Exact Algorithms for Maximum Independent Set

Xiao, Mingyu; Nagamochi, Hiroshi

doi:10.1007/978-3-642-45030-3_31

Cited by 39 publications

(30 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The complexity is reduced to O(2 0.276n ) by Robson [20] with a modified recursive algorithm. This result is recently improved by Xiao et al [26] to O (1.2002 n · n O (1) ). These exact methods are applicable to problem instances of very limited sizes.…”

Section: Related Workmentioning

confidence: 76%

“…The best approximation ratio known for the MIS problem is O(n(log log n) 2 /(log n) 3 ) [10]. In addition, there are several studies on exponential-time exact algorithms: Robson [20] solves the problem in time O(2 0.276n ) using exponential space, which is recently improved to O (1.2002 n · n O (1) ) by Xiao [26]. All the above algorithms require memory space linear in the size of the input graph.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Towards maximum independent sets on massive graphs

Liu

Hua

et al. 2015

Proc. VLDB Endow.

View full text Add to dashboard Cite

Maximum independent set (MIS) is a fundamental problem in graph theory and it has important applications in many areas such as social network analysis, graphical information systems and coding theory. The problem is NP-hard, and there has been numerous studies on its approximate solutions. While successful to a certain degree, the existing methods require memory space at least linear in the size of the input graph. This has become a serious concern in view of the massive volume of today's fast-growing graphs.In this paper, we study the MIS problem under the semi-external setting, which assumes that the main memory can accommodate all vertices of the graph but not all edges. We present a greedy algorithm and a general vertex-swap framework, which swaps vertices to incrementally increase the size of independent sets. Our solutions require only few sequential scans of graphs on the disk file, thus enabling in-memory computation without costly random disk accesses. Experiments on large-scale datasets show that our solutions are able to compute a large independent set for a massive graph with 59 million vertices and 151 million edges using a commodity machine, with a memory cost of 469MB and a time cost of three minutes, while yielding an approximation ratio that is around 99% of the theoretical optimum.

show abstract

Section: Related Workmentioning

confidence: 76%

Section: Introductionmentioning

confidence: 99%

Towards maximum independent sets on massive graphs

Liu

Hua

et al. 2015

Proc. VLDB Endow.

View full text Add to dashboard Cite

show abstract

“…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

“…Though these works apply reduction techniques as a preprocessing step, many works apply reductions as a natural step of the algorithm. Reductions were originally used by Tarjan and Trojanowski [41] to reduce the running time of the brute force O(n 2 2 n ) algorithm to time O(2 n/3 ), and reductions are further used to give the fastest known polynomial space algorithm with running time of O * (1.1996 n ) by Xiao and Nagamochi [46]. These algorithms apply reductions during recursion, only branching when the graph can no longer be reduced [19]-known as the branch-and-reduce method.…”

Section: Related Workmentioning

confidence: 99%

Scalable Kernelization for Maximum Independent Sets

Hespe¹,

Schulz²,

Strash³

2018

2018 Proceedings of the Twentieth Workshop on Algorithm Engineering and Experiments (ALENEX)

View full text Add to dashboard Cite

The most efficient algorithms for finding maximum independent sets in both theory and practice use reduction rules to obtain a much smaller problem instance called a kernel. The kernel can then be solved quickly using exact or heuristic algorithms-or by repeatedly kernelizing recursively in the branch-and-reduce paradigm. It is of critical importance for these algorithms that kernelization is fast and returns a small kernel. Current algorithms are either slow but produce a small kernel, or fast and give a large kernel. We attempt to accomplish both of these goals simultaneously, by giving an efficient parallel kernelization algorithm based on graph partitioning and parallel bipartite maximum matching.We combine our parallelization techniques with two techniques to accelerate kernelization further: dependency checking that prunes reductions that cannot be applied, and reduction tracking that allows us to stop kernelization when reductions become less fruitful. Our algorithm produces kernels that are orders of magnitude smaller than the fastest kernelization methods, while having a similar execution time. Furthermore, our algorithm is able to compute kernels with size comparable to the smallest known kernels, but up to two orders of magnitude faster than previously possible. Finally, we show that our kernelization algorithm can be used to accelerate existing state-of-the-art heuristic algorithms, allowing us to find larger independent sets faster on large real-world networks and synthetic instances.

show abstract

Exact Algorithms for Maximum Independent Set

Cited by 39 publications

References 14 publications

Towards maximum independent sets on massive graphs

Towards maximum independent sets on massive graphs

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

Scalable Kernelization for Maximum Independent Sets

Contact Info

Product

Resources

About