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graph [MATH] . The weights, as defined in ( ) are non-negative integers with a maximum value of [MATH] . This value depends on [MATH] and [MATH] but is independent of the number of vertices of [MATH] . In other words, fixing the other parameters, [MATH] gives us a traslation from [MATH] to a weighted graph, with intege... |
By composing Lemma 11 Theorem 12 and Theorem 20 with the appropriate parameters and combining it with the observation above we get the following. |
Theorem 21 For every [MATH] with [MATH] , and [MATH] , there is an [MATH] -interpretation [MATH] that maps instances of 3XOR to undirected graphs in such a way that, for every 3XOR |
instance [MATH] the following hold: 1. if [MATH] is satisfiable, then [MATH] 2. if [MATH] is not [MATH] -satisfiable, then [MATH] , where [MATH] |
Proof. Firt we define [MATH] and then check that it is an [MATH] -interpretation. Let [MATH] be a large enough integer so that the following inequality holds: |
[EQUATION] when [MATH] is small enough, and [MATH] is large enough, so that Theorem 20 applies. Such a [MATH] exists because [MATH] and [MATH] . Apply the bipartite reduction to [MATH] to obtain the instance [MATH] from Lemma 11 . The domain size of [MATH] |
is [MATH] . Apply the parallel repetition reduction of Theorem 12 to [MATH] with parameter [MATH] to get another instance [MATH] . Next apply the intermediate reduction of Lemma 19 to get a graph [MATH] . Finally, apply the Dinur-Safra long-code reduction of Theorem 20 to get a weighted graph [MATH] and convert it to a... |
We still need to check that [MATH] is an [MATH] -interpretation. As in the proof of Theorem 14 , producing [MATH] from [MATH] and [MATH] from [MATH] is straightforward. Producing [MATH] from [MATH] is equally straightforward: the definition of the intermediate reduction is explicit enough that this can be checked direc... |
requires some explanation. In the description of the vertex-cover long-code reduction we already described [MATH] as an alternative presentation ( of [MATH] in ( ). This alternative presentation suggests that the vertex-set of [MATH] be defined by an [MATH] -interpretation of dimension [MATH] through the method of fini... |
and [MATH] involve exactly the same elements from [MATH] and, in such a case, whether the unique permutation that takes [MATH] to [MATH] also takes the set of subsets of [MATH] coded by [MATH] to the set of subsets of [MATH] coded by [MATH] . The first part can be stated by means of a simple quantifier-free formula. Th... |
potential permutations of [MATH] Once the domain is defined as [MATH] in equation ( ), defining the edge-set is easy. Defining the weights is also straightforward given that [MATH] [MATH] [MATH] and [MATH] are all fixed constants independent of [MATH] , and as noted above, we can replace the weights with sets of unweig... |
Now we can state the improved version of Corollary 18 Composing Theorem Theorem 21 , and Lemma we get the following. Theorem 22 For any [MATH] there is a [MATH] such that 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] , where [MATH] |
In terms of algorithms, Theorem 22 says that no algorithm that can be expressed in [MATH] , or even [MATH] for [MATH] , can achieve an approximation ratio better than [MATH] . In particular, this means that [MATH] levels of the Lasserre hierarchy are necessary to give an approximation algorithm for vertex cover with an... |
5.3 Tight lower and upper bounds for [MATH] There are straightforward polynomial-time algorithms that yield a vertex cover in a graph with guaranteed approximation ratio [MATH] . It is conjectured that no polynomial-time algorithm can achieve an approximation ratio of [MATH] for any |
[MATH] ; it is even conjectured that achieving such an approximation ratio is NP-hard. It would be interesting to prove a version of this conjecture for algorithms expressible in [MATH] , and without the assumption that [MATH] . This could be established by a strengthened version of Theorem 22 with better ratios. We ne... |
Theorem 23 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. Proof. Let [MATH] be a family of [MATH] -regular expander graphs on [MATH] vertices, so that the largest independent set in [MATH] has size [MATH] . For the existence of such graphs see |
29 , Chapter 4] . It follows that the smallest vertex cover in [MATH] has size [MATH] . Hence, we can choose a value [MATH] such that [MATH] has no vertex cover smaller than [MATH] |
Let [MATH] be a [MATH] -regular bipartite graph on two sets of [MATH] vertices. Now, each part of a bipartite graph is a vertex cover, so [MATH] has a vertex cover of size [MATH] . However, it is known that [MATH] holds for any pair [MATH] and [MATH] of [MATH] -regular graphs with the same number of vertices, for any [... |
Essentially, Theorem 23 tells us that no algorithm that is invariant under [MATH] can determine [MATH] to an approximation better than [MATH] , and Theorem 22 |
tells us that no algorithm that is invariant under [MATH] for constant or even slowly growing [MATH] can determine [MATH] to an approximation better than [MATH] . A legitimate question at this point is whether there is any algorithm that is invariant under [MATH] , such as one expressible in [MATH] would be, that |
does achieve an approximation ratio of [MATH] . The natural polynomial-time algorithms that give a vertex cover with size at most [MATH] are not expressible in [MATH] . Indeed, we cannot expect a formula of [MATH] to define an actual vertex cover in a graph [MATH] as this is not invariant under automorphisms of [MATH] ... |
is tight: Theorem 24 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 [MATH] -separable. The proof of Theorem 24 proceeds through a series of lemmas. Lemma 25 If [MATH] is a [MATH] -regular graph on [MATH] vertices, for any [MATH] then [MATH] |
Proof. Let [MATH] be any set of vertices in [MATH] . Then the number of edges incident on vertices in [MATH] is at most [MATH] . Since the number of edges in [MATH] is [MATH] , if [MATH] is a vertex cover [MATH] |
and so [MATH] Let [MATH] be a graph and [MATH] be the partition of the vertices of [MATH] given by vertex refinement . So, there are constants [MATH] such that each [MATH] has exactly [MATH] neighbours in [MATH] . Since the graph is undirected, the number of edges from [MATH] to [MATH] is the same as in the other direc... |
Let [MATH] and [MATH] . Consider the undirected graph [MATH] with vertices [MATH] and edges [MATH] . Consider the instance [MATH] of weighted vertex cover obtained by taking the graph [MATH] and giving each vertex [MATH] the weight [MATH] Let [MATH] denote the value of the minimum weighted vertex cover of this instance... |
Lemma 26 If [MATH] then [MATH] Proof. The value [MATH] is determined entirely by the sizes of [MATH] in the vertex refinement of [MATH] and the corresponding values of [MATH] . Since [MATH] , these values are the same for [MATH] |
Lemma 27 [MATH] Proof. Let [MATH] be a minimum-weight vertex cover in [MATH] Take the set [MATH] defined by [MATH] . Note that the sets [MATH] and [MATH] are disjoint, [MATH] by definition, and [MATH] by construction. So [MATH] has exactly [MATH] vertices. We claim that [MATH] |
is a vertex cover in [MATH] . Let [MATH] be any edge of [MATH] with endpoints in [MATH] and [MATH] . If either [MATH] or [MATH] is in [MATH] , then the corresponding endpoint of [MATH] is in [MATH] since [MATH] for all [MATH] . If both [MATH] and [MATH] are not in [MATH] then both are in [MATH] |
and [MATH] . Thus, since [MATH] is a vertex cover for the graph [MATH] then one of [MATH] or [MATH] must be in [MATH] and again at least one endpoint of [MATH] is in [MATH] |
For the proof of the next lemma, we need the notion of a fractional vertex cover of a graph [MATH] . This is a function [MATH] satisfying the condition that for every [MATH] [MATH] . It is known that if [MATH] is a fractional vertex cover of [MATH] , then [MATH] (see 30 , Thm. 14.2] ). More generally, suppose we have a... |
Lemma 28 [MATH] Proof. Let [MATH] be any vertex cover of [MATH] . Let [MATH] and [MATH] and note that these sets are disjoint. We claim that [MATH] and [MATH] , and therefore [MATH] , establishing the result. |
First, consider [MATH] . Note that for any [MATH] , the subgraph of [MATH] induced by [MATH] is [MATH] -regular. Since [MATH] by definition of [MATH] , by Lemma 25 we have [MATH] and therefore [MATH] |
Secondly, consider the function [MATH] defined by [MATH] . We claim that this is a fractional vertex cover of the graph [MATH] . To verify this, we need to check that [MATH] whenever [MATH] . There are [MATH] |
edges between [MATH] and [MATH] . Each element of [MATH] can cover at most [MATH] of these edges and similarly each element of [MATH] covers at most [MATH] of them. Thus, since [MATH] |
is a vertex cover [MATH] . Substituting for [MATH] using the identity [MATH] gives [MATH] . Now dividing through by [MATH] gives [MATH] |
Thus, we have that the weighted vertex cover instance [MATH] admits the fractional solution [MATH] whose total weight is [EQUATION] |
Since [MATH] is the value of the minimum weight vertex cover of [MATH] , we have [MATH] , as was to be shown. Proof of Theorem 24 |
Suppose for contradiction that there is a [MATH] and [MATH] such that [MATH] . Since [MATH] and [MATH] must have the same number of vertices, we have [MATH] . But, by Lemma 28 we have [MATH] , by Lemma 27 we have [MATH] and by Lemma 26 we have [MATH] , giving a contradiction. |
Conclusions This paper introduces a new method for studying the hardness of approximability of [MATH] -hard optimization problems by showing that the approximation cannot be defined in a suitable logic such as [MATH] . This is done by showing that no class of bounded counting width can separate instances of the problem... |
? In other words, can we show that the class of graphs that have a vertex cover of density [MATH] is not separable from the class of graphs that do not have a vertex cover of density [MATH] , for some [MATH] and some constant [MATH] |
greater than [MATH] ? If this were achieved for [MATH] , for unbounded [MATH] , it would have major consequences in the study of semidefinite programming hierarchies of relaxations of vertex cover. A version of this question, with [MATH] being [MATH] and [MATH] |
being [MATH] for arbitrary small [MATH] , was stated as Open Problem 4.1 in . Indeed, similar questions can be posed for any optimization problem for which the exact inapproximability factor is not known, including MAX CUT sparsest cut, etc. |
Acknowledgments. The research reported here was initiated at the Simons Institute for the Theory of Computing during the programme on Logical Structures in Computation in autumn 2016. The first author was partially funded by European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation... |
# Source: arxiv 1806.11358 # Title: Systems of random linear equations and the phase transition in MacArthur's resource-competition model # Sections: all # Downloaded: 2026-03-03T05:15:57.723956+00:00 |
Population dynamics and ecological pattern formation Phase transitions Linear Algebra Systems of random linear equations and the phase transition in MacArthur’s resource-competition model |
Abstract Complex ecosystems generally consist of a large number of different species utilizing a large number of different resources. Several of their features cannot be captured by models comprising just a few species and resources. Recently, Tikhonov and Monasson have shown that a high-dimensional version of MacArthu... |
pacs: 87.23.Cc pacs: 05.70.Fh pacs: 02.10.Ud Introduction Ecosystems – from rainforests to the human gut – can harbor a surprisingly large number of different species |
. These species in general compete for a limited number of resources and possibly prey on each other. Inspired by this observation researchers from various fields examine the role biodiversity plays in complex ecosystems and how their community structure is shaped. However, mathematical studies of model ecosystems were... |
. Results obtained for small settings do not straightforwardly generalize to large systems characterized by collective phenomena and emergent properties |
. Starting with the pioneering work of May such phenomena are increasingly addressed by studying large models with random parameters. This is a sensible approach if self-averaging properties of the system may be identified that only depend on the features of the underlying distributions and not on the individual realiz... |
Along these lines, Tikhonov and Monasson recently investigated a high-dimensional version of MacArthur’s consumer-resource model |
. In this model, different species compete for a number of resources which are supplied by fixed influxes from the environment. An increasing population size of a species leads to a lower resource availability and consequentially to a reduced growth rate, creating a negative feedback loop. Even though the interactions ... |
. Moreover, by performing a stability analysis Altieri and Franz revealed that the collective phase exhibits marginally stable behavior. |
In the phase transition was examined by constructing a Lyapunov-function for the population dynamics and a subsequent replica calculation characterizing its extremum. In the present letter we show that the transition is more general and may be derived without reference to the actual dynamics of the model. We first poin... |
The model The version of MacArthur’s resource-competition model considered in consists of [MATH] species with abundancies [MATH] which can utilize [MATH] resources with availability [MATH] The resources are supplied from the outside by fixed influxes [MATH] that, depending on the overall demand [MATH] , are reduced to ... |
Each species [MATH] is characterized by a metabolic strategy vector [MATH] specifying which resources it can employ. Its entries [MATH] are one if it may consume resource [MATH] and zero otherwise. Moreover, each species needs a minimal resource supply [MATH] in order to survive: if the resource intake is smaller than ... |
[EQUATION] In order to allow for a fair competition between specialists and universalists the threshold [MATH] is chosen to increase with the number of utilizable resources |
[EQUATION] The resource availabilities [MATH] derive from the resource influxes [MATH] and are decreasing functions of the total demand [MATH] of resource [MATH] . Different models of resource supply differ in the form of these depletion functions [MATH] . We require that if influx and total demand balance for every re... |
[EQUATION] where the functions [MATH] are monotonically increasing functions of their argument and satisfy [MATH] . We do not need any further specification of the depletion functions in our analysis. |
In line with previous investigations it is assumed that the model parameters [MATH] and [MATH] are independent random variables. More specifically, [MATH] is taken to be one with probability [MATH] and zero with probability [MATH] . Small values of [MATH] therefore describe populations with many specialists whereas lar... |
For linearized depletion functions ( ) it was shown in that the system possesses two different phases. For small potential diversity, [MATH] , the system is in the vulnerable or V-phase. Here the inhomogeneity of resource influxes [MATH] penetrates down to the level of the resource availabilities [MATH] and the number ... |
In their calculation Tikhonov and Monasson construct a convex Lyapunov function [MATH] which is bounded from above and increases on every trajectory. They show that the stationary states of the system lie on the boundary of the so-called unsustainable region in the space of resource availabilities |
[EQUATION] A partition function of the system is then defined by [EQUATION] where the integration is performed over the unsustainable region and [MATH] is the Legendre transform of [MATH] . In the limits [MATH] the expectation value [MATH] is determined by application of the replica trick, thereby characterizing the st... |
Relation to systems of linear equations The phase transition observed in the resource-competition model can be related to a problem in linear algebra. Let us denote by [MATH] the matrix [MATH] of metabolic strategies, by [MATH] the vector of species abundancies, and by [MATH] the vector of resource influxes. The centra... |
[EQUATION] We therefore expect that the system is in the S-phase if these inhomogeneous linear equations possess a non-negative solution [MATH] and that it is in the V-phase if no such solution exists. Figure tests this assumption on the basis of numerical simulations. Shown is the fraction of random realizations of Eq... |
. It is clearly seen that the solution space of Eq. ( ) is separated into a phase in which typically no solution exists and a phase in which a solution can always be found. The dashed line marks the phase transition derived in |
. Similar behavior is found for other values of [MATH] The observed transition becomes sharper when the systems gets larger as shown by the finite size analysis of Fig. . As can be seen the steepness of the transition increases with increasing system size [MATH] and the extrapolated values of [MATH] converge to the ana... |
Analytical determination of the critical line The question whether the linear system ( ) for large [MATH] typically possesses a non-negative solution [MATH] can be analyzed analytically. To this end we first employ Farkas’ Lemma |
that stipulates that for given [MATH] and [MATH] either ( ) has a non-negative solution or there is a vector [MATH] such that [EQUATION] |
The intuitive meaning of this theorem is simple: the linear combinations of the row vectors [MATH] of [MATH] with non-negative coefficients form what is called the non-negative cone of these vectors. If [MATH] lies within this cone, there is a non-negative solution to Eq. ( ); if not, there must be a hyperplane (with n... |
The dual problem defined by ( ) is rather similar to the storage problem in the theory of feedforward neural networks and can be addressed by similar means. We define the fractional volume of vectors [MATH] that fulfil Eq. ( |
[EQUATION] with the indicator function [EQUATION] The spherical constraint [MATH] is introduced to lift the trivial degeneracy of solutions [MATH] for any positive [MATH] . If [MATH] is zero there are no solutions to ( ) and correspondingly there is a non-negative solution to ( ). Complementary, if [MATH] is larger tha... |
Due to the product structure of [MATH] the entropy [MATH] is expected to be self-averaging with respect to [MATH] and [MATH] . We may hence characterize the typical situation in a large system by considering the average entropy |
[EQUATION] With the help of the replica trick and using standard techniques this entropy may be expressed as a saddle-point integral over order parameters (see supplementary material) |
[EQUATION] Within the replica-symmetric ansatz we find [EQUATION] where the extremum is over [MATH] and [MATH] and the abbreviations [MATH] and [MATH] were used. |
At the transition the volume [MATH] shrinks to zero and the typical overlap [MATH] between two different solutions [MATH] approaches one. Keeping only the most divergent terms of ( 12 ) in this limit we find the following parametric representation of the critical line [MATH] (see supplementary material): |
[EQUATION] where [MATH] . This is the same result as found in exploiting the properties of the Lyapunov function. The expression for the entropy ( 12 ) is rather similar to the one for the average entropy in the storage problem of a perceptron as obtained by Gardner |
. In particular, for [MATH] we find from ( 13 [MATH] and therefore [MATH] , the classical result for the storage capacity of the perceptron. |
Conclusion We have shown that the phase transition in a high-dimensional version of MacArthur’s resource-competition model discovered recently by Tikhonov and Monasson is related to the existence of non-negative solutions of large random systems of linear equations. The starting point of our analysis is the observation... |
here we make – in the limit where the number of unknowns and the number of equations tend to infinity – also predictions for dense solutions. |
Using Farkas’ Lemma the question on the existence of non-negative solutions to linear systems can be mapped onto a dual problem involving a set of linear inequalities. Using methods from the statistical mechanics of disordered systems we have analytically analyzed the typical properties of this dual problem in the ther... |
Acknowledgements. We would like to thank Remi Monasson, Katharina Janzen and Mattes Heerwagen for fruitful discussions. Financial support from the German Science Foundation DFG under grant EN 278/10-1 is gratefully acknowledged. |
Supplementary material to Systems of random linear equations and the phase transition in MacArthur’s resource-competition model Stefan Landmann and Andreas Engel |
The problem The starting point of our considerations is the following question: Given an [MATH] random matrix [MATH] and a random vector [MATH] when does the system of linear equations |
[EQUATION] typically possess a solution [MATH] with all components non-negative, [MATH] ? The entries [MATH] of [MATH] are independently of each other one with probability [MATH] and zero with probability [MATH] . The components of the vector [MATH] are of the form [MATH] with the [MATH] ’s being independent Gaussian v... |
Farkas’ Lemma Farkas’ Lemma states that for a real matrix [MATH] and a real vector [MATH] always one but only one of the following systems has a solution: |
[EQUATION] System ( ) consists of [MATH] equations and [MATH] inequalities while system ( ) consists of [MATH] inequalities. With each solution [MATH] to ( ) also [MATH] with positive [MATH] is a solution. In order to eliminate this trivial degeneracy it is convenient to impose the spherical constraint [MATH] on the so... |
[EQUATION] If this volume is zero the linear equation system ( ) possesses a non-negative solution, if it is positive there is no such solution to ( ). |
The typical fractional volume The fractional volume [MATH] is a random quantity due to its dependence on the random parameters [MATH] and [MATH] . Because of its product structure its logarithm is expected to be self-averaging. Therefore, the central quantity of interest is the averaged intensive entropy |
[EQUATION] It may be calculated using the replica trick based on the identity: [EQUATION] [MATH] is determined for [MATH] and the result needs to be continued in a meaningful way to real [MATH] in order to perform the limit [MATH] . For [MATH] we have |
[EQUATION] where the replica index [MATH] runs from 1 to [MATH] and the denominator [MATH] is due to the normalization in ( ). Using standard techniques we replace the [MATH] - and [MATH] -functions by their integral representations: |
[EQUATION] The averages over [MATH] and [MATH] then yield: [EQUATION] Here [MATH] means that the diagonal terms, [MATH] , are excluded from the sum. The integrals over the [MATH] and those over the auxiliary parameters [MATH] and [MATH] can be decoupled by introducing the order parameters |
[EQUATION] via appropriate [MATH] -functions [EQUATION] The expression for the [MATH] -th power of the fractional volume then acquires the form |
[EQUATION] with the auxiliary functions [EQUATION] and [EQUATION] To determine the entropy ( ) we only need the asymptotics of this expression for [MATH] so that the integrals in ( 17 ) may be calculated by the saddle-point method. The term [MATH] can be neglected in this limit and will be dropped. |
We assume a replica-symmetric saddle-point and make the ansätze [EQUATION] Using this replica-symmetric structure at the saddle-point and keeping in mind that for the final limit [MATH] only terms up to order [MATH] are needed the expressions for [MATH] and [MATH] may be simplified. We find |
[EQUATION] Here the Gaussian measure [MATH] was introduced to perform the Hubbard-Stratonovich transform in the second line and the abbreviations |
[EQUATION] were used. Similar manipulations yield for [MATH] [EQUATION] Simplifying also the remaining terms in ( 17 ) for a replica-symmetric saddle-point and using ( ) and ( ) we finally get |
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