Improved upper bounds for vertex cover

Chen, Jianer; Kanj, Iyad A.; Xia, Ge

doi:10.1016/j.tcs.2010.06.026

Cited by 352 publications

(252 citation statements)

References 16 publications

Supporting

Mentioning

249

Contrasting

Order By: Relevance

“…However, there is a gap between the theoretically fastest algorithms (i.e., those currently having the best time complexity) and the empirically fastest algorithms (i.e., those currently with the best running time for popular benchmark instances). In the theoretical research on exponential complexity or parameterized complexity of branching algorithms, branch-and-reduce methods, which involve a plethora of branching and reduction rules without using any lower bounds, currently have the best time complexity for a number of important problems, such as Independent Set (or, equivalently, Vertex Cover) [22,4], Dominating Set [10], and Directed Feedback Vertex Set [16]. On the other hand, in practice, branch-and-bound methods that involve problemspecific lower bounds or LP-based branch-and-cut methods, which generate new cuts to improve the lower bounds, are often used.…”

Section: Introductionmentioning

confidence: 99%

Branch-and-reduce exponential/FPT algorithms in practice: A case study of vertex cover

Akiba

Iwata

2016

Theoretical Computer Science

View full text Add to dashboard Cite

We investigate the gap between theory and practice for exact branching algorithms. In theory, branch-and-reduce algorithms currently have the best time complexity for numerous important problems. On the other hand, in practice, state-of-the-art methods are based on different approaches, and the empirical efficiency of such theoretical algorithms have seldom been investigated probably because they are seemingly inefficient because of the plethora of complex reduction rules. In this paper, we design a branch-and-reduce algorithm for the vertex cover problem using the techniques developed for theoretical algorithms and compare its practical performance with other state-of-the-art empirical methods. The results indicate that branch-and-reduce algorithms are actually quite practical and competitive with other state-ofthe-art approaches for several kinds of instances, thus showing the practical impact of theoretical research on branching algorithms.

show abstract

Section: Introductionmentioning

confidence: 99%

Branch-and-reduce exponential/FPT algorithms in practice: A case study of vertex cover

Akiba

Iwata

2016

Theoretical Computer Science

View full text Add to dashboard Cite

show abstract

“…Fix some highly connected subgraph G[S] of order k if it exists. First compute a minimum vertex cover C for G in O(1.274 τ + τ n) time [6]. Enumerate all 2 τ possibilities for C := C ∩ S. Clearly, one branch contains the correct C .…”

Section: Set Multicovermentioning

confidence: 99%

Finding Highly Connected Subgraphs

Hüffner

Komusiewicz

Sorge

2015

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Abstract. A popular way of formalizing clusters in networks are highly connected subgraphs, that is, subgraphs of k vertices that have edge connectivity larger than k/2 (equivalently, minimum degree larger than k/2). We examine the computational complexity of finding highly connected subgraphs. We show that the problem is NP-hard. Thus, we explore possible parameterizations, such as the solution size, number of vertices in the input, the size of a vertex cover in the input, and the number of edges outgoing from the solution (edge isolation). For some parameters, we find strong intractability results; among the parameters yielding tractability, the edge isolation seems to provide the best trade-off between running time bounds and a small value of the parameter in relevant instances.

show abstract

“…The best known algorithm for V C k is due to Chen, Kanj, and Jia [15] and runs in time O(1.2738 k + kn).…”

Section: ) For P = Npmentioning

confidence: 99%

Classifying Problems into Complexity Classes

Gasarch

2014

Advances in Computers

View full text Add to dashboard Cite

A fundamental problem in computer science is, stated informally: Given a problem, how hard is it?. We measure hardness by looking at the following question: Given a set A whats is the fastest algorithm to determine if "x ∈ A?" We measure the speed of an algorithm by how long it takes to run on inputs of length n, as a function of n. For example, sorting a list of length n can be done in roughly n log n steps.Obtaining a fast algorithm is only half of the problem. Can you prove that there is no better algorithm? This is notoriously difficult; however, we can classify problems into complexity classes where those in the same class are of roughly the same complexity.In this chapter we define many complexity classes and describe natural problems that are in them. Our classes go all the way from regular languages to various shades of undecidable. We then summarize all that is known about these classes.

show abstract

Improved upper bounds for vertex cover

Cited by 352 publications

References 16 publications

Branch-and-reduce exponential/FPT algorithms in practice: A case study of vertex cover

Branch-and-reduce exponential/FPT algorithms in practice: A case study of vertex cover

Finding Highly Connected Subgraphs

Classifying Problems into Complexity Classes

Contact Info

Product

Resources

About