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Long Code Reductions In this section we show that certain reductions from the theory of inapproximability of MAX 3XOR and MAX 3SAT can be expressed as [MATH] -interpretations. While no reduction can provide an improvement on the already optimal inapproximability results that are implied by Theorems |
and , these [MATH] -interpretations have the merit of providing optimal gap pairs starting at any initial gap pair, provided the initial gap pair exhibits any constant gap separation whatsoever. In addition, the details of the [MATH] -interpretations that we work out here will also be useful when we discuss the reducti... |
4.1 Parallel repetition We begin by defining the LABEL COVER problem, a standard problem in the study of hardness of approximation. Indeed, it is described in the textbook , p. 494] as being “ubiquitous” in the PCP literature. For a full discussion of the problem, see , Chap. 22] , where it is called [MATH] |
with the projection property. An instance [MATH] of the LABEL COVER problem is given by two disjoint sets of variables [MATH] and [MATH] with domains of values [MATH] |
and [MATH] , respectively, a predicate [MATH] , and an assignment of weights [MATH] . If all the non-zero weights [MATH] are equal, then the instance is said to have uniform weights . If for all [MATH] the sums [MATH] of incident weights are equal, then the instance is called left-regular . A |
right-regular instance is defined analogously in terms of [MATH] . The instance is a projection game if for every [MATH] with [MATH] it holds that for every [MATH] there is exactly one [MATH] |
satisfying [MATH] . It is called a unique game if [MATH] and it is a projection game both ways: from [MATH] to [MATH] and from [MATH] to [MATH] . The instance is said to have parameters [MATH] if [MATH] [MATH] [MATH] and [MATH] . Its |
domain size is [MATH] A value-assignment for an instance [MATH] is a pair of functions [MATH] and [MATH] . The weight [MATH] of the value-assignment [MATH] is the total weight of the pairs [MATH] satisfying the constraint [MATH] ; i.e., |
[EQUATION] For [MATH] , we say that the instance is [MATH] -satisfiable if there is a value-assignment whose weight is at least [MATH] where [MATH] is the maximum possible weight. We call it satisfiable if it is [MATH] -satisfiable. |
The bipartite reduction takes an instance [MATH] of 3XOR and produces a projection game instance [MATH] of LABEL COVER defined as follows. The sets [MATH] and [MATH] are the set of equations in [MATH] and the set of variables in [MATH] , respectively. The weight [MATH] is [MATH] if [MATH] is one of the variables in the... |
and [MATH] for [MATH] , the set of pairs [MATH] satisfying [MATH] . In other words, [MATH] if, and only if, [MATH] appears in the equation [MATH] , and if [MATH] is [MATH] and [MATH] , then the (partial) assignment [MATH] , which satisfies the equation [MATH] by construction, agrees with the (partial) assignment [MATH]... |
Lemma 11 For every instance [MATH] of 3XOR and every [MATH] , the following hold: 1. if [MATH] is [MATH] -satisfiable, then [MATH] is [MATH] -satisfiable, |
2. if [MATH] is not [MATH] -satisfiable, then [MATH] is not [MATH] -satisfiable. Moreover, [MATH] is a left-regular projection game that has uniform weights. |
Proof. Let [MATH] be the number of equations in [MATH] , so [MATH] has exactly [MATH] pairs [MATH] of unit weight. Such pairs are called constraints. For proving 1, let [MATH] be an assignment for [MATH] that satisfies at least [MATH] of the [MATH] equations in [MATH] . For each equation [MATH] in [MATH] say [MATH] , d... |
if [MATH] satisfies [MATH] , and define [MATH] otherwise. For each variable [MATH] in [MATH] , define [MATH] . Each equation in [MATH] gives rise to exactly three constraints in [MATH] , and if the equation is satisfied by [MATH] , then all three constraints associated to it in [MATH] are satisfied by [MATH] . Thus [MA... |
satisfies at least [MATH] of the [MATH] constraints in [MATH] , so [MATH] is [MATH] -satisfiable. For proving 2, let [MATH] be an assigment for [MATH] that satisfies at least [MATH] of the [MATH] constraints in [MATH] . For each variable [MATH] in [MATH] , define [MATH] . Let [MATH] |
be the number of equations of [MATH] that are satisfied by [MATH] . In terms of [MATH] , the assignment [MATH] satisfies at most [MATH] of the [MATH] constraints of [MATH] . Thus [MATH] , so [MATH] |
is [MATH] -satisfiable. The parallel repetition reduction takes an instance [MATH] of LABEL COVER , and an integer [MATH] , and produces another instance [MATH] of LABEL COVER defined as follows. Let [MATH] and [MATH] be the sets of variables in [MATH] and let [MATH] be the weight assignment. The sets of variables of [... |
and [MATH] are [MATH] and [MATH] respectively. For [MATH] [MATH] [MATH] and [MATH] , the predicate [MATH] is defined as [MATH] . Observe that this definition guarantees that if [MATH] is a projection game, then so is [MATH] |
Theorem 12 (Parallel Repetition Theorem There exists a constant [MATH] such that for every instance [MATH] of LABEL COVER with domain size at most [MATH] , every [MATH] and every [MATH] the following hold: |
1. if [MATH] is satisfiable, then [MATH] is satisfiable, 2. if [MATH] is not [MATH] -satisfiable, then [MATH] is not [MATH] -satisfiable. |
Moreover, if [MATH] is a projection game, left-regular, right-regular, or has uniform weights, then so is [MATH] Although it is the case that the bipartite and the parallel repetition reductions are both [MATH] -interpretations, we do not need to formulate this. Instead, we show the [MATH] -definability of the composit... |
4.2 First long-code reduction The first long-code reduction takes a projection game instance [MATH] of LABEL COVER and a rational [MATH] and produces an instance [MATH] |
of 3XOR defined as follows. Let [MATH] and [MATH] be the sets of variables of sizes [MATH] and [MATH] , respectively, with associated domains of values [MATH] and [MATH] , let [MATH] be the weight assignment, let [MATH] be the predicate of [MATH] , and for each [MATH] with [MATH] and each [MATH] |
let [MATH] be the unique value [MATH] that satisfies [MATH] . The existence of such a function [MATH] is guaranteed from the assumption that [MATH] is a projection game. The set of variables of [MATH] includes one variable [MATH] for each [MATH] |
and [MATH] , and one variable [MATH] for each [MATH] and [MATH] , for a total of [MATH] variables. Before we are able to define the set of equations of [MATH] we need a piece of notation. For a vector [MATH] of dimension [MATH] , we write [MATH] and [MATH] . Note that [MATH] is a single field element, and [MATH] is a v... |
Theorem 13 (Håstad 3-Query Linear Test For every [MATH] with [MATH] and [MATH] and every projection game instance [MATH] of LABEL COVER , the following hold: |
1. if [MATH] is satisfiable, then [MATH] is [MATH] -satisfiable, 2. if [MATH] is not [MATH] -satisfiable, then [MATH] is not [MATH] -satisfiable. |
The proof of Theorem 13 follows from Lemmas 5.1 and 5.2 in . In order to see this, we need to explain how our notation matches the one in |
. Besides the obvious and minor correspondance between multiplicative and additive notation for [MATH] , with [MATH] and [MATH] there are three other noticeable differences between the statement of Theorem 13 and the statements of Lemmas 5.1 and 5.2 in |
The first difference is that Theorem 13 applies to arbitrary projection game instances of LABEL COVER , while the statements in are phrased only for the special cases of the problem that result from applying the parallel repetition construction to a suitable bipartite reduction applied to 3SAT instance. We chose to for... |
show, and also because this is how more recent expositions of these results are presented (see, e.g., ). The second difference is that the conclusion of our statement is phrased in terms of the [MATH] -satisfiability of a 3XOR |
instance, while the statements of Lemmas 5.1 and 5.2 in are phrased in terms of the acceptance rate of a probabilistic test that has the following form: given access to certain tables |
[MATH] and [MATH] , with [MATH] entries [MATH] and [MATH] for certain index sets [MATH] and [MATH] , respectively, choose a random 3-variables parity test on the [MATH] and [MATH] entries under a well-designed special-purpose distribution, and check if it is satisfied. This difference is only notational and minor: our ... |
and [MATH] tables. With this change, [MATH] -satisfiability of the instance translates into the probability of acceptance of the test being at least [MATH] , and vice-versa. |
The third difference in the notation is that our variables [MATH] and [MATH] , and the corresponding entries [MATH] and [MATH] of the tables [MATH] and [MATH] , are indexed by [MATH] |
and [MATH] instead of the more natural [MATH] and [MATH] , respectively. This is due to the fact that we implement the operations of folding over true and conditioning upon [MATH] from |
directly in our construction. In other words, our tables [MATH] and [MATH] are what calls [MATH] and [MATH] respectively. Folding over true as in [MATH] is achieved for [MATH] through the notation [MATH] and [MATH] defined above: we chose to partition [MATH] into [MATH] pairs of the form [MATH] , as [MATH] ranges over ... |
, by construction. Conditioning upon [MATH] as in [MATH] for [MATH] is achieved through the same mechanism as folding over true with the additional observation that the operation of conditioning upon [MATH] is necessary only if the instance of LABEL COVER fails to satisfy the property that for every [MATH] and every [M... |
conditions the table [MATH] upon [MATH] . In our case we do not require this since the given instance of LABEL COVER is a projection game instance, and, in particular, for every [MATH] there is exactly one [MATH] , and hence at least one [MATH] , such that [MATH] ; i.e., [MATH] is the constant [MATH] predicate. It shou... |
the values are taken as arbitrary truth assignments to the variables of a collection of clauses, and not all such assignments satisfy all the clauses. Our exposition is again more modular and also matches more recent expositions of the results in |
(again, see, e.g., ). With this notational correspondence, it is now easy to see that Lemma 5.1 in gives the first claim in Theorem 13 , and Lemma 5.2 in |
applied with [MATH] gives the second claim in Theorem 13 Next, by composing Lemma 11 Theorem 12 , and Theorem 13 with the appropriate parameters we get the following: |
Theorem 14 For every [MATH] with [MATH] and [MATH] there is an FO-interpretation [MATH] that maps instances of 3XOR to instances of 3XOR in such a way that, for every 3XOR instance [MATH] the following hold: |
Proof. First we define [MATH] and then check that this definition is an FO-interpretation. In anticipation for the proof, let [MATH] be a large enough integer so that the following inequality holds: |
[EQUATION] where [MATH] is the constant in Theorem 12 . Such a [MATH] exists because [MATH] and [MATH] . Apply the bipartite reduction to [MATH] to obtain the instance [MATH] from Lemma 11 . Observe that the domain size [MATH] |
of [MATH] is [MATH] . Next apply the parallel repetition reduction to [MATH] with parameter [MATH] to obtain a new instance [MATH] . Finally apply the long-code reduction to [MATH] with parameter [MATH] to obtain the system [MATH] . The parameters were chosen in a way that the system [MATH] satisfies properties 1 and 2... |
It remains to argue that [MATH] can be produced from [MATH] by an FO-interpretation. To define [MATH] from [MATH] there is no difficulty at all: the FO-interpretation is even linear. To define [MATH] from [MATH] |
we note that [MATH] is a constant, and that the weights [MATH] of [MATH] are [MATH] or [MATH] , so again there is no difficulty. In this case the FO-interpretation has dimension [MATH] , and it is [MATH] -bounded. To define [MATH] from [MATH] we note that the domain sizes [MATH] and [MATH] of the instance [MATH] are co... |
of [MATH] are still zeros or ones and both [MATH] and [MATH] are constants, the multiplicities of the equations of [MATH] are also constants, and hence [MATH] -definable. |
It is useful to compare Theorem 14 with Lemma . Both statements are reductions that take 3XOR instances to 3XOR instances, and they both preserve gaps. But the reductions differ in what happens to satisfiable instances. For statement , in which the extreme case is [MATH] , the reduction in Lemma preserves this extremal... |
4.3 Second long-code reduction The second long-code reduction takes a projection game instance [MATH] of LABEL COVER and a rational [MATH] |
and produces an instance [MATH] of 3SAT defined as follows. Before we define [MATH] , let us define an intermediate instance [MATH] of 3SAT that takes a different parameter [MATH] Let [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] and [MATH] be as in the first long-code reduction. The set of vari... |
and [MATH] from the first long-code reduction. Now the instance [MATH] includes [MATH] copies of the clause [MATH] for each [MATH] , each [MATH] and each [MATH] , where [MATH] is the denominator of [MATH] reduced to lowest terms, [MATH] is the number of positions [MATH] |
with [MATH] and [MATH] is the number of positions [MATH] with [MATH] and [MATH] for [MATH] if [MATH] , while [MATH] is the indicator for the event that in each position [MATH] with [MATH] we have [MATH] . Finally, to define the instance [MATH] set [MATH] and [MATH] and [MATH] for [MATH] , and let the instance be [MATH] |
Theorem 15 (Håstad 3-Query Disjunction Test There exists [MATH] such that for every [MATH] with [MATH] and every projection game instance [MATH] of LABEL COVER |
the following hold: 1. if [MATH] is satisfiable, then [MATH] is satisfiable, 2. if [MATH] is not [MATH] -satisfiable, then [MATH] is not [MATH] -satisfiable. |
For the proof of Theorem 15 , see Lemmas 6.12 and 6.13 in . As in the first long-code reduction, some explanation is needed for seeing this. |
Besides the notational differences that were already pointed out in the first long-code reduction, the second long-code reduction adds the following. First, the constants [MATH] and [MATH] in the definition of [MATH] come from setting [MATH] in the definition of Test F3S [MATH] in |
. According to Lemma 6.9 in , this is an acceptable setting of [MATH] . Second, the constant [MATH] in Theorem 15 is meant to be chosen small enough so as to ensure that, for each [MATH] satisfying [MATH] , we have [MATH] for [MATH] , where [MATH] is the constant hidden in the asymptotic [MATH] -notation of Lemma 6.13 ... |
. Such an [MATH] exists because [MATH] as [MATH] . With this notation, Lemma 6.12 in gives point 1, and Lemma 6.13 in with [MATH] gives point 2 in Theorem 15 |
By composing Lemma 11 Theorem 12 , and Theorem 15 with the appropriate parameters we get the following: Theorem 16 For every [MATH] with [MATH] and [MATH] there is an FO-interpretation [MATH] that maps instances of 3XOR to instances of 3SAT in such a way that, for every 3XOR instance [MATH] the following hold: |
Proof. First we define [MATH] and then check that this definition is an FO-interpretation. Let [MATH] be a large enough integer so that the following inequality holds: |
[EQUATION] where [MATH] is the constant in Theorem 12 and [MATH] is small enough as in Theorem 15 . Such [MATH] exists because [MATH] and [MATH] as well as [MATH] . Apply the bipartite reduction to [MATH] to obtain the instance [MATH] from Lemma 11 . Observe that the domain size [MATH] of [MATH] is [MATH] . Next apply ... |
to obtain the system [MATH] . The parameters were chosen so that the system [MATH] satisfies properties 1 and 2, through Theorem 15 . As in the proof of Theorem 14 this reduction is FO-definable. |
This gives us another route to Theorem Vertex Cover We investigate gap inexpressibility results for the vertex cover problem VC on graphs. Recall that a set [MATH] of vertices in a graph [MATH] is a vertex cover if every edge in [MATH] has at least one of its endpoints in [MATH] . If the graph comes with a weight funct... |
In the following we write [MATH] for the weight of a minimum weight vertex cover, and [MATH] , where [MATH] , for the vertex cover density . Analogously, we write [MATH] for the weight of a maximum weight independent set, and [MATH] . Clearly [MATH] holds for all weighted graphs. |
5.1 Direct reductions The standard reduction that proves the [MATH] -completeness of the vertex cover problem (see, e.g. 25 , Thm. 9.4] ) takes an instance [MATH] of 3SAT with [MATH] variables and [MATH] clauses and gives a graph [MATH] with [MATH] vertices in which the minimum vertex cover has size exactly [MATH] , if... |
We can improve on this by considering instead the so-called FGLSS reduction (see , and for what by now became standard terminology) from 3XOR to vertex-cover, which we describe next. |
Theorem 17 There is a linearly-bounded first-order reduction [MATH] that takes an instance [MATH] of 3XOR with [MATH] equations to a graph [MATH] |
with [MATH] vertices so that if [MATH] is the maximum number of equations of [MATH] that can be simultaneously satisfied, then [MATH] |
Proof. For each equation [MATH] in [MATH] , the graph [MATH] has a 4-clique of vertices, each labelled with a distinct assignment of values to the three variables that make the equation true. In addition, we have an edge between any pair of vertices that are labelled by inconsistent assignments. It is easily seen that ... |
From this, and Theorem , we immediately get the following result. Corollary 18 For any [MATH] , if [MATH] is the collection of graphs [MATH] |
with [MATH] and [MATH] is the collection of graphs [MATH] with [MATH] then [MATH] and [MATH] are not [MATH] -separable for any [MATH] such that [MATH] |
This improves the FPC inapproximability ratio from [MATH] to [MATH] Better lower bounds on the approximation ratio are known under the assumption that [MATH] . One such lower bound was achieved by Dinur and Safra |
who showed that, under this assumption, no polynomial-time algorithm for approximating vertex cover can achieve an approximation ratio better than [MATH] . In the next section we argue that this reduction is also an FO-interpretation, so we get the same inapproximability ratio for algorithms that are expressible in [MA... |
5.2 Dinur-Safra reduction As in the long-code reductions from Section , this reduction is also composed of several steps: we start with the bipartite reduction, continue with the parallel repetition reduction, then we apply an intermediate reduction to a technical variant of the independent set problem, and end with a ... |
The intermediate reduction takes a projection game instance [MATH] of LABEL COVER as input and produces an undirected graph [MATH] defined as follows. Let [MATH] [MATH] [MATH] [MATH] [MATH] |
and [MATH] determine the projection game instance [MATH] . The set of vertices of the graph [MATH] is [MATH] . There is an edge between [MATH] and [MATH] |
in [MATH] if, and only, if either [MATH] and [MATH] or [MATH] and there exists [MATH] such that [MATH] and [MATH] and [MATH] . This defines [MATH] . In the terminology of |
, the graph [MATH] is [MATH] -co-partite: its edge-set is the complement of an [MATH] -partite graph with all its parts of size [MATH] |
For an undirected (unweighted) graph [MATH] , recall that [MATH] denotes the size of a largest independent set in [MATH] . For an integer [MATH] , let [MATH] denote the size of a largest subset of vertices of [MATH] that does not contain any [MATH] -clique. Note that [MATH] , and [MATH] |
Lemma 19 For every integer [MATH] , every [MATH] with [MATH] and every instance [MATH] of LABEL COVER that is a left-regular projection game with uniform weights and parameters [MATH] , the following hold: |
1. if [MATH] is satisfiable, then [MATH] 2. if [MATH] is not [MATH] -satisfiable, then [MATH] Moreover, [MATH] is an [MATH] -co-partite graph. |
Note that, in the statement of Lemma 19 , the completeness case (point ) is about [MATH] but the soundness case (point ) is about [MATH] . For the proof of Lemma 19 , see the proof of Theorem 2.1 in |
The vertex-cover long-code reduction inputs a graph [MATH] , two rational parameters [MATH] and [MATH] satisfying [MATH] , and two integer parameters [MATH] and [MATH] , and, if [MATH] is an [MATH] -co-partite graph for some [MATH] , then it produces a (weighted) graph [MATH] |
defined as follows. Let the vertex-set [MATH] of [MATH] be [MATH] so that [MATH] forms an [MATH] -clique for each [MATH] . In abstract, the set of vertices of [MATH] is |
[EQUATION] where [MATH] , and [MATH] is an integer that depends only on [MATH] [MATH] , and [MATH] , and is independent of [MATH] , that is set as in Definition 2.3 of |
. Here, and in the following, [MATH] and [MATH] denote the collections of subsets of [MATH] of size exactly [MATH] and size at least [MATH] , respectively, and [MATH] denotes the collection of all subsets of [MATH] . Thus, if [MATH] is the number of vertices of [MATH] , then [MATH] has [MATH] vertices. Since we want to... |
Let [MATH] denote the set of [MATH] -tuples of pairwise distinct elements from [MATH] . Formally, [EQUATION] For each [MATH] let [MATH] be the natural bijection defined by [MATH] for [MATH] . The set |
[EQUATION] is a good proxy for the set [MATH] through the identification of [MATH] and [MATH] given by [MATH] . Now, turning [MATH] into a faithful copy of [MATH] |
is only a matter of taking a quotient with the appropriate equivalence relation, as we do next. Let [MATH] be the equivalence relation on [MATH] defined by [MATH] if and only if for each [MATH] there exists [MATH] with [MATH] |
and for each [MATH] there exists [MATH] with [MATH] Restricted to [MATH] , this is still an equivalence relation. Moreover, whenever [MATH] and [MATH] |
are [MATH] -equivalent tuples in [MATH] , there is a unique permutation [MATH] that sends [MATH] to [MATH] ; i.e., that satisfies [MATH] , or [MATH] |
for each [MATH] . Now we extend this equivalence relation [MATH] from the set [MATH] to the set [MATH] as follows: [MATH] if, and only if, [MATH] and the unique permutation [MATH] that sends [MATH] to [MATH] also sends [MATH] to [MATH] ; i.e., it satisfies [MATH] , where [MATH] denotes the natural action of [MATH] on [... |
[EQUATION] is an alternative presentation of the same set [MATH] . This alternative presentation of [MATH] is useful when we argue that the reduction is an [MATH] -interpretation in Theorem 21 |
below. We still need to define the vertex-weights and the edge-set of [MATH] . The weight of a vertex [MATH] in [MATH] is defined as |
[EQUATION] where [MATH] is the denominator of [MATH] reduced to lowest terms, and [MATH] . Next we define the edge-set: two vertices [MATH] and [MATH] in [MATH] are adjacent if, and only if, either [MATH] and [MATH] , or there exist an edge [MATH] of [MATH] and an [MATH] -element subset [MATH] of [MATH] such that [MATH... |
and [MATH] and, for all [MATH] , either [MATH] or [MATH] Theorem 20 (Dinur-Safra Vertex-Cover Test For any two rationals [MATH] and [MATH] satisfying [MATH] and [MATH] , any small enough [MATH] , any large enough integer [MATH] , and any [MATH] -co-partite graph [MATH] , the following hold: |
1. if [MATH] , then [MATH] 2. if [MATH] , then [MATH] , where [MATH] For the proof of Theorem 20 , see Theorem 2.2 in The reduction described above produces a weighted |
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