductions of L to M and M to L. E.g. assuming P ̸= NP\nthere is a language in any of these classes that is neither in P nor NP-complete,\nsince there is such a language in NP by Ladner’s celebrated result [14].\nThe expressive power of classes Φ(Forb(F′)) corresponds to many combi-\nnatorially studied problems and presents a combinatorial counterpart to the\ncelebrated result of Fagin [4] who expressed every NP problem in logical terms\nby means of an Existential Second Order formula.\nThe fact that the membership problem for classes Φ(Forb(F′)) and their\ninjective and full variants Φ(Forbinj(F′)) and Φ(Forbfull(F′)) have full compu-\ntational power is pleasing from the combinatorial point of view as these classes\ncover well known examples of hard combinatorial problems: Ramsey type prob-\nlems (where as in Theorem 3 we consider edge colored graphs), colorings of\nbounded degree graphs (defined by an injectivity condition as in Theorem 4)\nand structural partitions (studied e.g. in [8] as in Theorem 5). It follows that, in\n\nNP by means of lifts and shadows\n3\nthe full generality, one cannot expect dichotomies here. On the other side of the\nspectrum, Feder and Vardi have formulated the celebrated Dichotomy conjecture\nfor all coloring problems (CSP).\nOur main result is Theorem 9: we give an easy characterization of those\nlanguages Φ(Forb(F′)) which are coloring problems (CSP). This can be viewed\nas an extension of the duality characterization theorem for structures [6]. We\ndemonstrate the power of this theorem while reproving some theorems about\nthe local chromatic number. In contrast with this we show that the shadow\nΦ(Forb(F′)) of a vertex colored class of digraphs Forb(F′) is always a CSP\nlanguage when restricted to a bounded expansion class (this notion generalizes\nbounded degree and proper minor closed classes) [20]. Our main tools are finite\ndualities [23, 6], restricted dualities [21], and the Sparse Incomparability Lemma\n[22, 9]. The detailed proofs can be found in the full version of this paper [13].\n2\nPreliminaries\nWe consider finite relational structures although in most of the paper we only\ndeal with digraphs, i.e. relational structures with just one binary relation. This\nitself is one of the main features of this note: oriented graphs suffice. Digraphs\nwill be denoted by A, B, . . . (as we want to stress that they may be replaced by\nmore general structures).\nLet Γ denote a finite set we refer to as colors. A Γ-colored graph (structure) is\na graph (or structure) together with either a coloring of its vertices or a coloring\nof all pairs of vertices by colors from Γ. Only in Theorem 3 we shall consider\ncoloring of all pairs (but in Theorem 3 this will play an important role). Thus\nin the whole paper we shall undestand by a colored graph a graph with colored\nvertices. We denote colored digraphs (relational structures) by A′, B′ etc. Fol-\nlowing the more general notions in category theory we call A′ a lift of A and A is\ncalled the shadow of A′. Thus (vertex-) colored digraphs (structures) can be also\ndescribed as monadic lifts. A homomorphism of digraphs (relational structures)\npreserves all the edges (arcs). A homomorphism of colored digraphs (relational\nstructures) preserves the color of vertices (pairs of vertices), too. The Constraint\nSatisfaction Problem corresponding to the graph (relational structure) A is the\nmembership problem for the class of all graphs (structures) defined by {B : B is\nhomomorphic to A}. We call a mapping between two (colored) digraphs a full\nhomomorphism if in addition the preimage of an edge is an edge. Full homo-\nmorphisms have very easy structure, as every full homomorphism which is onto\nis a retraction. The other special homomorphisms we will be interested in are\ninjective homomorphisms.\nLet F′ be a finite set of colored relational structures (digraphs). By Forb(F′)\nwe denote the set of all colored relational structures (digraphs) A′ satisfying\nF′ ̸−→A′ for every F′ ∈F′. (If we use injective or full homomorphisms this will\nbe denoted by Forbinj(F′) or Forbfull(F′), respectively.)\nSimilarly (well, dually), for the finite set of colored relational structures (di-\ngraphs) D′ we denote by CSP(D′) the class of all colored digraphs A′ satisfying\n\n4\nKun, Neˇsetˇril\nA′ −→D′ for some D′ ∈D′. (This is sometimes denoted by →D.) Now suppose\nthat the classes Forb(F′) and CSP(D′) are equal. Then we say that the pair\n(F′, D′) is a finite duality. Explicitly, a finite duality means that the following\nequivalence holds for every (colored) relational structure (digraph):\n∀F′ ∈F′\nF′ ̸−→A′ ⇐⇒∃D′ ∈D′\nA′ −→D′.\nWe say that the structure A is core if every homomorphism A −→A is\nan automorphism. Every finite structure A contains (up to an isomorphism) a\nuniquely determined core substructure homomorphically equivalent to A, see\n[23] [9]. The following result was recently proved in [6] and [23]. It characterizes\nfinite dualities of digraphs (or more generally relational structures with a given\nsignature).\nTheorem 1. For every finite set F of (relational) forests there exists (up to ho-\nmomorphism equivalence) a finite uniquely determined set D of structures such\nthat (F, D) forms a finite duality, i.e. Forb(F) = CSP(D). Up to homomor-\nphism equivalence there are no other finite dualities.\nLet Φ denote the forgetful functor which corresponds to a Γ-colored relational\nstructure (digraph) the uncolored one, i.e. it forgets about the coloring. We will\ninvestigate classes of the form Φ(Forb(F′)). We call the pair (F′, D) shadow\nduality if Φ(Forb(F′)) = CSP(D). An example of shadow duality is the language\nof 3-colorable graphs discussed in the introduction (or, as can be seen easily, any\nCSP problem in general). Finite dualities became much more abundant when we\ndemand the validity of the above formula just for all graphs from a given class K.\nIn such a case we speak about K-restricted duality. It has been proved in [21] that\nso called Bounded Expansion classes (which include both proper minor closed\nclasses and classes of graphs with bounded degree) have a restricted duality for\nevery choice of F′.\nThe study of homomorphism properties of structures not containing short\ncycles (i.e. with a large girth) is a combinatorial problem studied intensively.\nThe following result has proved particularly useful in various applications. It is\noften called the Sparse Incomparability Lemma:\nTheorem 2. Let k, lbe positive integers and let A be a structure. Then there\nexists a structure B with the following properties:\n1. There exists a homomorphism f : B −→A;\n2. For every structure C with at most k points the following holds: there exists\na homomorphism A −→C if and only if there exists a homomorphism\nB −→C;\n3. B has girth ≥l.\nThis result was proved by probabilistic method in [22] [24], see also [9]. The\npolynomial time construction of B is possible, too: in the case of binary relations\n(digraphs) this was done in [18] and for relational structures in [12].\n\nNP by means of lifts and shadows\n5\n3\nStatement of Results\n3.1\nNP by means of finitely many forbidden lifts\nThe class SNP consists of all problems expressible by an existential second-order\nformula with a universal first-order part [4]. The class SNP is computationally\nequivalent to NP. Feder and Vardi [5] have proved that three syntactically defined\nsubclasses of the class SNP still have the full computational power of the class\nNP. We reformulate this result to our combinatorial setting of lifts and shadows.\nTheorem 3.\nFor every language L ∈NP there exist a finite set of colors Γ\nand a finite set of Γ-colored digraphs F′, where we color all pairs of vertices such\nthat L is computationally equivalent to the membership problem for Φ(Forb(F′)).\nTheorem 4.\nFor every language L ∈NP there exist a finite set of colors Γ\nand a finite set of Γ-colored digraphs F′, (where we color the vertices) such that\nL is computationally equivalent to the membership problem for Φ(Forbinj(F′)).\nTheorem 5.\nFor every language L ∈NP there exist a finite set of colors Γ\nand a finite set of Γ-colored digraphs F′, (where we color the vertices) such that\nL is computationally equivalent to the membership problem for Φ(Forbfull(F′)).\n3.2\nLifts and Shadows of Dualities\nIt follows from Section 3.1 that shadows of Forb of a finite set of colored digraphs,\nthis is classes Φ(Forb(F′)), where F′ is a finite set, have the computational\npower of the whole NP. What about finite dualities? Are the shadow dualities\nalso more frequent? The negative answer is expressed by Theorem 7 and shows\na remarkable stability of dualities. Towards this end we first observe that every\nduality (of lifted structures) implies a shadow duality:\nTheorem 6.\nLet Γ be a finite set of colors and F′ a finite set of Γ-colored\ndigraphs (relational structures), where we color all of the vertices. Suppose that\nthere exists a finite set of Γ-colored digraphs (relational structures) D′ such that\nForb(F′) = CSP(D′). Then Φ(Forb(F′)) = CSP(Φ(D′)).\nTheorem 6 may be sometimes reversed: Shadow dualities may be “lifted”\nin case that lifted graphs have colored vertices (this is sometimes described as\nmonadic lift). This is non-trivial and in fact Theorem 7 may be seen as the core\nof this paper.\nTheorem 7.\nLet Γ be a finite set of colors and F′ be a finite set of Γ-colored\ndigraphs (relational structures), where we color all of the vertices. Suppose that\nΦ(Forb(F′)) = CSP(D) for a finite set D of digraphs (relational structures).\nThen there exists a finite set D′ of Γ-colored digraphs (relational structures)\nsuch that Forb(F′) = CSP(D′).\n\n6\nKun, Neˇsetˇril\n4\nProofs\nThe proofs of Theorems 3, 4 and 5 are in the full version of this paper [13]. We do\nnot include them as they need some new definitions (and space) but nevertheless\nbasically follow the strategy of [5].\nBefore proving Theorems 6 and 7 we formulate first a simple lemma which\nwe shall use repeatedly:\nLemma 1. (lifting) Let A, B relational structures, homomorphism f : A −→\nB, a finite set of colors Γ and Φ(B′) = B be given. Then there exists a lift A′,\nsuch that Φ(A′) = A and the mapping f is a homomorphism A′ −→B′ (of\ncolored structures).\nProof. (of Theorem 6) Suppose that A ∈CSP(Φ(D′)), say A ∈CSP(Φ(D′)).\nNow for a homomorphism f : A −→Φ(D′) there is at least one lift A′ of A such\nthat the mapping f is a homomorphism A′ →D′ (here we use Lifting Lemma\n1). Since the pair(F′, D′) is a duality F′ ↛A′ holds for any F′ ∈F′ and thus\nin turn A ∈Φ(Forb(F′)).\nConversely, let us assume that A′ ∈Forb(F′) satisfies Φ(A′) = A. But then\nA′ ∈CSP(D′) and thus by the functorial property of Φ we have A = Φ(A′) ∈\nCSP(Φ(D′)).\nProof. (of Theorem 7) Assume Φ(Forb(F′)) = CSP(D). Our goal is to find D′\nsuch that Forb(F′) = CSP(D′). This will follow as a (non-trivial) combination of\nTheorems 1 and 2. By Theorem 1 we know that if F′ is a set of (relational) forests\nthen the set F′ has a dual set D′ (in the class of covering colored structures; we\njust list all covering colored substructures of the dual set guaranteed by Theorem\n1). It is Φ(D′) = D by Theorem 6. So assume to the contrary that one of the\nstructures, say F′\n0, fails to be a forest (i.e. we assume that one of the components\nof F′\n0 has a cycle). We proceed by a refined induction (which will allow us to use\nmore properties of F′\n0) to show that D′ does not exist. Let us introduce carefully\nthe setting of the induction.\nWe assume shadow duality Φ(Forb(F′)) = CSP(D). Let D be fixed through-\nout the proof. Clearly many sets F′ will do the job and we select the set F′ such\nthat F′ consists of cores of all homomorphic images (explicitly: we close F′ un-\nder homomorphic images and then take the set of cores of all these structures).\nAmong all such sets F′ we take a set of minimal cardinality. It will be again\ndenoted by F′. We proceed by induction on the size |F′| of F′.\nThe set Forb(F′) is clearly determined by the minimal elements of F′ (mini-\nmal in the homomorphism order). Thus let us assume that one of these minimal\nelements, say F′\n0, is not a forest. By the minimality of F′ we see that we have a\nproper inclusion Φ(Forb(F′ \\ {F′\n0})) ⊃CSP(D). Thus there exists a structure\nS in the difference. But this in turn means that there has to be a lift S′ of S\nsuch that F′\n0 −→S′ and S ̸→D for every D ∈D. In fact not only that: as F′\n0\nis a core, as Forb(F′) is homomorphism closed and as F′ has minimal size we\nconclude that there exist S and S′ such that any homomorphism F′\n0 −→S′ is a\n\nNP by means of lifts and shadows\n7\nmonomorphism (i.e. one-to-one, otherwise we could replace F′\n0 by a set of all its\nhomomorphic images - F′\n0 would not be needed).\nNow we apply (the second non-trivial ingredient) Theorem 2 to structure\nS and an l> |X(F′\n0)|: we find a structure S0 with the following properties:\nS0 −→S, S0 −→D if and only if S −→D for every D ∈D and S0 contains\nno cycles of length ≤l. It follows that S0 ̸∈CSP(D). Next we apply Lemma 1\nto obtain a structure S′\n0 with S′\n0 −→S′. Now we use that S′\n0 is a monadic lift\nand so does not contain cycles of length ≤l. Now for any F′ ∈F′, F′ ̸= F′\n0 we\nhave F′ ↛S′\n0 as S′\n0 →S′ and F′ ↛S′. As the only homomorphism F′\n0 −→S′\nis a monomorphism the only (hypothetical) homomorphism F′\n0 −→S′ is also\nmonomorphism. But this is a contradiction as F′\n0 contains a cycle while S′\n0 has\nno cycles of length ≤l. This completes the proof.\n5\nApplications\n5.1\nClasses with bounded expansion\nWe study the restriction of classes Φ(Forb(F′)) to a class of digraphs with\nbounded expansion recently introduced in [20]. These classes are a generaliza-\ntion of proper minor closed and bounded degree classes of graphs. Using the\ndecomposition technique of [20] [21] we can prove that any class Φ(Forb(F′))\n(for a finite set F′ of monadic lifts) when restricted to a bounded expansion class\nequals to a CSP class (when restricted to the same class).\nTheorem 8.\nConsider the finite set of colors Γ and the class Φ(Forb(F′)) for\na finite set F′ of Γ-colored digraphs. Let C be a class of digraphs of bounded\nexpansion. Then there is a finite set of digraphs D such that Φ(Forb(F′)) ∩C =\nCSP(D) ∩C.\nConsider a monotone, first-order definable class of colored digraphs C which\nis closed under homomorphism and disjoint union. By a combination with recent\nresults of [2] we also obtain (perhaps a bit surprisingly) that the shadow C is a\nCSP language of digraphs. It remains to be seen to which bounded expansion\nclasses (of graphs and structures) this result generalizes.\n5.2\nThe classes MMSNP and FP - a characterization\nWe conclude with an application to descriptive theory of complexity classes. Re-\ncall that the class of languages defined by monotone, monadic formulas without\ninequality is denoted by MMSNP (Monotone Monadic Strict Nondeterministic\nPolynomial). (Feder and Vardi proved that the class MMSNP is computationally\nequivalent to the class CSP in a random sense [5], this was later derandomized\nby the first author [12].) Madeleine [16] introduced the class FP of languages\ndefined similarly to our forbidden monadic lifts of structures.\nIt has been proved in [16] that the classes FP and MMSNP are equal. In fact\nthe class MMSNP contains exactly the languages defined by forbidden monadic\nlifts.\n\n8\nKun, Neˇsetˇril\nProposition 1. A language of relational structures L is in the class MMSNP if\nand only if there is a finite set of colors Γ and a finite set of Γ-colored relational\nstructures F′ such that L = Φ(Forb(F′)).\nMadelaine and Stewart [17] gave a long process to decide whether an FP\nlanguage is a finite union of CSP languages. We use Theorems 6 and 7 and the\ndescription of dualities for relational structures [6] to give a short characteriza-\ntion of a more general class of languages.\nTheorem 9.\nConsider the finite set of colors Γ and the language Φ(Forb(F′))\nfor a finite set F′ of Γ-colored digraphs (relational structures).\nIf no F′ ∈F′ contains a cycle then there is a finite set of digraphs (relational\nstructures) D such that Φ(Forb(F′)) = CSP(D). If one of the lifts F′ in a min-\nimal subfamily of F′ contains a cycle in its core then the language Φ(Forb(F′))\nis not a finite union of CSP languages.\nProof. If no F′ ∈Forb(F′) contains a cycle then the set F′ has a dual D′\nby Theorem 1, and the shadow of this set D′ gives the dual set D of the set\nΦ(Forb(F′)) (by Theorem 6). On the other side if one F′ ∈Forb(F′) contains\na cycle in its core and if F′ is minimal (i.e. F′ is needed) then Forb(F′) does\nnot have a dual. The shadow of the language Forb(F′) is the language L and\nconsequently this fails to be a finite union of CSP languages by Theorem 7.\nTheorem 9 may be interpreted as stability of dualities for finite structures.\nWhile shadows of the classes Forb(F′) are computationally equivalent to the\nwhole NP, the shadow dualities are not bringing anything new: these are just\nshadows of dualities. In other words: the coloring problems in the class MMSNP\nare just shadow dualities. This holds for graphs as well for relational structures.\n5.3\nOn the local chromatic number\nNow we apply Theorem 9 in the analysis of local chromatic number introduced\nin [3] (see also [26]): we say that a graph G is locally (a, b)-colorable if there\nexists a proper coloring of G by b colors so that every (closed) neighborhood of\na vertex of G gets at most a colors. It follows from [3] that the class of all locally\n(a, b)-colorable graphs is of the form CSP(U(a, b)) for an explicitely constructed\ngraph U(a, b). We conclude this paper with an indirect proof of this result with\nan application to complexity:\nProposition 2. Let a, b be integers and consider the membership problem for the\nclass of locally (a, b)-colorable graphs. This is actually a Constraint Satisfaction\nProblem which is NP-complete if a, b ≥3 and it is polynomial time solvable else.\nProof. Consider the color set Γ = {1, . . . , b} and the following set F′ of Γ-colored\nundirected graphs. Let F′ consist of all monochromatic edges (colored by any\nof the b colors) and all the stars with a + 1 vertices colored by at least a + 1\ncolors. The corresponding language is exactly the required one: a graph G is in\n\nNP by means of lifts and shadows\n9\nthe language iffit admits a proper Γ-coloring, this is no monochromatic edge is\nhomomorphic to the colored graph, such that the neighbourhood of every vertex\n(including the vertex itself) has at most a different colors, i.e. no star with a + 1\nvertices of different color is homomorphic to it. Since F′ consists of colored trees\nthis will be a CSP language by Theorem 9.\nHell and the second author proved that CSP problems defined by undirected\ngraphs are in P if the graph is bipartite and NP-complete else [9]. We do not\ndetermine which graph defines this particular CSP problem (of locally (a, b)-\ncolorable graphs). But if a, b ≥3 then we know that it contains the triangle if,\nso the problem is NP-complete. It is easy to see that this membership problem\nis in P if a < 3 or b < 3.\n6\nSummary and Future Work\nWe found a computationally equivalent formulation of the class NP by means of\nfinitely many forbidden lifts of very special type. An ambitious project would be\nto find an equivalent digraph coloring problem for a given NP language really\neffectively (in human sense, our results provide a polynomial time algorithm).\nFor example it would be nice to exhibit a vertex coloring problem that is poly-\nnomially equivalent to the graph isomorphism problem. In general this mainly\ndepends on how to express the problem in terms of logic. The next class we\nseem to be able to deal with are coloring problems of structures with an equiva-\nlence relation. Another good candidate are lifts using linear order. This promises\nseveral interesting applications which were studied earlier in a different setting.\nWe also proved that shadow dualities and lifted monadic dualities are in 1−1\ncorrespondence. This abstract result has several consequences and streamlines\nsome earlier results in descriptive complexity theory (related to MMSNP and\nCSP classes). The simplicity of this approach suggests some other problems. It\nis tempting to try to relate Ladner’s diagonalization method [14] in this setting\n(as it was pioneered by Lov ́asz and G ́acs [7] for NP∩coNP in a similar context).\nThe characterization of Lifted Dualities is beyond reach but particular cases are\ninteresting as they generalize results of [23] [6] and as the corresponding duals\npresent polynomial instances of CSP.\nBut perhaps more importantly, our approach to the complexity subclasses\nof NP is based on lifts and shadows as a combination of algebra, combinatorics\nand logic. We believe that it has further applications and that it forms a useful\nparadigm.\nReferences\n1. A. Atserias: On Digraph Coloring Problems and Treewidth Duality. In: 20th IEEE\nSymposium on Logic in Computer Science (LICS), 2005, pp. 106–115.\n2. A. Atserias, A. Dawar, Ph. G. Kolaitis: On Preservation under Homomorphisms\nand Conjunctive Queries, Journal of the ACM 53, 2 (2006), 208–237.\n\n10\nKun, Neˇsetˇril\n3. P. Erd ̋os, Z. F ̈uredi, A. Hajnal, P. Komj ́ath, V. R ̈odl, ́A. Seress: Coloring graphs\nwith locally few colors, Discrete Math. 59, (1986), 21-34.\n4. R. Fagin: Generalized first-order spectra and polynomial-time recognizable sets.\nin: Complexity of Computation (ed. R. Karp), SIAM-AMS Proceedings 7, 1974,\npp. 43–73.\n5. T. Feder, M. Vardi: The computational structure of monotone monadic SNP and\nconstraint satisfaction: A study through Datalog and group theory, SIAM J. Com-\nput. 28, 1 (1999), 57–104.\n6. J. Foniok, J. Neˇsetˇril, C. Tardif: Generalized dualities and maximal finite antichains\nin the homomorphism order of relational structures, KAM-DIMATIA Series 2006-\n766 (to appear in European J. Comb.).\n7. P. G ́acs, L. Lov ́asz: Some remarks on generalized spectra, Z. Math. Log. Grdl. 23,\n(1977), no. 6, 547–554.\n8. T. Feder, P. Hell, S. Klein, and R. Motwani, Complexity of graph partition prob-\nlems, 31st Annual ACM STOC (1999) 464–472.\n9. P. Hell, J. Neˇsetˇril: Graphs and Homomorphism, Oxford University Press,\n2004.\n10. N. Immerman: Languages that capture complexity classes, SIAM J. Comput. 16\n(1987), 760–778.\n11. G. Kun: On the complexity of Constraint Satisfaction Problem, PhD thesis (in\nHungarian), 2006.\n12. G. Kun: Constraints, MMSNP and expander structures, Combinatorica, submit-\nted, 2007.\n13. G. Kun, J. Neˇsetˇril: Forbidden lifts (NP and CSP for combinatorists), KAM-\nDIMATIA Series 2006-775 (to appear in European J. Comb.)\n14. R. E. Ladner: On the structure of Polynomial Time Reducibility, Journal of the\nACM, 22,1 (1975), 155–171.\n15. T. Luczak, J. Neˇsetˇril: A probabilistic approach to the dichotomy problem, SIAM\nJ. Comp. 36, 3 (2006), 835–843.\n16. F. Madelaine: Constraint satisfaction problems and related logic, PhD thesis, 2003.\n17. F. Madelaine and I. A. Stewart: Constraint satisfaction problems and related logic,\nmanuscript, 2005.\n18. J. Matouˇsek, J. Neˇsetˇril: Constructions of sparse graphs with given homomor-\nphisms (to appear)\n19. J. Neˇsetˇril, A. Pultr: On classes of relations and graphs determined by subobjects\nand factorobjects, Discrete Math. 22 (1978), 287–300.\n20. J. Neˇsetˇril, P. Ossona de Mendez: Low tree-width decompositions and algorithmic\nconsequences. In STOC’06, Proceedings of the 38th Annual ACM Symposium on\nTheory of Computing, ACM Press 2006, pp. 391–400.\n21. J. Neˇsetˇril, P. Ossona de Mendez: Grad and Classes with bounded expansion III.\n- Restricted Dualities, KAM-DIMATIA Series 2005-741 (to appear in European J.\nComb.).\n22. J. Neˇsetˇril, V. R ̈odl: Chromatically optimal rigid graphs, J. Comb. Th. B 46 (1989),\n133–141.\n23. J. Neˇsetˇril and C. Tardif, Duality theorems for finite structures (characterising\ngaps and good characterizations), J. Combin. Theory B 80 (2000), 80–97.\n24. J. Neˇsetˇril, X. Zhu: On sparse graphs with given colorings and homomorphisms,\nJ. Comb. Th. B, 90,1 (2004), 161–172.\n25. B. Rossman: Existential positive types and preservation under homomorphisms, In:\n20th IEEE Symposium on Logic in Computer Science (LICS),2005, pp. 467–476.\n\nNP by means of lifts and shadows\n11\n26. G. Simonyi, G. Tardos: Local chromatic number, Ky Fan’s theorem and circular\ncolorings, Combinatorica 26 (2006), 589-626.\n27. M. Y. Vardi: The complexity of relational query languages. In: Proceedings of 14th\nACM Symposium on Theory of Computing, 1982, pp. 137–146.","paragraphs":[{"paragraph_id":"p1","order":1,"text":"arXiv:0706.3459v1 [cs.CC] 23 Jun 2007\nNP by means of lifts and shadows\nG ́abor Kun and Jaroslav Neˇsetˇril\nDepartment of Mathematics, University of Memphis\n373 Dunn Hall, Memphis, TN 38152;\nDepartment of Applied Mathematics (KAM) and\nInstitute of Theoretical Computer Science (ITI),\nCharles University, Malostransk ́e n ́am 22, Praha;\nE-mail: kungabor@cs.elte.hu\nE-mail: nesetril@kam.mff.cuni.cz\n1\nAbstract. We show that every NP problem is polynomially equivalent\nto a simple combinatorial problem: the membership problem for a special\nclass of digraphs. These classes are defined by means of shadows (pro-\njections) and by finitely many forbidden colored (lifted) subgraphs. Our\ncharacterization is motivated by the analysis of syntactical subclasses\nwith the full computational power of NP, which were first studied by\nFeder and Vardi.\nOur approach applies to many combinatorial problems and it induces\nthe characterization of coloring problems (CSP) defined by means of\nshadows. This turns out to be related to homomorphism dualities. We\nprove that a class of digraphs (relational structures) defined by finitely\nmany forbidden colored subgraphs (i.e. lifted substructures) is a CSP\nclass if and only if all the the forbidden structures are homomorphically\nequivalent to trees. We show a surprising richness of coloring problems\nwhen restricted to most frequent graph classes. Using results of Neˇsetˇril\nand Ossona de Mendez for bounded expansion classes (which include\nbounded degree and proper minor closed classes) we prove that the re-\nstriction of every class defined as the shadow of finitely many colored\nsubgraphs equals to the restriction of a coloring (CSP) class.\nKey words: digraph, homomorphism, duality, NP, Constraint Satisfac-\ntion Problem\n1\nIntroduction, Background and Previous Work\nThink of 3-colorability of a graph G. This is a well known hard (and a canonical\nNP-complete) problem. From the combinatorial point of view there is a stan-\ndard way how to approach this problem (and monotone properties in general):\n1 Part of this work was supported by ITI and DIMATIA of Charles University Prague\nunder grant 1M0021620808, by OTKA Grant no. T043671, NK 67867, by NKTH\n(National Office for Research and Technology, Hungary), AEOLUS and also by Isaac\nNewton Institute (INI) Cambridge."},{"paragraph_id":"p2","order":2,"text":"2\nKun, Neˇsetˇril\ninvestigate minimal graphs without this property, denote by F the language of\nall such critical graphs and define the set Forb(F) of all structures which do not\n“contain” any F ∈F. Then the language Forb(F) coincides with the language\nof 3-colorable graphs. Unfortunately, in the most cases the set F is infinite. How-\never the properties characterized by a finite set F are very interesting if we allow\nlifts and shadows.\nLet us briefly illustrate this by our example of 3-colorability. Instead of a\ngraph G = (V, E) we consider the graph G together with three unary relations\nC1, C2, C3 (i.e. colors of vertices) which cover the vertex set V ; this structure will\nbe denoted by G′ and called a lift of G (thus G′ has one binary and three unary\nrelations). There are 3 forbidden substructures: For each i = 1, 2, 3 the single edge\ngraph K2 together with cover Ci = {1, 2} and Cj = ∅for j ̸= i form structure\nF′\ni (where the signature of F′\ni contains one binary and three unary relations).\nThe language of all 3-colorable graphs is just the language Φ(Forb(F′\n1, F′\n2, F′\n3)),\nwhere Φ is the forgetful functor which transforms G′ to G. We call G the shadow\nof G′.\nClearly this situation can be generalized and one of the main results of this\npaper is Theorem 3 which states that every NP problem is polynomially equiv-\nalent to the membership problem for a class Φ(Forb(F′)). Here F′ is a finite set\nof (vertex pair)-colored digraphs, Forb(F′) is the class of all lifted graphs G′ for\nwhich there is no homomorphism F ′ −→G′ for an F ′ ∈F′. Thus Forb(F′) is\nthe class of all graphs G′ with forbidden homomorphisms from F′. (See Section\n2 for definitions.) Theorems 4 and 5 provide similar results for forbidden colored\nsubgraphs and for forbidden induced subgraphs (in both cases vertex colorings\nsuffice).\nWe should add one more remark. We of course do not only claim that every\nproblem in NP can be polynomially reduced to a problem in any of these classes.\nThis would only mean that each of these classes contains an NP-complete prob-\nlem. What we claim is that these classes have the computational power of the\nwhole NP class. More precisely, to each language L in NP there exists a language\nM in any of these three classes such that M is polynomially equivalent to L, i.e.\nthere exist polynomial reductions of L to M and M to L. E.g. assuming P ̸= NP\nthere is a language in any of these classes that is neither in P nor NP-complete,\nsince there is such a language in NP by Ladner’s celebrated result [14].\nThe expressive power of classes Φ(Forb(F′)) corresponds to many combi-\nnatorially studied problems and presents a combinatorial counterpart to the\ncelebrated result of Fagin [4] who expressed every NP problem in logical terms\nby means of an Existential Second Order formula.\nThe fact that the membership problem for classes Φ(Forb(F′)) and their\ninjective and full variants Φ(Forbinj(F′)) and Φ(Forbfull(F′)) have full compu-\ntational power is pleasing from the combinatorial point of view as these classes\ncover well known examples of hard combinatorial problems: Ramsey type prob-\nlems (where as in Theorem 3 we consider edge colored graphs), colorings of\nbounded degree graphs (defined by an injectivity condition as in Theorem 4)\nand structural partitions (studied e.g. in [8] as in Theorem 5). It follows that, in"},{"paragraph_id":"p3","order":3,"text":"NP by means of lifts and shadows\n3\nthe full generality, one cannot expect dichotomies here. On the other side of the\nspectrum, Feder and Vardi have formulated the celebrated Dichotomy conjecture\nfor all coloring problems (CSP).\nOur main result is Theorem 9: we give an easy characterization of those\nlanguages Φ(Forb(F′)) which are coloring problems (CSP). This can be viewed\nas an extension of the duality characterization theorem for structures [6]. We\ndemonstrate the power of this theorem while reproving some theorems about\nthe local chromatic number. In contrast with this we show that the shadow\nΦ(Forb(F′)) of a vertex colored class of digraphs Forb(F′) is always a CSP\nlanguage when restricted to a bounded expansion class (this notion generalizes\nbounded degree and proper minor closed classes) [20]. Our main tools are finite\ndualities [23, 6], restricted dualities [21], and the Sparse Incomparability Lemma\n[22, 9]. The detailed proofs can be found in the full version of this paper [13].\n2\nPreliminaries\nWe consider finite relational structures although in most of the paper we only\ndeal with digraphs, i.e. relational structures with just one binary relation. This\nitself is one of the main features of this note: oriented graphs suffice. Digraphs\nwill be denoted by A, B, . . . (as we want to stress that they may be replaced by\nmore general structures).\nLet Γ denote a finite set we refer to as colors. A Γ-colored graph (structure) is\na graph (or structure) together with either a coloring of its vertices or a coloring\nof all pairs of vertices by colors from Γ. Only in Theorem 3 we shall consider\ncoloring of all pairs (but in Theorem 3 this will play an important role). Thus\nin the whole paper we shall undestand by a colored graph a graph with colored\nvertices. We denote colored digraphs (relational structures) by A′, B′ etc. Fol-\nlowing the more general notions in category theory we call A′ a lift of A and A is\ncalled the shadow of A′. Thus (vertex-) colored digraphs (structures) can be also\ndescribed as monadic lifts. A homomorphism of digraphs (relational structures)\npreserves all the edges (arcs). A homomorphism of colored digraphs (relational\nstructures) preserves the color of vertices (pairs of vertices), too. The Constraint\nSatisfaction Problem corresponding to the graph (relational structure) A is the\nmembership problem for the class of all graphs (structures) defined by {B : B is\nhomomorphic to A}. We call a mapping between two (colored) digraphs a full\nhomomorphism if in addition the preimage of an edge is an edge. Full homo-\nmorphisms have very easy structure, as every full homomorphism which is onto\nis a retraction. The other special homomorphisms we will be interested in are\ninjective homomorphisms.\nLet F′ be a finite set of colored relational structures (digraphs). By Forb(F′)\nwe denote the set of all colored relational structures (digraphs) A′ satisfying\nF′ ̸−→A′ for every F′ ∈F′. (If we use injective or full homomorphisms this will\nbe denoted by Forbinj(F′) or Forbfull(F′), respectively.)\nSimilarly (well, dually), for the finite set of colored relational structures (di-\ngraphs) D′ we denote by CSP(D′) the class of all colored digraphs A′ satisfying"},{"paragraph_id":"p4","order":4,"text":"4\nKun, Neˇsetˇril\nA′ −→D′ for some D′ ∈D′. (This is sometimes denoted by →D.) Now suppose\nthat the classes Forb(F′) and CSP(D′) are equal. Then we say that the pair\n(F′, D′) is a finite duality. Explicitly, a finite duality means that the following\nequivalence holds for every (colored) relational structure (digraph):\n∀F′ ∈F′\nF′ ̸−→A′ ⇐⇒∃D′ ∈D′\nA′ −→D′.\nWe say that the structure A is core if every homomorphism A −→A is\nan automorphism. Every finite structure A contains (up to an isomorphism) a\nuniquely determined core substructure homomorphically equivalent to A, see\n[23] [9]. The following result was recently proved in [6] and [23]. It characterizes\nfinite dualities of digraphs (or more generally relational structures with a given\nsignature).\nTheorem 1. For every finite set F of (relational) forests there exists (up to ho-\nmomorphism equivalence) a finite uniquely determined set D of structures such\nthat (F, D) forms a finite duality, i.e. Forb(F) = CSP(D). Up to homomor-\nphism equivalence there are no other finite dualities.\nLet Φ denote the forgetful functor which corresponds to a Γ-colored relational\nstructure (digraph) the uncolored one, i.e. it forgets about the coloring. We will\ninvestigate classes of the form Φ(Forb(F′)). We call the pair (F′, D) shadow\nduality if Φ(Forb(F′)) = CSP(D). An example of shadow duality is the language\nof 3-colorable graphs discussed in the introduction (or, as can be seen easily, any\nCSP problem in general). Finite dualities became much more abundant when we\ndemand the validity of the above formula just for all graphs from a given class K.\nIn such a case we speak about K-restricted duality. It has been proved in [21] that\nso called Bounded Expansion classes (which include both proper minor closed\nclasses and classes of graphs with bounded degree) have a restricted duality for\nevery choice of F′.\nThe study of homomorphism properties of structures not containing short\ncycles (i.e. with a large girth) is a combinatorial problem studied intensively.\nThe following result has proved particularly useful in various applications. It is\noften called the Sparse Incomparability Lemma:\nTheorem 2. Let k, lbe positive integers and let A be a structure. Then there\nexists a structure B with the following properties:\n1. There exists a homomorphism f : B −→A;\n2. For every structure C with at most k points the following holds: there exists\na homomorphism A −→C if and only if there exists a homomorphism\nB −→C;\n3. B has girth ≥l.\nThis result was proved by probabilistic method in [22] [24], see also [9]. The\npolynomial time construction of B is possible, too: in the case of binary relations\n(digraphs) this was done in [18] and for relational structures in [12]."},{"paragraph_id":"p5","order":5,"text":"NP by means of lifts and shadows\n5\n3\nStatement of Results\n3.1\nNP by means of finitely many forbidden lifts\nThe class SNP consists of all problems expressible by an existential second-order\nformula with a universal first-order part [4]. The class SNP is computationally\nequivalent to NP. Feder and Vardi [5] have proved that three syntactically defined\nsubclasses of the class SNP still have the full computational power of the class\nNP. We reformulate this result to our combinatorial setting of lifts and shadows.\nTheorem 3.\nFor every language L ∈NP there exist a finite set of colors Γ\nand a finite set of Γ-colored digraphs F′, where we color all pairs of vertices such\nthat L is computationally equivalent to the membership problem for Φ(Forb(F′)).\nTheorem 4.\nFor every language L ∈NP there exist a finite set of colors Γ\nand a finite set of Γ-colored digraphs F′, (where we color the vertices) such that\nL is computationally equivalent to the membership problem for Φ(Forbinj(F′)).\nTheorem 5.\nFor every language L ∈NP there exist a finite set of colors Γ\nand a finite set of Γ-colored digraphs F′, (where we color the vertices) such that\nL is computationally equivalent to the membership problem for Φ(Forbfull(F′)).\n3.2\nLifts and Shadows of Dualities\nIt follows from Section 3.1 that shadows of Forb of a finite set of colored digraphs,\nthis is classes Φ(Forb(F′)), where F′ is a finite set, have the computational\npower of the whole NP. What about finite dualities? Are the shadow dualities\nalso more frequent? The negative answer is expressed by Theorem 7 and shows\na remarkable stability of dualities. Towards this end we first observe that every\nduality (of lifted structures) implies a shadow duality:\nTheorem 6.\nLet Γ be a finite set of colors and F′ a finite set of Γ-colored\ndigraphs (relational structures), where we color all of the vertices. Suppose that\nthere exists a finite set of Γ-colored digraphs (relational structures) D′ such that\nForb(F′) = CSP(D′). Then Φ(Forb(F′)) = CSP(Φ(D′)).\nTheorem 6 may be sometimes reversed: Shadow dualities may be “lifted”\nin case that lifted graphs have colored vertices (this is sometimes described as\nmonadic lift). This is non-trivial and in fact Theorem 7 may be seen as the core\nof this paper.\nTheorem 7.\nLet Γ be a finite set of colors and F′ be a finite set of Γ-colored\ndigraphs (relational structures), where we color all of the vertices. Suppose that\nΦ(Forb(F′)) = CSP(D) for a finite set D of digraphs (relational structures).\nThen there exists a finite set D′ of Γ-colored digraphs (relational structures)\nsuch that Forb(F′) = CSP(D′)."},{"paragraph_id":"p6","order":6,"text":"6\nKun, Neˇsetˇril\n4\nProofs\nThe proofs of Theorems 3, 4 and 5 are in the full version of this paper [13]. We do\nnot include them as they need some new definitions (and space) but nevertheless\nbasically follow the strategy of [5].\nBefore proving Theorems 6 and 7 we formulate first a simple lemma which\nwe shall use repeatedly:\nLemma 1. (lifting) Let A, B relational structures, homomorphism f : A −→\nB, a finite set of colors Γ and Φ(B′) = B be given. Then there exists a lift A′,\nsuch that Φ(A′) = A and the mapping f is a homomorphism A′ −→B′ (of\ncolored structures).\nProof. (of Theorem 6) Suppose that A ∈CSP(Φ(D′)), say A ∈CSP(Φ(D′)).\nNow for a homomorphism f : A −→Φ(D′) there is at least one lift A′ of A such\nthat the mapping f is a homomorphism A′ →D′ (here we use Lifting Lemma\n1). Since the pair(F′, D′) is a duality F′ ↛A′ holds for any F′ ∈F′ and thus\nin turn A ∈Φ(Forb(F′)).\nConversely, let us assume that A′ ∈Forb(F′) satisfies Φ(A′) = A. But then\nA′ ∈CSP(D′) and thus by the functorial property of Φ we have A = Φ(A′) ∈\nCSP(Φ(D′)).\nProof. (of Theorem 7) Assume Φ(Forb(F′)) = CSP(D). Our goal is to find D′\nsuch that Forb(F′) = CSP(D′). This will follow as a (non-trivial) combination of\nTheorems 1 and 2. By Theorem 1 we know that if F′ is a set of (relational) forests\nthen the set F′ has a dual set D′ (in the class of covering colored structures; we\njust list all covering colored substructures of the dual set guaranteed by Theorem\n1). It is Φ(D′) = D by Theorem 6. So assume to the contrary that one of the\nstructures, say F′\n0, fails to be a forest (i.e. we assume that one of the components\nof F′\n0 has a cycle). We proceed by a refined induction (which will allow us to use\nmore properties of F′\n0) to show that D′ does not exist. Let us introduce carefully\nthe setting of the induction.\nWe assume shadow duality Φ(Forb(F′)) = CSP(D). Let D be fixed through-\nout the proof. Clearly many sets F′ will do the job and we select the set F′ such\nthat F′ consists of cores of all homomorphic images (explicitly: we close F′ un-\nder homomorphic images and then take the set of cores of all these structures).\nAmong all such sets F′ we take a set of minimal cardinality. It will be again\ndenoted by F′. We proceed by induction on the size |F′| of F′.\nThe set Forb(F′) is clearly determined by the minimal elements of F′ (mini-\nmal in the homomorphism order). Thus let us assume that one of these minimal\nelements, say F′\n0, is not a forest. By the minimality of F′ we see that we have a\nproper inclusion Φ(Forb(F′ \\ {F′\n0})) ⊃CSP(D). Thus there exists a structure\nS in the difference. But this in turn means that there has to be a lift S′ of S\nsuch that F′\n0 −→S′ and S ̸→D for every D ∈D. In fact not only that: as F′\n0\nis a core, as Forb(F′) is homomorphism closed and as F′ has minimal size we\nconclude that there exist S and S′ such that any homomorphism F′\n0 −→S′ is a"},{"paragraph_id":"p7","order":7,"text":"NP by means of lifts and shadows\n7\nmonomorphism (i.e. one-to-one, otherwise we could replace F′\n0 by a set of all its\nhomomorphic images - F′\n0 would not be needed).\nNow we apply (the second non-trivial ingredient) Theorem 2 to structure\nS and an l> |X(F′\n0)|: we find a structure S0 with the following properties:\nS0 −→S, S0 −→D if and only if S −→D for every D ∈D and S0 contains\nno cycles of length ≤l. It follows that S0 ̸∈CSP(D). Next we apply Lemma 1\nto obtain a structure S′\n0 with S′\n0 −→S′. Now we use that S′\n0 is a monadic lift\nand so does not contain cycles of length ≤l. Now for any F′ ∈F′, F′ ̸= F′\n0 we\nhave F′ ↛S′\n0 as S′\n0 →S′ and F′ ↛S′. As the only homomorphism F′\n0 −→S′\nis a monomorphism the only (hypothetical) homomorphism F′\n0 −→S′ is also\nmonomorphism. But this is a contradiction as F′\n0 contains a cycle while S′\n0 has\nno cycles of length ≤l. This completes the proof.\n5\nApplications\n5.1\nClasses with bounded expansion\nWe study the restriction of classes Φ(Forb(F′)) to a class of digraphs with\nbounded expansion recently introduced in [20]. These classes are a generaliza-\ntion of proper minor closed and bounded degree classes of graphs. Using the\ndecomposition technique of [20] [21] we can prove that any class Φ(Forb(F′))\n(for a finite set F′ of monadic lifts) when restricted to a bounded expansion class\nequals to a CSP class (when restricted to the same class).\nTheorem 8.\nConsider the finite set of colors Γ and the class Φ(Forb(F′)) for\na finite set F′ of Γ-colored digraphs. Let C be a class of digraphs of bounded\nexpansion. Then there is a finite set of digraphs D such that Φ(Forb(F′)) ∩C =\nCSP(D) ∩C.\nConsider a monotone, first-order definable class of colored digraphs C which\nis closed under homomorphism and disjoint union. By a combination with recent\nresults of [2] we also obtain (perhaps a bit surprisingly) that the shadow C is a\nCSP language of digraphs. It remains to be seen to which bounded expansion\nclasses (of graphs and structures) this result generalizes.\n5.2\nThe classes MMSNP and FP - a characterization\nWe conclude with an application to descriptive theory of complexity classes. Re-\ncall that the class of languages defined by monotone, monadic formulas without\ninequality is denoted by MMSNP (Monotone Monadic Strict Nondeterministic\nPolynomial). (Feder and Vardi proved that the class MMSNP is computationally\nequivalent to the class CSP in a random sense [5], this was later derandomized\nby the first author [12].) Madeleine [16] introduced the class FP of languages\ndefined similarly to our forbidden monadic lifts of structures.\nIt has been proved in [16] that the classes FP and MMSNP are equal. In fact\nthe class MMSNP contains exactly the languages defined by forbidden monadic\nlifts."},{"paragraph_id":"p8","order":8,"text":"8\nKun, Neˇsetˇril\nProposition 1. A language of relational structures L is in the class MMSNP if\nand only if there is a finite set of colors Γ and a finite set of Γ-colored relational\nstructures F′ such that L = Φ(Forb(F′)).\nMadelaine and Stewart [17] gave a long process to decide whether an FP\nlanguage is a finite union of CSP languages. We use Theorems 6 and 7 and the\ndescription of dualities for relational structures [6] to give a short characteriza-\ntion of a more general class of languages.\nTheorem 9.\nConsider the finite set of colors Γ and the language Φ(Forb(F′))\nfor a finite set F′ of Γ-colored digraphs (relational structures).\nIf no F′ ∈F′ contains a cycle then there is a finite set of digraphs (relational\nstructures) D such that Φ(Forb(F′)) = CSP(D). If one of the lifts F′ in a min-\nimal subfamily of F′ contains a cycle in its core then the language Φ(Forb(F′))\nis not a finite union of CSP languages.\nProof. If no F′ ∈Forb(F′) contains a cycle then the set F′ has a dual D′\nby Theorem 1, and the shadow of this set D′ gives the dual set D of the set\nΦ(Forb(F′)) (by Theorem 6). On the other side if one F′ ∈Forb(F′) contains\na cycle in its core and if F′ is minimal (i.e. F′ is needed) then Forb(F′) does\nnot have a dual. The shadow of the language Forb(F′) is the language L and\nconsequently this fails to be a finite union of CSP languages by Theorem 7.\nTheorem 9 may be interpreted as stability of dualities for finite structures.\nWhile shadows of the classes Forb(F′) are computationally equivalent to the\nwhole NP, the shadow dualities are not bringing anything new: these are just\nshadows of dualities. In other words: the coloring problems in the class MMSNP\nare just shadow dualities. This holds for graphs as well for relational structures.\n5.3\nOn the local chromatic number\nNow we apply Theorem 9 in the analysis of local chromatic number introduced\nin [3] (see also [26]): we say that a graph G is locally (a, b)-colorable if there\nexists a proper coloring of G by b colors so that every (closed) neighborhood of\na vertex of G gets at most a colors. It follows from [3] that the class of all locally\n(a, b)-colorable graphs is of the form CSP(U(a, b)) for an explicitely constructed\ngraph U(a, b). We conclude this paper with an indirect proof of this result with\nan application to complexity:\nProposition 2. Let a, b be integers and consider the membership problem for the\nclass of locally (a, b)-colorable graphs. This is actually a Constraint Satisfaction\nProblem which is NP-complete if a, b ≥3 and it is polynomial time solvable else.\nProof. Consider the color set Γ = {1, . . . , b} and the following set F′ of Γ-colored\nundirected graphs. Let F′ consist of all monochromatic edges (colored by any\nof the b colors) and all the stars with a + 1 vertices colored by at least a + 1\ncolors. The corresponding language is exactly the required one: a graph G is in"},{"paragraph_id":"p9","order":9,"text":"NP by means of lifts and shadows\n9\nthe language iffit admits a proper Γ-coloring, this is no monochromatic edge is\nhomomorphic to the colored graph, such that the neighbourhood of every vertex\n(including the vertex itself) has at most a different colors, i.e. no star with a + 1\nvertices of different color is homomorphic to it. Since F′ consists of colored trees\nthis will be a CSP language by Theorem 9.\nHell and the second author proved that CSP problems defined by undirected\ngraphs are in P if the graph is bipartite and NP-complete else [9]. We do not\ndetermine which graph defines this particular CSP problem (of locally (a, b)-\ncolorable graphs). But if a, b ≥3 then we know that it contains the triangle if,\nso the problem is NP-complete. It is easy to see that this membership problem\nis in P if a < 3 or b < 3.\n6\nSummary and Future Work\nWe found a computationally equivalent formulation of the class NP by means of\nfinitely many forbidden lifts of very special type. An ambitious project would be\nto find an equivalent digraph coloring problem for a given NP language really\neffectively (in human sense, our results provide a polynomial time algorithm).\nFor example it would be nice to exhibit a vertex coloring problem that is poly-\nnomially equivalent to the graph isomorphism problem. In general this mainly\ndepends on how to express the problem in terms of logic. The next class we\nseem to be able to deal with are coloring problems of structures with an equiva-\nlence relation. Another good candidate are lifts using linear order. This promises\nseveral interesting applications which were studied earlier in a different setting.\nWe also proved that shadow dualities and lifted monadic dualities are in 1−1\ncorrespondence. This abstract result has several consequences and streamlines\nsome earlier results in descriptive complexity theory (related to MMSNP and\nCSP classes). The simplicity of this approach suggests some other problems. It\nis tempting to try to relate Ladner’s diagonalization method [14] in this setting\n(as it was pioneered by Lov ́asz and G ́acs [7] for NP∩coNP in a similar context).\nThe characterization of Lifted Dualities is beyond reach but particular cases are\ninteresting as they generalize results of [23] [6] and as the corresponding duals\npresent polynomial instances of CSP.\nBut perhaps more importantly, our approach to the complexity subclasses\nof NP is based on lifts and shadows as a combination of algebra, combinatorics\nand logic. We believe that it has further applications and that it forms a useful\nparadigm.\nReferences\n1. A. Atserias: On Digraph Coloring Problems and Treewidth Duality. In: 20th IEEE\nSymposium on Logic in Computer Science (LICS), 2005, pp. 106–115.\n2. A. Atserias, A. Dawar, Ph. G. Kolaitis: On Preservation under Homomorphisms\nand Conjunctive Queries, Journal of the ACM 53, 2 (2006), 208–237."},{"paragraph_id":"p10","order":10,"text":"10\nKun, Neˇsetˇril\n3. P. Erd ̋os, Z. F ̈uredi, A. Hajnal, P. Komj ́ath, V. R ̈odl, ́A. Seress: Coloring graphs\nwith locally few colors, Discrete Math. 59, (1986), 21-34.\n4. R. Fagin: Generalized first-order spectra and polynomial-time recognizable sets.\nin: Complexity of Computation (ed. R. Karp), SIAM-AMS Proceedings 7, 1974,\npp. 43–73.\n5. T. Feder, M. Vardi: The computational structure of monotone monadic SNP and\nconstraint satisfaction: A study through Datalog and group theory, SIAM J. Com-\nput. 28, 1 (1999), 57–104.\n6. J. Foniok, J. Neˇsetˇril, C. Tardif: Generalized dualities and maximal finite antichains\nin the homomorphism order of relational structures, KAM-DIMATIA Series 2006-\n766 (to appear in European J. Comb.).\n7. P. G ́acs, L. Lov ́asz: Some remarks on generalized spectra, Z. Math. Log. Grdl. 23,\n(1977), no. 6, 547–554.\n8. T. Feder, P. Hell, S. Klein, and R. Motwani, Complexity of graph partition prob-\nlems, 31st Annual ACM STOC (1999) 464–472.\n9. P. Hell, J. Neˇsetˇril: Graphs and Homomorphism, Oxford University Press,\n2004.\n10. N. Immerman: Languages that capture complexity classes, SIAM J. Comput. 16\n(1987), 760–778.\n11. G. Kun: On the complexity of Constraint Satisfaction Problem, PhD thesis (in\nHungarian), 2006.\n12. G. Kun: Constraints, MMSNP and expander structures, Combinatorica, submit-\nted, 2007.\n13. G. Kun, J. Neˇsetˇril: Forbidden lifts (NP and CSP for combinatorists), KAM-\nDIMATIA Series 2006-775 (to appear in European J. Comb.)\n14. R. E. Ladner: On the structure of Polynomial Time Reducibility, Journal of the\nACM, 22,1 (1975), 155–171.\n15. T. Luczak, J. Neˇsetˇril: A probabilistic approach to the dichotomy problem, SIAM\nJ. Comp. 36, 3 (2006), 835–843.\n16. F. Madelaine: Constraint satisfaction problems and related logic, PhD thesis, 2003.\n17. F. Madelaine and I. A. Stewart: Constraint satisfaction problems and related logic,\nmanuscript, 2005.\n18. J. Matouˇsek, J. Neˇsetˇril: Constructions of sparse graphs with given homomor-\nphisms (to appear)\n19. J. Neˇsetˇril, A. Pultr: On classes of relations and graphs determined by subobjects\nand factorobjects, Discrete Math. 22 (1978), 287–300.\n20. J. Neˇsetˇril, P. Ossona de Mendez: Low tree-width decompositions and algorithmic\nconsequences. In STOC’06, Proceedings of the 38th Annual ACM Symposium on\nTheory of Computing, ACM Press 2006, pp. 391–400.\n21. J. Neˇsetˇril, P. Ossona de Mendez: Grad and Classes with bounded expansion III.\n- Restricted Dualities, KAM-DIMATIA Series 2005-741 (to appear in European J.\nComb.).\n22. J. Neˇsetˇril, V. R ̈odl: Chromatically optimal rigid graphs, J. Comb. Th. B 46 (1989),\n133–141.\n23. J. Neˇsetˇril and C. Tardif, Duality theorems for finite structures (characterising\ngaps and good characterizations), J. Combin. Theory B 80 (2000), 80–97.\n24. J. Neˇsetˇril, X. Zhu: On sparse graphs with given colorings and homomorphisms,\nJ. Comb. Th. B, 90,1 (2004), 161–172.\n25. B. Rossman: Existential positive types and preservation under homomorphisms, In:\n20th IEEE Symposium on Logic in Computer Science (LICS),2005, pp. 467–476."},{"paragraph_id":"p11","order":11,"text":"NP by means of lifts and shadows\n11\n26. G. Simonyi, G. Tardos: Local chromatic number, Ky Fan’s theorem and circular\ncolorings, Combinatorica 26 (2006), 589-626.\n27. M. Y. Vardi: The complexity of relational query languages. In: Proceedings of 14th\nACM Symposium on Theory of Computing, 1982, pp. 137–146."}],"pages":[{"page":1,"text":"arXiv:0706.3459v1 [cs.CC] 23 Jun 2007\nNP by means of lifts and shadows\nG ́abor Kun and Jaroslav Neˇsetˇril\nDepartment of Mathematics, University of Memphis\n373 Dunn Hall, Memphis, TN 38152;\nDepartment of Applied Mathematics (KAM) and\nInstitute of Theoretical Computer Science (ITI),\nCharles University, Malostransk ́e n ́am 22, Praha;\nE-mail: kungabor@cs.elte.hu\nE-mail: nesetril@kam.mff.cuni.cz\n1\nAbstract. We show that every NP problem is polynomially equivalent\nto a simple combinatorial problem: the membership problem for a special\nclass of digraphs. These classes are defined by means of shadows (pro-\njections) and by finitely many forbidden colored (lifted) subgraphs. Our\ncharacterization is motivated by the analysis of syntactical subclasses\nwith the full computational power of NP, which were first studied by\nFeder and Vardi.\nOur approach applies to many combinatorial problems and it induces\nthe characterization of coloring problems (CSP) defined by means of\nshadows. This turns out to be related to homomorphism dualities. We\nprove that a class of digraphs (relational structures) defined by finitely\nmany forbidden colored subgraphs (i.e. lifted substructures) is a CSP\nclass if and only if all the the forbidden structures are homomorphically\nequivalent to trees. We show a surprising richness of coloring problems\nwhen restricted to most frequent graph classes. Using results of Neˇsetˇril\nand Ossona de Mendez for bounded expansion classes (which include\nbounded degree and proper minor closed classes) we prove that the re-\nstriction of every class defined as the shadow of finitely many colored\nsubgraphs equals to the restriction of a coloring (CSP) class.\nKey words: digraph, homomorphism, duality, NP, Constraint Satisfac-\ntion Problem\n1\nIntroduction, Background and Previous Work\nThink of 3-colorability of a graph G. This is a well known hard (and a canonical\nNP-complete) problem. From the combinatorial point of view there is a stan-\ndard way how to approach this problem (and monotone properties in general):\n1 Part of this work was supported by ITI and DIMATIA of Charles University Prague\nunder grant 1M0021620808, by OTKA Grant no. T043671, NK 67867, by NKTH\n(National Office for Research and Technology, Hungary), AEOLUS and also by Isaac\nNewton Institute (INI) Cambridge."},{"page":2,"text":"2\nKun, Neˇsetˇril\ninvestigate minimal graphs without this property, denote by F the language of\nall such critical graphs and define the set Forb(F) of all structures which do not\n“contain” any F ∈F. Then the language Forb(F) coincides with the language\nof 3-colorable graphs. Unfortunately, in the most cases the set F is infinite. How-\never the properties characterized by a finite set F are very interesting if we allow\nlifts and shadows.\nLet us briefly illustrate this by our example of 3-colorability. Instead of a\ngraph G = (V, E) we consider the graph G together with three unary relations\nC1, C2, C3 (i.e. colors of vertices) which cover the vertex set V ; this structure will\nbe denoted by G′ and called a lift of G (thus G′ has one binary and three unary\nrelations). There are 3 forbidden substructures: For each i = 1, 2, 3 the single edge\ngraph K2 together with cover Ci = {1, 2} and Cj = ∅for j ̸= i form structure\nF′\ni (where the signature of F′\ni contains one binary and three unary relations).\nThe language of all 3-colorable graphs is just the language Φ(Forb(F′\n1, F′\n2, F′\n3)),\nwhere Φ is the forgetful functor which transforms G′ to G. We call G the shadow\nof G′.\nClearly this situation can be generalized and one of the main results of this\npaper is Theorem 3 which states that every NP problem is polynomially equiv-\nalent to the membership problem for a class Φ(Forb(F′)). Here F′ is a finite set\nof (vertex pair)-colored digraphs, Forb(F′) is the class of all lifted graphs G′ for\nwhich there is no homomorphism F ′ −→G′ for an F ′ ∈F′. Thus Forb(F′) is\nthe class of all graphs G′ with forbidden homomorphisms from F′. (See Section\n2 for definitions.) Theorems 4 and 5 provide similar results for forbidden colored\nsubgraphs and for forbidden induced subgraphs (in both cases vertex colorings\nsuffice).\nWe should add one more remark. We of course do not only claim that every\nproblem in NP can be polynomially reduced to a problem in any of these classes.\nThis would only mean that each of these classes contains an NP-complete prob-\nlem. What we claim is that these classes have the computational power of the\nwhole NP class. More precisely, to each language L in NP there exists a language\nM in any of these three classes such that M is polynomially equivalent to L, i.e.\nthere exist polynomial reductions of L to M and M to L. E.g. assuming P ̸= NP\nthere is a language in any of these classes that is neither in P nor NP-complete,\nsince there is such a language in NP by Ladner’s celebrated result [14].\nThe expressive power of classes Φ(Forb(F′)) corresponds to many combi-\nnatorially studied problems and presents a combinatorial counterpart to the\ncelebrated result of Fagin [4] who expressed every NP problem in logical terms\nby means of an Existential Second Order formula.\nThe fact that the membership problem for classes Φ(Forb(F′)) and their\ninjective and full variants Φ(Forbinj(F′)) and Φ(Forbfull(F′)) have full compu-\ntational power is pleasing from the combinatorial point of view as these classes\ncover well known examples of hard combinatorial problems: Ramsey type prob-\nlems (where as in Theorem 3 we consider edge colored graphs), colorings of\nbounded degree graphs (defined by an injectivity condition as in Theorem 4)\nand structural partitions (studied e.g. in [8] as in Theorem 5). It follows that, in"},{"page":3,"text":"NP by means of lifts and shadows\n3\nthe full generality, one cannot expect dichotomies here. On the other side of the\nspectrum, Feder and Vardi have formulated the celebrated Dichotomy conjecture\nfor all coloring problems (CSP).\nOur main result is Theorem 9: we give an easy characterization of those\nlanguages Φ(Forb(F′)) which are coloring problems (CSP). This can be viewed\nas an extension of the duality characterization theorem for structures [6]. We\ndemonstrate the power of this theorem while reproving some theorems about\nthe local chromatic number. In contrast with this we show that the shadow\nΦ(Forb(F′)) of a vertex colored class of digraphs Forb(F′) is always a CSP\nlanguage when restricted to a bounded expansion class (this notion generalizes\nbounded degree and proper minor closed classes) [20]. Our main tools are finite\ndualities [23, 6], restricted dualities [21], and the Sparse Incomparability Lemma\n[22, 9]. The detailed proofs can be found in the full version of this paper [13].\n2\nPreliminaries\nWe consider finite relational structures although in most of the paper we only\ndeal with digraphs, i.e. relational structures with just one binary relation. This\nitself is one of the main features of this note: oriented graphs suffice. Digraphs\nwill be denoted by A, B, . . . (as we want to stress that they may be replaced by\nmore general structures).\nLet Γ denote a finite set we refer to as colors. A Γ-colored graph (structure) is\na graph (or structure) together with either a coloring of its vertices or a coloring\nof all pairs of vertices by colors from Γ. Only in Theorem 3 we shall consider\ncoloring of all pairs (but in Theorem 3 this will play an important role). Thus\nin the whole paper we shall undestand by a colored graph a graph with colored\nvertices. We denote colored digraphs (relational structures) by A′, B′ etc. Fol-\nlowing the more general notions in category theory we call A′ a lift of A and A is\ncalled the shadow of A′. Thus (vertex-) colored digraphs (structures) can be also\ndescribed as monadic lifts. A homomorphism of digraphs (relational structures)\npreserves all the edges (arcs). A homomorphism of colored digraphs (relational\nstructures) preserves the color of vertices (pairs of vertices), too. The Constraint\nSatisfaction Problem corresponding to the graph (relational structure) A is the\nmembership problem for the class of all graphs (structures) defined by {B : B is\nhomomorphic to A}. We call a mapping between two (colored) digraphs a full\nhomomorphism if in addition the preimage of an edge is an edge. Full homo-\nmorphisms have very easy structure, as every full homomorphism which is onto\nis a retraction. The other special homomorphisms we will be interested in are\ninjective homomorphisms.\nLet F′ be a finite set of colored relational structures (digraphs). By Forb(F′)\nwe denote the set of all colored relational structures (digraphs) A′ satisfying\nF′ ̸−→A′ for every F′ ∈F′. (If we use injective or full homomorphisms this will\nbe denoted by Forbinj(F′) or Forbfull(F′), respectively.)\nSimilarly (well, dually), for the finite set of colored relational structures (di-\ngraphs) D′ we denote by CSP(D′) the class of all colored digraphs A′ satisfying"},{"page":4,"text":"4\nKun, Neˇsetˇril\nA′ −→D′ for some D′ ∈D′. (This is sometimes denoted by →D.) Now suppose\nthat the classes Forb(F′) and CSP(D′) are equal. Then we say that the pair\n(F′, D′) is a finite duality. Explicitly, a finite duality means that the following\nequivalence holds for every (colored) relational structure (digraph):\n∀F′ ∈F′\nF′ ̸−→A′ ⇐⇒∃D′ ∈D′\nA′ −→D′.\nWe say that the structure A is core if every homomorphism A −→A is\nan automorphism. Every finite structure A contains (up to an isomorphism) a\nuniquely determined core substructure homomorphically equivalent to A, see\n[23] [9]. The following result was recently proved in [6] and [23]. It characterizes\nfinite dualities of digraphs (or more generally relational structures with a given\nsignature).\nTheorem 1. For every finite set F of (relational) forests there exists (up to ho-\nmomorphism equivalence) a finite uniquely determined set D of structures such\nthat (F, D) forms a finite duality, i.e. Forb(F) = CSP(D). Up to homomor-\nphism equivalence there are no other finite dualities.\nLet Φ denote the forgetful functor which corresponds to a Γ-colored relational\nstructure (digraph) the uncolored one, i.e. it forgets about the coloring. We will\ninvestigate classes of the form Φ(Forb(F′)). We call the pair (F′, D) shadow\nduality if Φ(Forb(F′)) = CSP(D). An example of shadow duality is the language\nof 3-colorable graphs discussed in the introduction (or, as can be seen easily, any\nCSP problem in general). Finite dualities became much more abundant when we\ndemand the validity of the above formula just for all graphs from a given class K.\nIn such a case we speak about K-restricted duality. It has been proved in [21] that\nso called Bounded Expansion classes (which include both proper minor closed\nclasses and classes of graphs with bounded degree) have a restricted duality for\nevery choice of F′.\nThe study of homomorphism properties of structures not containing short\ncycles (i.e. with a large girth) is a combinatorial problem studied intensively.\nThe following result has proved particularly useful in various applications. It is\noften called the Sparse Incomparability Lemma:\nTheorem 2. Let k, lbe positive integers and let A be a structure. Then there\nexists a structure B with the following properties:\n1. There exists a homomorphism f : B −→A;\n2. For every structure C with at most k points the following holds: there exists\na homomorphism A −→C if and only if there exists a homomorphism\nB −→C;\n3. B has girth ≥l.\nThis result was proved by probabilistic method in [22] [24], see also [9]. The\npolynomial time construction of B is possible, too: in the case of binary relations\n(digraphs) this was done in [18] and for relational structures in [12]."},{"page":5,"text":"NP by means of lifts and shadows\n5\n3\nStatement of Results\n3.1\nNP by means of finitely many forbidden lifts\nThe class SNP consists of all problems expressible by an existential second-order\nformula with a universal first-order part [4]. The class SNP is computationally\nequivalent to NP. Feder and Vardi [5] have proved that three syntactically defined\nsubclasses of the class SNP still have the full computational power of the class\nNP. We reformulate this result to our combinatorial setting of lifts and shadows.\nTheorem 3.\nFor every language L ∈NP there exist a finite set of colors Γ\nand a finite set of Γ-colored digraphs F′, where we color all pairs of vertices such\nthat L is computationally equivalent to the membership problem for Φ(Forb(F′)).\nTheorem 4.\nFor every language L ∈NP there exist a finite set of colors Γ\nand a finite set of Γ-colored digraphs F′, (where we color the vertices) such that\nL is computationally equivalent to the membership problem for Φ(Forbinj(F′)).\nTheorem 5.\nFor every language L ∈NP there exist a finite set of colors Γ\nand a finite set of Γ-colored digraphs F′, (where we color the vertices) such that\nL is computationally equivalent to the membership problem for Φ(Forbfull(F′)).\n3.2\nLifts and Shadows of Dualities\nIt follows from Section 3.1 that shadows of Forb of a finite set of colored digraphs,\nthis is classes Φ(Forb(F′)), where F′ is a finite set, have the computational\npower of the whole NP. What about finite dualities? Are the shadow dualities\nalso more frequent? The negative answer is expressed by Theorem 7 and shows\na remarkable stability of dualities. Towards this end we first observe that every\nduality (of lifted structures) implies a shadow duality:\nTheorem 6.\nLet Γ be a finite set of colors and F′ a finite set of Γ-colored\ndigraphs (relational structures), where we color all of the vertices. Suppose that\nthere exists a finite set of Γ-colored digraphs (relational structures) D′ such that\nForb(F′) = CSP(D′). Then Φ(Forb(F′)) = CSP(Φ(D′)).\nTheorem 6 may be sometimes reversed: Shadow dualities may be “lifted”\nin case that lifted graphs have colored vertices (this is sometimes described as\nmonadic lift). This is non-trivial and in fact Theorem 7 may be seen as the core\nof this paper.\nTheorem 7.\nLet Γ be a finite set of colors and F′ be a finite set of Γ-colored\ndigraphs (relational structures), where we color all of the vertices. Suppose that\nΦ(Forb(F′)) = CSP(D) for a finite set D of digraphs (relational structures).\nThen there exists a finite set D′ of Γ-colored digraphs (relational structures)\nsuch that Forb(F′) = CSP(D′)."},{"page":6,"text":"6\nKun, Neˇsetˇril\n4\nProofs\nThe proofs of Theorems 3, 4 and 5 are in the full version of this paper [13]. We do\nnot include them as they need some new definitions (and space) but nevertheless\nbasically follow the strategy of [5].\nBefore proving Theorems 6 and 7 we formulate first a simple lemma which\nwe shall use repeatedly:\nLemma 1. (lifting) Let A, B relational structures, homomorphism f : A −→\nB, a finite set of colors Γ and Φ(B′) = B be given. Then there exists a lift A′,\nsuch that Φ(A′) = A and the mapping f is a homomorphism A′ −→B′ (of\ncolored structures).\nProof. (of Theorem 6) Suppose that A ∈CSP(Φ(D′)), say A ∈CSP(Φ(D′)).\nNow for a homomorphism f : A −→Φ(D′) there is at least one lift A′ of A such\nthat the mapping f is a homomorphism A′ →D′ (here we use Lifting Lemma\n1). Since the pair(F′, D′) is a duality F′ ↛A′ holds for any F′ ∈F′ and thus\nin turn A ∈Φ(Forb(F′)).\nConversely, let us assume that A′ ∈Forb(F′) satisfies Φ(A′) = A. But then\nA′ ∈CSP(D′) and thus by the functorial property of Φ we have A = Φ(A′) ∈\nCSP(Φ(D′)).\nProof. (of Theorem 7) Assume Φ(Forb(F′)) = CSP(D). Our goal is to find D′\nsuch that Forb(F′) = CSP(D′). This will follow as a (non-trivial) combination of\nTheorems 1 and 2. By Theorem 1 we know that if F′ is a set of (relational) forests\nthen the set F′ has a dual set D′ (in the class of covering colored structures; we\njust list all covering colored substructures of the dual set guaranteed by Theorem\n1). It is Φ(D′) = D by Theorem 6. So assume to the contrary that one of the\nstructures, say F′\n0, fails to be a forest (i.e. we assume that one of the components\nof F′\n0 has a cycle). We proceed by a refined induction (which will allow us to use\nmore properties of F′\n0) to show that D′ does not exist. Let us introduce carefully\nthe setting of the induction.\nWe assume shadow duality Φ(Forb(F′)) = CSP(D). Let D be fixed through-\nout the proof. Clearly many sets F′ will do the job and we select the set F′ such\nthat F′ consists of cores of all homomorphic images (explicitly: we close F′ un-\nder homomorphic images and then take the set of cores of all these structures).\nAmong all such sets F′ we take a set of minimal cardinality. It will be again\ndenoted by F′. We proceed by induction on the size |F′| of F′.\nThe set Forb(F′) is clearly determined by the minimal elements of F′ (mini-\nmal in the homomorphism order). Thus let us assume that one of these minimal\nelements, say F′\n0, is not a forest. By the minimality of F′ we see that we have a\nproper inclusion Φ(Forb(F′ \\ {F′\n0})) ⊃CSP(D). Thus there exists a structure\nS in the difference. But this in turn means that there has to be a lift S′ of S\nsuch that F′\n0 −→S′ and S ̸→D for every D ∈D. In fact not only that: as F′\n0\nis a core, as Forb(F′) is homomorphism closed and as F′ has minimal size we\nconclude that there exist S and S′ such that any homomorphism F′\n0 −→S′ is a"},{"page":7,"text":"NP by means of lifts and shadows\n7\nmonomorphism (i.e. one-to-one, otherwise we could replace F′\n0 by a set of all its\nhomomorphic images - F′\n0 would not be needed).\nNow we apply (the second non-trivial ingredient) Theorem 2 to structure\nS and an l> |X(F′\n0)|: we find a structure S0 with the following properties:\nS0 −→S, S0 −→D if and only if S −→D for every D ∈D and S0 contains\nno cycles of length ≤l. It follows that S0 ̸∈CSP(D). Next we apply Lemma 1\nto obtain a structure S′\n0 with S′\n0 −→S′. Now we use that S′\n0 is a monadic lift\nand so does not contain cycles of length ≤l. Now for any F′ ∈F′, F′ ̸= F′\n0 we\nhave F′ ↛S′\n0 as S′\n0 →S′ and F′ ↛S′. As the only homomorphism F′\n0 −→S′\nis a monomorphism the only (hypothetical) homomorphism F′\n0 −→S′ is also\nmonomorphism. But this is a contradiction as F′\n0 contains a cycle while S′\n0 has\nno cycles of length ≤l. This completes the proof.\n5\nApplications\n5.1\nClasses with bounded expansion\nWe study the restriction of classes Φ(Forb(F′)) to a class of digraphs with\nbounded expansion recently introduced in [20]. These classes are a generaliza-\ntion of proper minor closed and bounded degree classes of graphs. Using the\ndecomposition technique of [20] [21] we can prove that any class Φ(Forb(F′))\n(for a finite set F′ of monadic lifts) when restricted to a bounded expansion class\nequals to a CSP class (when restricted to the same class).\nTheorem 8.\nConsider the finite set of colors Γ and the class Φ(Forb(F′)) for\na finite set F′ of Γ-colored digraphs. Let C be a class of digraphs of bounded\nexpansion. Then there is a finite set of digraphs D such that Φ(Forb(F′)) ∩C =\nCSP(D) ∩C.\nConsider a monotone, first-order definable class of colored digraphs C which\nis closed under homomorphism and disjoint union. By a combination with recent\nresults of [2] we also obtain (perhaps a bit surprisingly) that the shadow C is a\nCSP language of digraphs. It remains to be seen to which bounded expansion\nclasses (of graphs and structures) this result generalizes.\n5.2\nThe classes MMSNP and FP - a characterization\nWe conclude with an application to descriptive theory of complexity classes. Re-\ncall that the class of languages defined by monotone, monadic formulas without\ninequality is denoted by MMSNP (Monotone Monadic Strict Nondeterministic\nPolynomial). (Feder and Vardi proved that the class MMSNP is computationally\nequivalent to the class CSP in a random sense [5], this was later derandomized\nby the first author [12].) Madeleine [16] introduced the class FP of languages\ndefined similarly to our forbidden monadic lifts of structures.\nIt has been proved in [16] that the classes FP and MMSNP are equal. In fact\nthe class MMSNP contains exactly the languages defined by forbidden monadic\nlifts."},{"page":8,"text":"8\nKun, Neˇsetˇril\nProposition 1. A language of relational structures L is in the class MMSNP if\nand only if there is a finite set of colors Γ and a finite set of Γ-colored relational\nstructures F′ such that L = Φ(Forb(F′)).\nMadelaine and Stewart [17] gave a long process to decide whether an FP\nlanguage is a finite union of CSP languages. We use Theorems 6 and 7 and the\ndescription of dualities for relational structures [6] to give a short characteriza-\ntion of a more general class of languages.\nTheorem 9.\nConsider the finite set of colors Γ and the language Φ(Forb(F′))\nfor a finite set F′ of Γ-colored digraphs (relational structures).\nIf no F′ ∈F′ contains a cycle then there is a finite set of digraphs (relational\nstructures) D such that Φ(Forb(F′)) = CSP(D). If one of the lifts F′ in a min-\nimal subfamily of F′ contains a cycle in its core then the language Φ(Forb(F′))\nis not a finite union of CSP languages.\nProof. If no F′ ∈Forb(F′) contains a cycle then the set F′ has a dual D′\nby Theorem 1, and the shadow of this set D′ gives the dual set D of the set\nΦ(Forb(F′)) (by Theorem 6). On the other side if one F′ ∈Forb(F′) contains\na cycle in its core and if F′ is minimal (i.e. F′ is needed) then Forb(F′) does\nnot have a dual. The shadow of the language Forb(F′) is the language L and\nconsequently this fails to be a finite union of CSP languages by Theorem 7.\nTheorem 9 may be interpreted as stability of dualities for finite structures.\nWhile shadows of the classes Forb(F′) are computationally equivalent to the\nwhole NP, the shadow dualities are not bringing anything new: these are just\nshadows of dualities. In other words: the coloring problems in the class MMSNP\nare just shadow dualities. This holds for graphs as well for relational structures.\n5.3\nOn the local chromatic number\nNow we apply Theorem 9 in the analysis of local chromatic number introduced\nin [3] (see also [26]): we say that a graph G is locally (a, b)-colorable if there\nexists a proper coloring of G by b colors so that every (closed) neighborhood of\na vertex of G gets at most a colors. It follows from [3] that the class of all locally\n(a, b)-colorable graphs is of the form CSP(U(a, b)) for an explicitely constructed\ngraph U(a, b). We conclude this paper with an indirect proof of this result with\nan application to complexity:\nProposition 2. Let a, b be integers and consider the membership problem for the\nclass of locally (a, b)-colorable graphs. This is actually a Constraint Satisfaction\nProblem which is NP-complete if a, b ≥3 and it is polynomial time solvable else.\nProof. Consider the color set Γ = {1, . . . , b} and the following set F′ of Γ-colored\nundirected graphs. Let F′ consist of all monochromatic edges (colored by any\nof the b colors) and all the stars with a + 1 vertices colored by at least a + 1\ncolors. The corresponding language is exactly the required one: a graph G is in"},{"page":9,"text":"NP by means of lifts and shadows\n9\nthe language iffit admits a proper Γ-coloring, this is no monochromatic edge is\nhomomorphic to the colored graph, such that the neighbourhood of every vertex\n(including the vertex itself) has at most a different colors, i.e. no star with a + 1\nvertices of different color is homomorphic to it. Since F′ consists of colored trees\nthis will be a CSP language by Theorem 9.\nHell and the second author proved that CSP problems defined by undirected\ngraphs are in P if the graph is bipartite and NP-complete else [9]. We do not\ndetermine which graph defines this particular CSP problem (of locally (a, b)-\ncolorable graphs). But if a, b ≥3 then we know that it contains the triangle if,\nso the problem is NP-complete. It is easy to see that this membership problem\nis in P if a < 3 or b < 3.\n6\nSummary and Future Work\nWe found a computationally equivalent formulation of the class NP by means of\nfinitely many forbidden lifts of very special type. An ambitious project would be\nto find an equivalent digraph coloring problem for a given NP language really\neffectively (in human sense, our results provide a polynomial time algorithm).\nFor example it would be nice to exhibit a vertex coloring problem that is poly-\nnomially equivalent to the graph isomorphism problem. In general this mainly\ndepends on how to express the problem in terms of logic. The next class we\nseem to be able to deal with are coloring problems of structures with an equiva-\nlence relation. Another good candidate are lifts using linear order. This promises\nseveral interesting applications which were studied earlier in a different setting.\nWe also proved that shadow dualities and lifted monadic dualities are in 1−1\ncorrespondence. This abstract result has several consequences and streamlines\nsome earlier results in descriptive complexity theory (related to MMSNP and\nCSP classes). The simplicity of this approach suggests some other problems. It\nis tempting to try to relate Ladner’s diagonalization method [14] in this setting\n(as it was pioneered by Lov ́asz and G ́acs [7] for NP∩coNP in a similar context).\nThe characterization of Lifted Dualities is beyond reach but particular cases are\ninteresting as they generalize results of [23] [6] and as the corresponding duals\npresent polynomial instances of CSP.\nBut perhaps more importantly, our approach to the complexity subclasses\nof NP is based on lifts and shadows as a combination of algebra, combinatorics\nand logic. We believe that it has further applications and that it forms a useful\nparadigm.\nReferences\n1. A. Atserias: On Digraph Coloring Problems and Treewidth Duality. In: 20th IEEE\nSymposium on Logic in Computer Science (LICS), 2005, pp. 106–115.\n2. A. Atserias, A. Dawar, Ph. G. Kolaitis: On Preservation under Homomorphisms\nand Conjunctive Queries, Journal of the ACM 53, 2 (2006), 208–237."},{"page":10,"text":"10\nKun, Neˇsetˇril\n3. P. Erd ̋os, Z. F ̈uredi, A. Hajnal, P. Komj ́ath, V. R ̈odl, ́A. Seress: Coloring graphs\nwith locally few colors, Discrete Math. 59, (1986), 21-34.\n4. R. Fagin: Generalized first-order spectra and polynomial-time recognizable sets.\nin: Complexity of Computation (ed. R. Karp), SIAM-AMS Proceedings 7, 1974,\npp. 43–73.\n5. T. Feder, M. Vardi: The computational structure of monotone monadic SNP and\nconstraint satisfaction: A study through Datalog and group theory, SIAM J. Com-\nput. 28, 1 (1999), 57–104.\n6. J. Foniok, J. Neˇsetˇril, C. Tardif: Generalized dualities and maximal finite antichains\nin the homomorphism order of relational structures, KAM-DIMATIA Series 2006-\n766 (to appear in European J. Comb.).\n7. P. G ́acs, L. Lov ́asz: Some remarks on generalized spectra, Z. Math. Log. Grdl. 23,\n(1977), no. 6, 547–554.\n8. T. Feder, P. Hell, S. Klein, and R. Motwani, Complexity of graph partition prob-\nlems, 31st Annual ACM STOC (1999) 464–472.\n9. P. Hell, J. Neˇsetˇril: Graphs and Homomorphism, Oxford University Press,\n2004.\n10. N. Immerman: Languages that capture complexity classes, SIAM J. Comput. 16\n(1987), 760–778.\n11. G. Kun: On the complexity of Constraint Satisfaction Problem, PhD thesis (in\nHungarian), 2006.\n12. G. Kun: Constraints, MMSNP and expander structures, Combinatorica, submit-\nted, 2007.\n13. G. Kun, J. Neˇsetˇril: Forbidden lifts (NP and CSP for combinatorists), KAM-\nDIMATIA Series 2006-775 (to appear in European J. Comb.)\n14. R. E. Ladner: On the structure of Polynomial Time Reducibility, Journal of the\nACM, 22,1 (1975), 155–171.\n15. T. Luczak, J. Neˇsetˇril: A probabilistic approach to the dichotomy problem, SIAM\nJ. Comp. 36, 3 (2006), 835–843.\n16. F. Madelaine: Constraint satisfaction problems and related logic, PhD thesis, 2003.\n17. F. Madelaine and I. A. Stewart: Constraint satisfaction problems and related logic,\nmanuscript, 2005.\n18. J. Matouˇsek, J. Neˇsetˇril: Constructions of sparse graphs with given homomor-\nphisms (to appear)\n19. J. Neˇsetˇril, A. Pultr: On classes of relations and graphs determined by subobjects\nand factorobjects, Discrete Math. 22 (1978), 287–300.\n20. J. Neˇsetˇril, P. Ossona de Mendez: Low tree-width decompositions and algorithmic\nconsequences. In STOC’06, Proceedings of the 38th Annual ACM Symposium on\nTheory of Computing, ACM Press 2006, pp. 391–400.\n21. J. Neˇsetˇril, P. Ossona de Mendez: Grad and Classes with bounded expansion III.\n- Restricted Dualities, KAM-DIMATIA Series 2005-741 (to appear in European J.\nComb.).\n22. J. Neˇsetˇril, V. R ̈odl: Chromatically optimal rigid graphs, J. Comb. Th. B 46 (1989),\n133–141.\n23. J. Neˇsetˇril and C. Tardif, Duality theorems for finite structures (characterising\ngaps and good characterizations), J. Combin. Theory B 80 (2000), 80–97.\n24. J. Neˇsetˇril, X. Zhu: On sparse graphs with given colorings and homomorphisms,\nJ. Comb. Th. B, 90,1 (2004), 161–172.\n25. B. Rossman: Existential positive types and preservation under homomorphisms, In:\n20th IEEE Symposium on Logic in Computer Science (LICS),2005, pp. 467–476."},{"page":11,"text":"NP by means of lifts and shadows\n11\n26. G. Simonyi, G. Tardos: Local chromatic number, Ky Fan’s theorem and circular\ncolorings, Combinatorica 26 (2006), 589-626.\n27. M. Y. Vardi: The complexity of relational query languages. In: Proceedings of 14th\nACM Symposium on Theory of Computing, 1982, pp. 137–146."}]},"section_tree":[],"assets":{"figures":[],"tables":[],"images":[]},"math_expressions":{"inline":[],"block":[{"equation_id":"eq1","equation_number":null,"raw_text":"graph G = (V, E) we consider the graph G together with three unary relations","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq2","equation_number":null,"raw_text":"relations). There are 3 forbidden substructures: For each i = 1, 2, 3 the single edge","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq3","equation_number":null,"raw_text":"graph K2 together with cover Ci = {1, 2} and Cj = ∅for j ̸= i form structure","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq4","equation_number":null,"raw_text":"there exist polynomial reductions of L to M and M to L. E.g. assuming P ̸= NP","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq5","equation_number":null,"raw_text":"that (F, D) forms a finite duality, i.e. Forb(F) = CSP(D). Up to homomor-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq6","equation_number":null,"raw_text":"duality if Φ(Forb(F′)) = CSP(D). An example of shadow duality is the language","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq7","equation_number":null,"raw_text":"Forb(F′) = CSP(D′). Then Φ(Forb(F′)) = CSP(Φ(D′)).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq8","equation_number":null,"raw_text":"Φ(Forb(F′)) = CSP(D) for a finite set D of digraphs (relational structures).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq9","equation_number":null,"raw_text":"such that Forb(F′) = CSP(D′).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq10","equation_number":null,"raw_text":"B, a finite set of colors Γ and Φ(B′) = B be given. Then there exists a lift A′,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq11","equation_number":null,"raw_text":"such that Φ(A′) = A and the mapping f is a homomorphism A′ −→B′ (of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq12","equation_number":null,"raw_text":"Conversely, let us assume that A′ ∈Forb(F′) satisfies Φ(A′) = A. But then","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq13","equation_number":null,"raw_text":"A′ ∈CSP(D′) and thus by the functorial property of Φ we have A = Φ(A′) ∈","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq14","equation_number":null,"raw_text":"Proof. (of Theorem 7) Assume Φ(Forb(F′)) = CSP(D). Our goal is to find D′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq15","equation_number":null,"raw_text":"such that Forb(F′) = CSP(D′). This will follow as a (non-trivial) combination of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq16","equation_number":null,"raw_text":"1). It is Φ(D′) = D by Theorem 6. So assume to the contrary that one of the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq17","equation_number":null,"raw_text":"We assume shadow duality Φ(Forb(F′)) = CSP(D). Let D be fixed through-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq18","equation_number":null,"raw_text":"and so does not contain cycles of length ≤l. Now for any F′ ∈F′, F′ ̸= F′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq19","equation_number":null,"raw_text":"expansion. Then there is a finite set of digraphs D such that Φ(Forb(F′)) ∩C =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq20","equation_number":null,"raw_text":"structures F′ such that L = Φ(Forb(F′)).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq21","equation_number":null,"raw_text":"structures) D such that Φ(Forb(F′)) = CSP(D). If one of the lifts F′ in a min-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq22","equation_number":null,"raw_text":"Proof. Consider the color set Γ = {1, . . . , b} and the following set F′ of Γ-colored","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"}],"equation_refs":[]},"symbol_table":[],"references":{"in_text_markers":[],"bibliography":[]},"claims":[],"flow_graph":{"nodes":[],"edges":[]},"quality":{"text_char_count":28995,"parse_confidence":0.5,"equation_parse_rate_proxy":1.0,"citation_resolution_rate_proxy":0.0,"structure_coverage_proxy":0.0,"asset_coverage_proxy":0.0,"accepted_for_training":true}}