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The parameterized complexity of kBiclique
 In Proc. 26th SODA
, 2014
"... Given a graph G and a parameter k, the kBiclique problem asks whether G contains a complete bipartite subgraph Kk,k. This is one of the most easily stated problems on graphs whose parameterized complexity has been long unknown. We prove that kBiclique is W[1]hard by giving an fptreduction from k ..."
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Given a graph G and a parameter k, the kBiclique problem asks whether G contains a complete bipartite subgraph Kk,k. This is one of the most easily stated problems on graphs whose parameterized complexity has been long unknown. We prove that kBiclique is W[1]hard by giving an fptreduction from kClique to kBiclique, 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 n, s and t, we consider a bipartite graph G = (A ∪ ̇ B,E) such that A can be partitioned into A = V1 ∪ ̇ V2 ∪̇, · · · , ∪ ̇ Vn and for every s distinct indices i1, · · · , is, there exist vi1 ∈ Vi1, · · · , vis ∈ Vis such that vi1, · · · , vis have at least t+ 1 common neighbors in B; on the other hand, every s+1 distinct vertices in A have at most t common neighbors in B. We prove that given such threshold bipartite graphs, we can construct an fptreduction from kClique to kBiclique. Using the Paleytype graphs and Weil’s character sum theorem, we show that for t = (s+1)! and n large enough, such threshold bipartite graphs can be computed in polynomial time. One corollary of our reduction is that there is no f(k) ·no(k) time algorithm to decide whether a graph contains a subgraph isomorphic to Kk!,k! unless the Exponential Time Hypothesis (ETH) fails. We also provide a probabilistic construction with better parameters t = Θ(s2), which indicates that kBiclique has no f(k) · no( k)time algorithm unless 3SAT with m clauses can be solved in 2o(m)time with high probability. Besides the lower bound for exact computation of kBiclique, our result also implies a dichotomy classification of the parameterized complexity of cardinality constraint satisfaction problems and the inapproximability of the maximum kintersection problem.
New exact algorithms for the 2constraint satisfaction problem
"... Many optimization problems can be phrased in terms of constraint satisfaction. In particular MAX2SAT and MAX2CSP are known to generalize many hard combinatorial problems on graphs. Algorithms solving the problem exactly have been designed but the running time is improved over trivial bruteforce ..."
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Many optimization problems can be phrased in terms of constraint satisfaction. In particular MAX2SAT and MAX2CSP are known to generalize many hard combinatorial problems on graphs. Algorithms solving the problem exactly have been designed but the running time is improved over trivial bruteforce solutions only for very sparse instances. Despite many efforts, the only known algorithm [29] solving MAX2CSP over n variables in less than O ∗ (2 n) steps uses exponential space. Several authors have designed algorithms with running time O ∗ (2 nf(d) ) where f: R + → (0, 1) is a slowly growing function and d is the average variable degree of the input formula. The current best known algorithm for MAX2CSP [26] runs in time O ∗ n(1 − 2 (2 d+1) ) and polynomial space. In this paper we continue this line of research and design new algorithms for the MAX2SAT and MAX2CSP problems. First, we present a general technique for obtaining new bounds on the running time of a simple algorithm for MAX2CSP analyzed with respect to the number of vertices from algorithms that are analyzed with respect to the number of constraints. The best known bound for the problem is improved to O ∗ n(1 − 3 (2 d+1) ) for d ≥ 3. We further improve the bound for MAX2SAT, in particular for d ≥ 6 we n(1 − 3.677 achieve O ∗ (2 d+1)). As a second result we present an algorithm with asymptotically better running time for the case when the input instance is not very sparse. Building on recent work of Feige and Kogan we derive an upper bound on the size of a vertex separator for graphs in terms of the average degree of the graph. We then design a simple algorithm solving MAX2CSP in time O ∗ (2 cdn 2α ln d), cd = 1 − for some α < 1 and d d = o(n).