Multitasking Capacity: Hardness Results and Improved Constructions

We consider the problem of determining the maximal α∈ (0,1] such that every matching M of size k (or at most k) in a bipartite graph G contains an induced matching of size at least α |M|. This measure was recently introduced in Alon et al. (NIPS 2018) and is motivated by connectionist models of cognition as well as modeling interference in wireless and communication networks. We prove various hardness results for computing α either exactly or approximately. En route to our results, we also consider the maximum connected matching problem: determining the largest matching N in a graph G such that every two edges in N are connected by an edge. We prove a nearly optimal n^1-ϵ hardness of approximation result (under randomized reductions) for connected matching in bipartite graphs (with both sides of cardinality n). Towards this end we define bipartite half-covers: A new combinatorial object that may be of independent interest. To the best of our knowledge, the best previous hardness result for the connected matching problem was some constant β>1. Finally, we demonstrate the existence of bipartite graphs with n vertices on each side of average degree d, that achieve α=1/2-ϵ for matchings of size sufficiently smaller than n/poly(d). This nearly matches the trivial upper bound of 1/2 on α which holds for any graph containing a path of length 3.

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1 Introduction

A matching in an undirected graph is a set of vertex disjoint edges. Matchings have been used in studying interference effects in parallel and distributed systems. The object of study is typically a set of units that transmit or receive information. For example, in the communication setting, there is a bipartite network consisting of senders () and receivers ()111To simplify matters, we consider the synchronous setting where transmissions occur in discrete time slots.. Given a set every sender wishes to send a message to its neighbor . The main assumption is that in order to successfully receive a message, a receiver can have only a single incident edge carrying a message at a given time, as messages arriving on multiple incident edges create interference with each other. This can formalized by a condition which we term the matching condition: A subset can be used for concurrent interference-free communication if it forms a matching in . However, in several communication settings such as radio and wireless networks [BLM93, CK85, AMS12], a more constrained setting is considered: the senders cannot choose which edges to broadcast, but instead, if they choose to transmit, then they automatically broadcast on all their incident edges. This leads to the stronger induced matching condition: A subset can be used for concurrent interference-free communication if it forms an induced matching in , namely no two edges in are connected by an edge in .

A similar interference model, directed towards understanding multitasking constraints in neural systems, has been proposed recently in computational neuroscience [FSGC14, MDO16b, MDO16a]. These works seek to understand the reason behind multitasking limitations: The limited ability of people to execute several actions concurrently, a ubiquitous finding in cognitive psychology [SS77]. The main idea in these works is that such limitations arise from interference effects between interrelated units and not because of limited resources (e.g., limited attention or constrained working memory). The models in  [FSGC14, MDO16b, MDO16a] follow the connectionist approach to cognition [RHM86, RMG87]

which strives to explain cognition in terms of a large network of interconnected units that operate in a parallel and distributed manner and have also played a pivotal role in the study of neural networks

[Hin90]. In [FSGC14, MDO16b, MDO16a] a formal model to study multitasking is provided where given a bipartite graph , every vertex is associated with a set of inputs , every vertex is associated with a set of outputs and the edge is associated with a function (“task”) . As before, it is assumed that every unit in transmits its computed value to all adjacent neighbors in , and that a value is stored without interference in a node only if it receives at most one value from a single unit in . In other words, given a set of of edges the set of functions can be performed concurrently (“multitasked”) if is an induced matching. These works are noteworthy as they relate graph theoretic properties of interconnected units and cognitive performance (multitasking). Perhaps surprisingly, hardly any works originating from connectionist models have studied how graph-theoretic properties relate to cognitive models arising from experimental findings or computer simulations.

Based on these interference assumptions, a new measure has been proposed to capture how well such networks allow for interference-free processing [ARS17]. The idea behind this measure is to consider a parameter , and ask whether every matching of size (or of size at most ) contains a large induced matching . Unless stated otherwise we will always assume that graphs are bipartite and that both sides of the bipartition have cardinality .

Definition 1.1.

Let be a bipartite graph, and let be a parameter. For we say that is a -multitasker if for every matching in of size , there exists an induced matching such that

Define to be the maximal such that is a -multitasker if contains a matching of size , and define if does not contain a matching of size . We call the parameter the multitasking capacity of for matchings of size .

Also, define and call it the multitasking capacity of for matchings of size at most .222Since we consider the minimum, the definition of ensures that values of for which there is no matching of size have no influence on .

The parameters measure how resilient is to interferences. The larger these parameters are, the better is considered as a multitasker.

One motivation for this definition is that it is sometimes assumed (e.g., [FSGC14]) that the set of tasks (edges) that need to be multitasked are restricted to be a matching, a restriction which is imposed by limitation on the parallelism of the network. In this case, the multitasking capacity quantifies for every set of allowable tasks what fraction of tasks from are guaranteed to be achieved without interference. Another motivation is the distinction between interference effects that result from a violation of the matching condition to those that result from a violation of the induced matching condition. In particular, the above multitasking measure allows us to assess the fraction of tasks that can be performed concurrently conditioned on not violating the matching condition.

In [ARS17] several properties of have been proven. For example, it was shown that for -regular graphs, and that for graphs of average degree . This upper bound supports the previous hypothesis [FSGC14] suggesting that as the average degree increases, the multitasking capacity inevitably decreases, regardless of the structure of the network – a phenomenon referred to as the “multiplexing versus multitasking trade-off”333In the irregular case this holds assuming the average degree satisfies .. It was also shown in [ARS17] how to construct graphs with desirable multitasking properties. Namely graphs for which for provided that , where is the average degree of .

The results in [ARS17] leave several questions.

Question 1.2.

Given a graph and a parameter , can we compute or efficiently?

Indeed, if we are to use or to evaluate how prone to interference parallel architectures are, then a natural question is whether it is possible to compute or approximate these quantities in polynomial time. For example, computer simulations are frequently used in developing connectionless models and these models often consist of networks consisting of dozens (or more) of units. Hence to evaluate the usefulness of in connectionist models of multitasking it is desirable to have efficient methods to compute exactly or approximately.

Another question is whether it is possible construct multitaskers with near-optimal capacity. While [ARS17] provide multitaskers with for (and show that the upper bound on is tight up to the degree of the term), the best constant value of they achieve is bounded away from the natural barrier444Observe that if a network contains a path of length then trivially for all . . We thus raise the following question.

Question 1.3.

Is there an infinite family of graphs of average degree such that for arbitrarily small and for some function ?

In this paper we address these two questions. For creftype 1.2 we show that under standard complexity theoretic assumptions and cannot be computed efficiently, thus giving a negative answer to this question. For creftype 1.3 we give a positive answer, by showing how to construct bipartite graphs with multitasking capacity approaching .

1.1 Our results

As it turns out, a useful notion in studying the computational hardness of computing the multitasking capacity is that of a connected matching, which is a matching in which every two edges are connected by a third edge (see Definition 2.3 for a formal definition). Connected matchings have been studied in several contexts, such as Hadwiger’s conjecture [KPT05, PST03, FGS05]. Motivated by applications to other optimization problems [JKW14], algorithms for finding connected matchings of maximum cardinality have been studied in special families of graphs such as chordal [Cam03] and bipartite chordal graphs [JKW14]555Observe that bipartite chordal graphs are not necessarily chordal. See [JKW14] for details. and bipartite permutation graphs [GHvHP14].

In Section 3 we establish hardness of approximation for the size of the largest connected matching to within a factor of assuming . Previously, this problem was known to be -hard to approximate within some constant factor [PST03] for general (non-bipartite) graphs. We also prove that deciding whether a bipartite graph with contains a connected matching of size is -hard.

In Section 4 we prove several hardness results for computing the multitasking capacity. To be more precise, we define the decision problem of computing the multitasking capacity as follows:

Definition 1.4.

Let be the problem of deciding whether for a given graph , a positive integer and a rational number it holds that .

The problem belongs to the second level of the polynomial hierarchy, , since the statement can be expressed as , where is the predicate checking that is a matching in of size , and is an induced matching, which is clearly computable in time . We note that it is not clear whether it belongs to or to , and in fact, we give evidence that belongs to neither of the classes. Specifically, we show that is both -hard and -hard; thus, if , then the polynomial hierarchy collapses to the first level.

Furthermore, we show various hardness of approximation results for computing and . Most notably, we show under standard complexity theoretic assumptions that (1) is inapproximable to within for any , and, (2) inapproximable to within any constant for for any . Furthermore, under a stronger assumption, we improve the inapproximability ratio for to for . Our hardness results are summarized in Table 1.

Variant Assumption (approximation factor) Remarks
for any
has maximum degree
some constant
arbitrarily large constant
ETH
Table 1: Hardness of approximation results for computing the multitasking capacity. In each row, the stated variant of the multitasking capacity (either or ) is hard to approximate under the stated assumption up to a multiplicative factor , for the stated values of and .

In Section 6, we prove the existence of multitaskers with near-optimal capacity. For integers with and , we show how to construct multitasker graph on vertices with average degree and , where . In particular, for this implies that can be taken to be , and thus tends to its natural barrier as grows.

1.2 Our techniques

Hardness results.

With respect to multitasking, connected matchings are the worst possible multitasking configuration for a matching of size . In particular, it holds trivially that and , and the equality holds if and only if contains a connected matching of size . This fact, together with extremal Ramsey bound on the size of independent sets, turns out to be instrumental in proving hardness results for computing the multitasking capacity.

Construction of multitaskers.

The starting point of our multitaskers with nearly optimal multitasking capacity is based on locally sparse graphs, similarly to [ARS17]. They used the local sparsity with Turan’s lower bound on independent sets in graphs with a given average degree in order to establish the existence of sufficiently large independent sets (which translate to induced matchings). However, the use of Turan’s bound necessarily entails a constant loss, which makes the final multitasking capacity bounded away from . We circumvent this roadblock by also requiring that the graph has large girth, and use this fact in order to carefully construct a large independent set.

2 Preliminaries

All graphs considered in this work are undirected. A matching in a graph is a collection of vertex disjoint edges. We say that a vertex is covered by if it is one of the endpoints of an edge in . We say that a matching is induced in if no two edges in are connected by an edge in , i.e., the vertices in span only the edges in and no other edges. Given a graph and an edge , we define the contraction of to be the operation that produced the graph , whose vertex set is , the vertex is connected to all vertices in neighboring or , and for all other vertices , they form an edge in if and only if they were connected in . Contracting a set of edges, and in particular contracting a matching, means contracting the edges one by one in an arbitrary order666We remark that the graph obtained from contracting a set of edges, indeed, does not depend on the order..

Below we define two combinatorial optimization problems that we will relate to when proving hardness of approximation results for the parameters

and .

Definition 2.1.

Given an undirected graph , an independent set in is a set of vertices that spans no edges. The Maximum Independent Set Problem () is the problem of finding a maximum cardinality of an independent set in .

Definition 2.2.

Given a graph , we say that two disjoint subsets of the vertices form a bipartite clique (biclique) in if for all and . We say that the biclique is balanced if . In the Maximum Balanced Biclique Problem we are given a bipartite graph and a parameter , and the goal is to decide whether contains a balanced biclique with vertices on each size.

Definition 2.3.

Given a graph , a connected matching in is a matching such that every two edges in are connected by an edge in . We use to denote the size of the maximum cardinality of a connected matching in . In the Connected Matching Problem, we are given graph and parameter and our goal is to determine whether .

Given an optimization (minimization or maximization) problem over graphs, we denote by the value of the optimal solution of for . An algorithm for a maximization (minimization) problem is said to achieve an approximation ratio if for every input the algorithm returns a solution such that (resp. ).

We assume familiarity with complexity classes such as , and the polynomial-time hierarchy. Precise definitions of these terms are omitted, and can be found, e.g., in [Pap03].

3 Hardness results for maximum connected matchings

In this section, we prove hardness results for finding large connected matchings in graphs.

3.1 Hardness of approximating the size of a maximum connected matching

We start by showing an almost optimal hardness of approximation result for the connected matching problem.

Theorem 3.1.

Given a bipartite graph with vertices on each side, it is -hard to approximate within a factor of for any under a randomized polynomial time reduction.

More precisely, given a bipartite graph with vertices on each side, it is -hard to distinguish between the case where and the case where for any .

A natural approach to prove hardness of approximation results for connected matching is to reduce the clique problem to it. Namely given a graph for which we wish to determine if contains a -clique, replace every vertex by an edge and add two edges and for every edge in . Call the resulting graph after these transformation . While it is clear that a large clique in translates to a large connected matching in , it is not clear that a large connected matching in implies a large clique in . The difficulty is that a connected matching might contain “bad” edges of the form where . An illustrative example is the case where is a biclique; in this case, the largest clique in has size only but the resulting graph contains a large connected matching of size as large as .

To overcome this problem, we first observe that instead of adding both and to the graph for every edge in . It suffices to add only one of the two to retain a large connected matching in the YES case. Then, the insight is that, when we choose the edge to add independently at random for each , we can control the number of bad edges in every connected matching in

We formalize the described ideas below, starting with the main gadget of our reduction:

Definition 3.2.

Fix . A bipartite graph is said to be a bipartite half-cover of if (1) for every , or , and (2) for every , .

The reduction used in the proof of Theorem 3.1 uses the existence of such bipartite half-covers of that do not contain a large connected matching. Such graphs can be easily constructed using a randomized algorithm as shown below.

Claim 3.3.

There is an -time randomized algorithm that on input outputs a graph , which is a bipartite half-cover of such that

with probability

.

Proof.

We construct by choosing for each to add to either or independently with probability 1/2. Clearly, is a bipartite half-cover of . Below we show that with probability . We prove this in two steps: first, we will prove the upper bound on a special class of connected matching and, then, we will show that any connected matching contains a large (constant fraction) matching of this type.

Let be any matching in . We say that the matching is non-repetitive if, for each , at most one of or appears in . We will now argue that with probability , any connected non-repetitive matching has size less than . To do so, consider any ordered tuple where are all distinct. The probability that is a connected matching is at most

where the first two equalities use the fact that are distinct, meaning that the events considered are all independent. Hence, by union bound over all such sequences, we can conclude that the probability that contains a connected non-repetitive matching of size is at most .

Finally, observe that any matching contains a non-repetitive matching of size at least . Indeed, given a matching we can construct iteratively by picking an arbitrary edge , remove and all edges touching or from and add to . We repeat this procedure until . Since we add one edge to while removing at most three edges from , we arrive at a non-repetitive of size at least . As a result, the graph does not contain any connected matching of size at least with probability . ∎

Remarks.

  1. We remark that a deterministic polynomial time construction of such graphs would imply that the hardness result in Theorem 3.1 holds under a deterministic reduction (as oppose to the randomized reduction, currently stated).

  2. We comment that there is a connection between Ramsey graphs and half-cover of with small . Specifically, if we can deterministically construct half-cover for with , then we can deterministically construct -vertex -Ramsey graphs. This is because, we can think of half-cover as a bichromatic where for is colored red if and it is colored blue otherwise (i.e. ). It is easy to check that any monochromatic clique of size in implies a connected matching of size in . While there are explicit constructions of Ramsey graphs, it is unclear (to us) how to construct such half-cover from these constructions.

  3. Using a different approach we can show that it is -hard to compute under a deterministic reduction. See Appendix A for details.

3.1.1 Proof of Theorem 3.1

With the gadget from creftype 3.3 we are ready to prove Theorem 3.1. This is done in the following claim.

Claim 3.4.

Let be an -vertex graph, and let be a balanced bipartite graph. Let be the balanced bipartite graph with vertices on each side, where (1) for every , if and only if and , and (2) for every , .

Then, for any such we have where denotes the clique number of . Furthermore, if is a bipartite half-cover of , then .

creftype 3.4 immediately implies Theorem 3.1. Indeed, by [Hås01, Zuc06] given an -vertex graph it is NP-hard to decide between the case where , and the case where . Therefore, we can define a randomized reduction that given an -vertex graph constructs (with high probability) , the bipartite half-cover of , with , and outputs , which can be clearly constructed in time that is linear in the size of . In the YES case, if , then by the “furthermore” part of creftype 3.4 we have , and in the NO case, if , then by creftype 3.4 we have . This completes the proof of Theorem 3.1.

We now turn to the proof of creftype 3.4.

Proof of creftype 3.4.

First, we will show that . Let be any connected matching in . We partition into two disjoint sets and where and . We will show that and .

To show that , suppose that . By the definition if is connected to in , then . Therefore, induces a clique in and follows.

Next, we show that . Let us first define non-repetitive matching in the same way as that in the proof of creftype 3.3. Using the same argument as in that proof, we can conclude that contains a non-repetitive connected matching of size at least . We claim that is also a connected matching in . Indeed, since every edge in belongs to , the non-repetitiveness implies that any pair of edges in is connected by an edge that also belongs to . As a result, we can conclude that .

Combining the above two bounds yields as desired.

Finally, assume that is a bipartite half-cover of . For any clique in , it is not hard to see that the matching is a connected matching in . Indeed, for each distinct we have either or (from definition of bipartite half-cover of ), and hence either or belongs to . Therefore, , which completes our proof. ∎

3.2 Hardness of finding a connected perfect matching

In this section we show that given a bipartite graph with vertices on each side, it is -hard to find a connected matching of size .

Theorem 3.5.

Given a bipartite graph with it is -hard to determine whether .

Proof.

By Theorem 3.1 given a graph with vertices of each side it is -hard to decide whether contains a connected matching of size . Consider the reduction that given a graph outputs as follows. The sets and are two disjoint sets that are also disjoint from with . The set of edges is defined as . That is, the graph contains the graph as the induced graph on the vertices , and in addition, every vertex in is connected to all vertices in , and every vertex in is connected to all vertices in ,

The graph is a balanced bi-partite graph with vertices on each side. We claim that if and only if .

In one direction, suppose that has a connected matching of size . We construct a matching of size as follows. For each vertex not covered by , we pick a distinct element that is a neighbor of . Define a matching in to be , where . By the construction of , each edge in is connected to every other edge in using an edge between and . Every pair of edges in are connected since is a connected matching in . Thus, is a connected matching of size in .

Conversely, suppose has a connected matching of size . Then, there must be is a submatching of size such that no edge in contains a vertex in . Thus, is a matching in , and since is a connected matching so is . It follows that has a connected matching of size , as required. ∎

4 Hardness results for computing

In this section we study the computational complexity both of the decision problem as well as the problem of computing exactly or approximately. We first show an almost optimal inapproximability result for , which is stated and proved below.

Theorem 4.1.

For any , given a bipartite graph with vertices in each part, it is -hard to approximate within a factor .

Furthermore, given a bipartite graph with vertices in each part, where the degree of each vertex is at most it is -hard to approximate within a factor and it is -hard to approximate within a factor .

Proof.

The proof is by a reduction from the Maxium Independent Set problem. Given an vertex graph instance of the we construct a bipartite graph as follows. Denote the vertices of by . Then the vertices of the bipartite graph are defined by and , and the edges of are . Note that the only perfect matching in , i.e., a matching of size , is the matching . Indeed, suppose there exists another matching with . Then has at least one edge of the form with and suppose that is such that is minimal (where the minimum is taken with respect to all edges not in ). If any edge in covers , then it cannot belong to as is a a matching. By the definition of there cannot be an edge in that covers by the minimality of . As all vertices of must be matched in order for , we get a contradiction showing that is indeed the unique matching of size .

We claim that contains an independent set of size at least if and only if . Indeed, a set is an independent set in if and only if is an induced matching contained in . Hence if contains an independent set of size then contains an induced matching of size . Conversely, If contains an induced matching of size then has an independent set of size . It is well known that for any it is -hard to distinguish between -vertex graphs that contain an independent set of size at least (YES case) and graph that do not contain an independent set of size at least (NO-case) [Hås01, Zuc06]. By the reduction described above it is -hard to distinguish between a bipartite graph with sides of cardinality satisfying to a graph satisfying as this would enable to distinguish between the YES and NO cases described above. The result now follows by taking to equal .

The result for graphs of maximum degree follows by noting that if the maximal degree of is at most , then the maximal degree of is upper bounded by . Therefore, since it is -hard to approximate in graphs of maximum degree within a factor of [Cha16] and -hard to approximate in graphs of maximum degree within a factor of [AKS09], the analogous hardness computing also follows. ∎

We remark that by adding isolated vertices to the graph, the above hardness result also implies hardness of approximating to within factor of for every and every for any constant .

Recall the decision problem from Definition 1.4. As mentioned in the introduction, clearly belongs to the class . We show the following:

Theorem 4.2.

The decision problem is -hard and -hard.

Proof of Theorem 4.2.

By Theorem 4.1 if follows that that there is a reduction from any problem in that produces a graph and a parameter such that in the YES case , and in the NO case . In particular, this implies that is -hard.

In order to prove that is -hard we use Theorem 3.5. Indeed, observe that if and only if contains a connected matching of size , and hence there is a reduction from any problem in that produces a graph and such that in the YES case , and in the NO case . This completes the proof of Theorem 4.2

Using Theorem 3.5, we demonstrate that it is unlikely that belongs to .

Corollary 4.3.

If the decision problem belongs to , then the polynomial-time hierarchy collapses to the first level.

Indeed, this follows from the fact that if , then (see e.g., Proposition 10.2 in [Pap03]), and hence the polynomial hierarchy collapses to the first level.

We end this section with several remarks.

  1. Note that the proof of Theorem 4.1 shows that the problem of computing is -hard on graphs with vertices on each side that even if contains a unique perfect matching.

  2. Note also that the hardness result in Theorem 4.1 for bounded degree graphs is unlikely to hold for regular graphs (as opposed to graphs with degree at most ) This is because in [ARS17] it is shown that for every -regular graph . In particular, this implies that it is easy to approximate within a factor of for -regular graphs.

5 Hardness results for computing

Here we prove that it is hard to calculate the parameter .

5.1 Hardness results for computing

We first consider the case.

Theorem 5.1.

Given a bipartite graph with , it is -hard to compute .

Proof.

It is immediate that and that equality holds if and only if contains a connected matching of size . The theorem follows from Theorem 3.5. ∎

We proceed and consider approximating .

Theorem 5.2.

Unless , there is no polynomial algorithm for approximating within some constant factor.

Proof.

We first use the fact that it is -hard to distinguish between -vertex graphs with cliques of size to graphs with no clique of size where are some constants satisfying . Indeed it is well known that there are such that it is -hard to distinguish between -vertex graphs with cliques of size and graphs with no clique of size (e.g. [Hås01]). The fact now follows by taking a graph of vertices, adding to it a clique of size and connecting all vertices in this clique to all vertices of .

Given a graph apply the reduction in creftype 3.4 (with being the random graph described in creftype 3.3) and call the resulting graph . If there is a clique of size then clearly . Suppose there is no clique of size in . Then by creftype 3.3, with high probability there is no connected matching in of size greater than where can be taken to be arbitrarily small. It follows that for , every connected matching in contains a induced matching of size at least . Therefore, for we have that conditioned on the existence of a matching of size , . Indeed, as . As for it clearly holds that we have that in this case . This implies that approximating within a ratio smaller than in polynomial time would allow one to determine whether contains a clique of size or no clique of size . Taking such that concludes the proof.

5.2 Hardness results for computing for

We now turn to the problem of proving hardness of approximation results for for ; for certain values of , we show that is -hard to approximate to within any constant factor under randomized reduction. One approach to prove this is to use the reduction in Theorem 4.1. However, this approach does not seem to work, as it allows one to consider also matchings that contain “diagonal edges” of the form and it is not clear how to apply the analysis in Theorem 4.1 to such matchings. Instead, we build upon the hardness of the connected matching problem given in Theorem 3.1. We claim that the reduction in Theorem 3.1 shows that it is hard to approximate for . Note that in the YES-case, if , then . The NO-case is a bit subtle, and it is, a priori, not clear why implies that any matching of size at most contains a large induced matching. We resolve this problem using the following Ramsey-theoretic fact (see e.g., [BH92, ES35]).

Fact 5.3.

Let be an -vertex graph not containing a clique of size and suppose . Then contains an independent set of size at least .

Coupled with Theorem 4.1 we prove the following result.

Theorem 5.4.

For any constants and , it is -hard (under randomized reduction) to approximate within a factor of on bipartite graphs with vertices on each side for .

Proof.

By Theorem 3.1 given a bipartite graph it is -hard to distinguish between the case where , and the case where for .

For the YES-case if , then clearly for .

In the NO-case suppose that , and consider an arbitrary matching of size with . If then clearly contains an induced matching of size at least . Otherwise, contract all edges in . Denote by the subgraph induced by the contracted nodes. Observe that a subset of nodes in forms a clique if and only if their corresponding edges in form a connected matching. Otherwise, by the assumption that we get that contains no clique of size . Hence, by creftype 5.3 we conclude that contains an independent set of size at least (assuming is sufficiently large).

Therefore, given a bipartite graph with vertices on each side, and it is -hard to distinguish between the YES-case of , and the NO-case of . This concludes the proof. ∎

We can achieve stronger hardness results under stronger assumptions than -hardness. Recall that the Exponential Time Hypothesis (ETH) postulates that no algorithm of running time can decide whether an -variable SAT formula has a satisfying assignment. Assuming ETH we have the following hardness result:

Theorem 5.5.

Assuming ETH there exists a such that given with there is no polynomial time algorithm that approximates within a factor of where is a universal constant independent of .

We will rely on the following simple lower bound on independent sets in graphs of average degree due to Turan.

Lemma 5.6.

Every -vertex graph with average degree contains an independent set of size at least .

Proof of Theorem 5.5.

It is known [Man17] that assuming ETH for there is no polynomial algorithm that distinguishes between the case where contains a bipartite clique with vertices on each side (YES-case) to the case where every subgraph contained in with vertices satisfies (NO-case). In the first case . In the second case, given a matching with and we claim that contains an induced matching of size . The claim is trivially true if