Experimental Evaluation of Approximation and Heuristic Algorithms for Maximum Distance-Bounded Subgraph Problems

In this paper, we consider two distance-based relaxed variants of the maximum clique problem (Max Clique), named Maxd-Clique and Maxd-Club for positive integers d. Max 1-Clique and Max 1-Club cannot be efficiently approximated within a factor of n1−ε for any real ε>0 unless P=NP , since they are identical to Max Clique (Håstad in Acta Math 182(1):105–142, 1999; Zuckerman in Theory Comput 3:103–128, 2007). In addition, it is NP -hard to approximate Maxd-Clique and Maxd-Club to within a factor of n1/2−ε for any fixed integer d≥2 and any real ε>0 (Asahiro et al. in Approximating maximum diameter-bounded subgraphs. In: Proc of LATIN 2010, Springer, pp 615–626, 2010; Asahiro et al. in Optimal approximation algorithms for maximum distance-bounded subgraph problems. In: Proc of COCOA, Springer, pp 586–600, 2015). As for approximability of Maxd-Clique and Maxd-Club, a polynomial-time algorithm, called ReFindStar d, that achieves an optimal approximation ratio of O(n1/2) for Maxd-Clique and Maxd-Club was designed for any integer d≥2 in Asahiro et al. (2015, Algorithmica 80(6):1834–1856, 2018). Moreover, a simpler algorithm, called ByFindStar d, was proposed and it was shown in Asahiro et al. (2010, 2018) that although the approximation ratio of ByFindStar d is much worse for any odd d≥3, its time complexity is better than ReFindStar d. In this paper, we implement those approximation algorithms and evaluate their quality empirically for random graphs. The experimental results show that (1) ReFindStar d can find larger d-clubs (d-cliques) than ByFindStar d for odd d, (2) the size of d-clubs (d-cliques) output by ByFindStar d is the same as ones by ReFindStar d for even d, and (3) ByFindStar d can find the same size of d-clubs (d-cliques) much faster than ReFindStar d. Furthermore, we propose and implement two new heuristics, Hclub d for Maxd-Club and Hclique d for Maxd-Clique. Then, we present the experimental evaluation of the solution size of ReFindStar d, Hclub d, Hclique d and previously known heuristic algorithms for random graphs and Erdős collaboration graphs

On combinatorial optimisation in analysis of protein-protein interaction and protein folding networks

Abstract: Protein-protein interaction networks and protein folding networks represent prominent research topics at the intersection of bioinformatics and network science. In this paper, we present a study of these networks from combinatorial optimisation point of view. Using a combination of classical heuristics and stochastic optimisation techniques, we were able to identify several interesting combinatorial properties of biological networks of the COSIN project. We obtained optimal or near-optimal solutions to maximum clique and chromatic number problems for these networks. We also explore patterns of both non-overlapping and overlapping cliques in these networks. Optimal or near-optimal solutions to partitioning of these networks into non-overlapping cliques and to maximum independent set problem were discovered. Maximal cliques are explored by enumerative techniques. Domination in these networks is briefly studied, too. Applications and extensions of our findings are discussed

Approximate Closest Community Search in Networks

Recently, there has been significant interest in the study of the community search problem in social and information networks: given one or more query nodes, find densely connected communities containing the query nodes. However, most existing studies do not address the "free rider" issue, that is, nodes far away from query nodes and irrelevant to them are included in the detected community. Some state-of-the-art models have attempted to address this issue, but not only are their formulated problems NP-hard, they do not admit any approximations without restrictive assumptions, which may not always hold in practice. In this paper, given an undirected graph G and a set of query nodes Q, we study community search using the k-truss based community model. We formulate our problem of finding a closest truss community (CTC), as finding a connected k-truss subgraph with the largest k that contains Q, and has the minimum diameter among such subgraphs. We prove this problem is NP-hard. Furthermore, it is NP-hard to approximate the problem within a factor $(2-\varepsilon)$, for any $\varepsilon >0$. However, we develop a greedy algorithmic framework, which first finds a CTC containing Q, and then iteratively removes the furthest nodes from Q, from the graph. The method achieves 2-approximation to the optimal solution. To further improve the efficiency, we make use of a compact truss index and develop efficient algorithms for k-truss identification and maintenance as nodes get eliminated. In addition, using bulk deletion optimization and local exploration strategies, we propose two more efficient algorithms. One of them trades some approximation quality for efficiency while the other is a very efficient heuristic. Extensive experiments on 6 real-world networks show the effectiveness and efficiency of our community model and search algorithms