The Parameterized Complexity of k-Biclique

The Parameterized Complexity of -Biclique

Bingkai Lin Department of Computer Science, The University of Tokyo. ERATO Kawarabayashi Large Graph Project.
Abstract

Given a graph and a parameter , the -Biclique problem asks whether contains a complete bipartite subgraph . This is one of the most easily stated problems on graphs whose parameterized complexity has been long unknown. We prove that -Biclique is -hard by giving an fpt-reduction from -Clique to -Biclique, thus solving this longstanding open problem.

Our reduction uses a class of bipartite graphs with a certain threshold property, which might be of some independent interest. More precisely, for positive integers , and , we consider a bipartite graph such that can be partitioned into and for every distinct indices , there exist such that have at least common neighbors in ; on the other hand, every distinct vertices in have at most common neighbors in .

We prove that given such threshold bipartite graphs, we can construct an fpt-reduction from -Clique to -Biclique. Using the Paley-type graphs and Weil’s character sum theorem, we show that for and large enough, such threshold bipartite graphs can be computed in polynomial time. One corollary of our reduction is that there is no time algorithm to decide whether a graph contains a subgraph isomorphic to unless the Exponential Time Hypothesis () fails. We also provide a probabilistic construction with better parameters , which indicates that -Biclique has no -time algorithm unless with clauses can be solved in -time with high probability. Besides the lower bound for exact computation of -Biclique, our result also implies a dichotomy classification of the parameterized complexity of cardinality constraint satisfaction problems and the inapproximability of the maximum -intersection problem.

1 Introduction

The Subgraph Isomorphism is a basic problem in algorithms and graph theory. Due to its generality, we do not expect it to have a polynomial time algorithm. However, this does not rule out the possibility that there exist efficient algorithms to solve this problem on some special class of graphs. For example, it is well known that whether is a subgraph of can be decided in time using the color-coding technique in [2], where denotes the tree-width of and is a computable function. Hence, if is a class of graphs with tree-width bounded by some constant, the subgraph isomorphism problem with is fixed parameter tractable, and this is believed to be optimal. In [16], Martin Grohe conjectured that the subgraph embedding problem with is -hard if and only if has unbounded tree-width. Under the assumption of , this would imply that there is no -time algorithm to decide whether contains a subgraph isomorphic to , because the class of balanced complete bipartite graphs has unbounded tree-width. In other words, we can not prove Grohe’s conjecture without answering the parameterized complexity of -Biclique. Although -Biclique is believed to be -hard, despite many attempts[6, 10, 15, 21], no -reduction from -Clique to -Biclique has previously been found. Let us not fail to mention that a polynomial reduction is given in [19], however, since such reduction requires the size of the clique instance to be , it is not an fpt-reduction.

A possible line of attack is to consider the Partitioned Subgraph Isomorphism problem, in which each vertex of the smaller graph has a distinct color and the vertices of are partitioned into subsets, each set is corresponding to one color. The problem is to find an injective mapping from to such that: (1) for all , and have the same color; (2) if and are adjacent in , then and are adjacent in . It is not hard to see that Partitioned Subgraph Isomorphism problem on the graph class C is -hard if C has unbounded tree-width[16]. An interesting fact is that if the graph has no homomorphism to any of its proper induced subgraphs, then the colored and uncolored version of Subgraph Isomorphism of are equivalent[22]. Unfortunately, this approach does not work for -Biclique because any bipartite graph has a homomorphism to any of its edges.

Therefore, resolving the complexity of -Biclique would significantly improve our understanding of the Subgraph Isomorphism problem. In addition, -Biclique also has connections with the cardinality constraints satisfaction problem. Bulatov and Marx obtained a trichotomy classification of the parameterized complexity of the constraint satisfaction problem with cardinality constraints(CCSP) in [8]. They showed that for any set of relations closed under substitution of constants, CCSP with the relations restricted in (denoted as ) is fixed parameterized tractable, Biclique-hard or -hard. By the well known dichotomy conjecture of Feder and Vardi, it is reasonable to believe that is either or -hard. Thus giving further incentive for the study of -Biclique.

We remark that the parameterized complexity of -biclique has received heavy attention from the parameterized complexity community[4, 8, 14, 16, 17]. It is the first problem on the “most infamous” list(page 677) in a new text book[11] by Downey and Fellows. “Almost everyone considers that this problem should obviously be -hard, and… it is rather an embarrassment to the field that the question remains open after all these years!”

In the rest of this section, we state our main results with some further applications and corollaries.

1.1 Our Results

{theorem}

For any -vertex graph and positive integer with , we can compute a graph in -time such that contains a if and only if contains a , where .

Corollary 1.1

-Biclique is -hard.

{theorem}

For any -vertex graph and positive integer with and , we can compute a random graph in -time such that, with probability at least , contains a if and only if contains a .

The core of our reduction is the construction of a bipartite graph with a -threshold property: every distinct vertices in have at most common neighbors in ; while there exist many sets of distinct vertices in having at least common neighbors in , where . An explicit construction of similar threshold bipartite graphs has been given in [5], in which they show that a certain fraction of distinct vertices in have this property(see Lemma 3.7 of [5]). Our contribution is proving that we can partition into several sets and guarantee that for any distinct sets, it is possible to choose one vertex from each set, the resulting vertices satisfying the property.

1.2 Lower Bound for Computing k-Biclique

One corollary of our main results is the lower bound for exact computation of -Biclique under the well-known -conjecture made by Impagliazzo, Paturi and Zane [18]:

Conjecture 1.2 (Exponential Time Hypothesis)

cannot be solved in time , where is the number of clauses in the input formula.

The result in [9] implies that for any instance of with clauses, we can construct an instance of -Clique in -time such that is an yes-instance of if and only if contains a -Clique. If the -Clique problem has -time algorithm, we can solve such instance in -time. That is: Assuming , -Clique problem has no -time algorithm for any computable function . With Theorem 1.1, we have the following lower bound: Assuming , there is no -time algorithm to decide whether a given graph with order contains a subgraph isomorphic to .

An interesting question is to find a linear fpt-reduction from -Clique to -Biclique, that is given and , computing a new graph in -time such that if and only if , where for some constant . The existence of such reduction would imply that -Biclique has no -time algorithm under the . However, since our reduction causes a quadratic blow-up of the size of solution, is the best we may achieve. If we assume a stronger version of , then Theorem 1.1 yields a better lower bound for -Biclique:

Corollary 1.3

Unless -clause can be solved in -time with high probability, there is no algorithm for any computable function to decide whether a given graph with order contains a subgraph isomorphic to .

1.3 Maximum k-Intersection Problem

In our reduction from -Clique to -Biclique, we actually prove that {theorem} For an -vertex graph and a positive integer with , let be the minimum integer such that and , let , we can compute a bipartite graph in -time such that:

  1. if , then there are vertices in with at least common neighbors in ;

  2. if , then every vertices in have at most common neighbors in .

This gap allows us to deduce an inapproximation result for the Maximum k-Intersection Problem:

Maximum -Intersection Problem Input: A family of sets with and a number . Parameter: . Problem: Find sets with maximum

It is not hard to see that, our reduction implies

Corollary 1.4

Assuming , there is no -time algorithm approximating Maximum -Intersection Problem with -approximation ratio for .

The polynomial time inapproximability of Maxinum -Intersection has been proved in [25] basing on the inapproximability of Maxinum Edge Biclique [3].

1.4 Cardinality Constraints Satisfaction Problem

Fix a domain , an instance of the constraint satisfaction problem(CSP) is a pair , where is a set of variables and is a set of constraints. Each constraint of can be written as , where is an -ary relation on for some positive integer and , an assignment satisfies a constraint if . The goal is to find an assignment satisfying all the constraints in . In the research of complexity of CSP, we usually fix a set of relation , and denote the CSP problem in which all the relations of the constraints are in .

It is well-known that many hard problems including satisfiability and graph coloring can be expressed under the CSP framework, hence solving constraint satisfaction problems is -hard. One way to cope with this -hard problem is to introduce a parameter and consider the parameterized version of such problem. In [8], Andrei A. Bulatov and Dániel Marx introduced two parameterized versions of CSP. More specifically, they assume that the domain contain a “free” value, say and other non-zero values, which are “expensive”. The goal is find an assignment with limited number of variables assigning expensive values. One way to reflect this goal is to take the number of nonzero values used in an assignment as parameter, which leads to the definition of the CSP with size constraints(OCSP); another more refined way is to prescribe how many variables have to be assigned each particular nonzero value, this leads to the definition of CSP with cardinality constraints. They provide a complete characterization of the fixed-parameter tractable cases of and show that all the remaining problems are -hard.

For CSP with cardinality constraints, the situation is strange. An simple observation shows that the -Biclique problem can be express as a CCSP instance. Without lose of generality, consider the -Biclique on bipartite graph, let , for any bipartite graph , we construct a CCSP instance with and , then we ask for an assignment with variables assigning and variables assigning . It is easy to check that for a bipartite graph , if the corresponding CCSP instance has such an assignment, then the bipartite complement of contains a . Therefore, without settling the parameterized complexity of -Biclique, they can only show that is fixed-parameter tractable, Biclique-hard or -hard. Combining our result and Theorem 1.2 in [8], we finally obtain a dichotomy theorem for the parameterized complexity of : {theorem} For every finite closed under substitution of constants, is either or -hard.


Organization of the Paper. The main idea of the reduction is presented in Section 3 after introducing the class of threshold bipartite graphs. To complete the reduction, we provide efficient constructions of the bipartite graph with threshold property. A probabilistic construction is given in Section 4, while the explicit construction can be found in Section 5. The explicit construction uses the Paley-type graph defined in [5] and a generalization of Lemma 3.8 in [5], whose proof is given in the Appendix. Finally, we discuss some interesting topics and open questions in Section 6.

2 Preliminaries

We use , and to denote the sets of nonnegative integers, positive integers and complex numbers respectively. For any number , let . For any real numbers , we use the notation to denote the numbers between and . For any prime power , is the Galois field with size , is the multiplicative group of . For every set we use to denote its size. Moreover, for any , we let be the set of all -element subsets of .

2.1 Parameterized Complexity

We denote the alphabet by and identify problems with subsets of . A parameterized problem is a pair consisting of a classical problem and a polynomial time computable parameterization . For example, the parameterized clique problem is defined in the form:

Input: A graph and a positive integer . Parameter: . Problem: Does contains a subgraph isomorphic to ?

An algorithm is an fpt-algorithm with respect to a parameterization if for every the running time of on is bounded by for a computable function . A parameterized problem is fixed-parameter tractable (or for short) if it has an fpt-algorithm.

Let and be two parameterized problems. An fpt-reduction from to is a mapping such that:

  1. For every we have if and only if .

  2. is computable by an fpt-algorithm with respect to ;

  3. There is a computable function such that for all .

A fpt-reduction is linear if . We write if there is an fpt-reduction from to ; if and . Suppose , it is easy to see that if is , then so is ; in particular, if , then it follows that is -hard (for the definition of -hardness, see [12, 14]). Obviously, if and is -hard, then so is .

2.2 Graphs

Every graph is determined by a nonempty vertex set and an edge set . Every nonempty subset induces a subgraph with the vertex set and the edge set . And is a clique in , if for every distinct we have . A clique with vertices is denoted as or -clique. A graph is bipartite if admits a partition into two classes such that every edge has its ends in different classes. A complete bipartite graph or biclique is a bipartite graph such that every two vertices from different partition classes are adjacent. We use the notation to denote the complete bipartite graph with vertices on one side and vertices on the other side. In the bipartite graph , for , let .

3 Reduction

We first define -Biclique, an imbalanced version of Biclique. Then we prove that -Biclique and -Biclique are equivalent under linear fpt-reductions. Hence, to prove Theorem 1.1, we only need to prove Theorem 1.3. To this end, we introduce the threshold graphs. Theorem 1.3 then follows by the reduction in Lemma 3.3 and the efficient construction of threshold graphs given in Lemma 3.4. Also, Theorem 1.1 follows in analogy with Theorem 1.3, but calling on Lemma 4.4, a probabilistic analog to Lemma 3.4. Lemma 3.5 and Lemma 4.4 are proved in Section 4 and 5.

-Biclique Input: A bipartite graph and two positive integers . Parameter: . Problem: Find a in with the left vertices in and the right vertices in .

Lemma 3.1

and the reductions of both directions are linear.

{proof}

We need to check two directions:

  1. : given a -Biclique instance , construct a bipartite graph , with and are two copies of and . It is routine to check that , so with is an instance of ;

  2. : suppose is an instance of -Biclique, where and . Construct a new bipartite graph by adding vertices into and connect all of these new vertices with vertices in . Then contains a iff contains a with vertices in and vertices in .

Definition 3.2 (-threshold property)

Suppose , a bipartite graph with a partition satisfy the -threshold property if:

  1. Every distinct vertices in have at most common neighbors in , i.e.

  2. For every distinct indices , there exist such that have at least common neighbors in , i.e.

Lemma 3.3 (reduction)

Given an -threshold bipartite graph . Let . For any n vertices graph , we can construct a new graph in -time, such that:

  1. if , then , ;

  2. if , then , .

{proof}

Suppose is a graph with vertices, our goal is to construct a bipartite graph satisfying (H1) and (H2).

Let , . We associate to each a vertex with the same index . Let be the function that for each , .

Then we construct the bipartite graph with:

  • ;

  • ;

  • .

We show that satisfies (H1) and (H2):

  1. If , let us say induces a in , then by (T2), there exists such that has at least common neighbors in , let and , we have and . Let , since for all distinct , we have , hence for all and , . So induces a complete bipartite subgraph in . It follows that satisfies (H1) because and ;

  2. If but , s.t. . Let , . We have and . Consider . By the definition of the edge set , in the graph , . Since and contains no , we have ; on the other hand, it is not hard to see that , hence implies . Thus and for any distinct , . It follows that induces a in , this is impossible.

By Lemma 3.3, to prove Theorem 1.3, we only need to compute the threshold bipartite graphs efficiently. Our main technical lemma is:

Lemma 3.4

For with for some and , a bipartite graph with the -threshold property can be computed in -time.

{proof}

[of Theorem 1.3] Given and , let be the minimum integer such that and , we have . Then we add a new clique with vertices into and connect them with every vertex in . It is easy to see that the new graph contains a -clique if and only if contains a -clique. Since , we have . Apply Lemma 3.4 on and , we obtain a -threshold bipartite graph. The result then follows from Lemma 3.3.

Theorem 1.1 can be easily deduced from Theorem 1.3 and Lemma 3.1. To prove Theorem 1.1, we show:

Lemma 3.5

For with , and , we can compute in -time a bipartite random graph satisfying the threshold property with probability at least .

4 Probabilistic Construction

The Erdős-Rényi random graph is constructed on vertices by joining every distinct pair of vertices independently with an edge with probability . An interesting property of these random graphs is that there is a parameter such that if a graph is balanced (i.e. every subgraph of has .), then for , contains a subgraph isomorphic to with high probability; and for , contains no subgraph isomorphic to with high probability (See [1] Chapter 4.4).

This suggests that we may construct the threshold bipartite graph defined in Section 3 using random graph. For and , define a bipartite random graph with and every pair of vertices and is joined by an edge with probability , randomly and independently. We will show that with high probability satisfies the -threshold property for some constant . To bound the probability of containing a subgraph , we need the following lemma, which is a simple consequence of Markov’s Inequality.:

Lemma 4.1

Let be a nonnegative integral valued random variable, then .

Let , the value of will be determined later. It follows that:

Lemma 4.2

.

{proof}

Let be the number of in , then

We have . Hence, when , , contains no with high probability.

Suppose are disjoint subsets of and for each , , where is a constant. Let be the number of in with the restriction that each contains exactly one vertex from the left side of such . It is easy to see that:

Let and , then . As goes large, . Of course, does not mean that . By the Chebyshev’s Inequality, is upper bounded by: {theorem}[Theorem 4.3.1 in [1]] . To show that is very close to zero, we need to prove that is . This can be easily deduced from the fact that is balanced(See [1] Chapter 4.4), however, since we want to upper bound the probability of does not satisfy (T2), we need to show a slightly stronger result saying that is .

Let , , where and . We can rewrite as , where is the indicator random variable for event . Denote for if and , are not independent. Let , then and it is not hard to see that if and only if , and (See the discussion in Chapter 4.3 of [1]). Then

We have

Lemma 4.3

For , .

{proof}

Now suppose are disjoint subsets of with , where . We know that each can be further partitioned into with and for all , . Let be the number of in such that each contains exactly one vertex from the left side of such and for , be the number of in such that for each , contains exactly one vertex from the left side of such . It is not hard to see that , and for any distinct , and are independent. It follows that:

Given a bipartite random graph , we partition into sets with . Then the probability that with such partition does not satisfy (T2) for parameter is bounded by

It follows that

So when , is an threshold bipartite graph with high probability. We have

Lemma 4.4

For any , and