The Dense k -Subgraph Problem

Feige, Uriel; Kortsarz, Guy; Peleg, David

doi:10.1007/s004530010050

Cited by 487 publications

(430 citation statements)

References 8 publications

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“…Given a graph G and a parameter k, the densest k-subgraph problem is to find a set of k vertices with maximum number of induced edges. The densest k-subgraph problem is well-studied in the literature [18,19]. The best known approximation factor for the densest k-subgraph problem is O(min{n δ , n/k})) for some δ < 1/3 and improvement seems to be hard [20,19].…”

Section: Hardness Results For Mkcspmentioning

confidence: 99%

Minimum Multicolored Subgraph Problem in Multiplex PCR Primer Set Selection and Population Haplotyping

Hajiaghayi¹,

Jain

Lau

et al. 2006

Computational Science – ICCS 2006

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Abstract. In this paper we consider the minimum weight multicolored subgraph problem (MWMCSP), which is a common generalization of minimum cost multiplex PCR primer set selection and maximum likelihood population haplotyping. In this problem one is given an undirected graph G with non-negative vertex weights and a color function that assigns to each edge one or more of n given colors, and the goal is to find a minimum weight set of vertices inducing edges of all n colors. We obtain improved approximation algorithms and hardness results for MWMCSP and its variant in which the goal is to find a minimum number of vertices inducing edges of at least k colors for a given integer k ≤ n.

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Section: Hardness Results For Mkcspmentioning

confidence: 99%

Minimum Multicolored Subgraph Problem in Multiplex PCR Primer Set Selection and Population Haplotyping

Hajiaghayi¹,

Jain

Lau

et al. 2006

Computational Science – ICCS 2006

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“…It can be shown that MaWISh is NP-complete by reduction from maximum-clique, by assigning unit weight to edges and −∞ to nonedges. This problem is closely related to the maximum edge subgraph (Feige et al, 2001) and maximum dispersion problems (Hassin et al, 1997), which are also NP-complete. However, the positive weight restriction on these problems limits the application of existing algorithms to the maximum weight induced subgraph problem.…”

Section: Alignment Graph and The Maximum-weight Induced Subgraph Problemmentioning

confidence: 99%

Pairwise Alignment of Protein Interaction Networks

Koyutürk

Kim

Topkara

et al. 2006

Journal of Computational Biology

243

189

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With an ever-increasing amount of available data on protein-protein interaction (PPI) networks and research revealing that these networks evolve at a modular level, discovery of conserved patterns in these networks becomes an important problem. Although available data on protein-protein interactions is currently limited, recently developed algorithms have been shown to convey novel biological insights through employment of elegant mathematical models. The main challenge in aligning PPI networks is to define a graph theoretical measure of similarity between graph structures that captures underlying biological phenomena accurately. In this respect, modeling of conservation and divergence of interactions, as well as the interpretation of resulting alignments, are important design parameters. In this paper, we develop a framework for comprehensive alignment of PPI networks, which is inspired by duplication/divergence models that focus on understanding the evolution of protein interactions. We propose a mathematical model that extends the concepts of match, mismatch, and gap in sequence alignment to that of match, mismatch, and duplication in network alignment and evaluates similarity between graph structures through a scoring function that accounts for evolutionary events. By relying on evolutionary models, the proposed framework facilitates interpretation of resulting alignments in terms of not only conservation but also divergence of modularity in PPI networks. Furthermore, as in the case of sequence alignment, our model allows flexibility in adjusting parameters to quantify underlying evolutionary relationships. Based on the proposed model, we formulate PPI network alignment as an optimization problem and present fast algorithms to solve this problem. Detailed experimental results from an implementation of the proposed framework show that our algorithm is able to discover conserved interaction patterns very effectively, in terms of both accuracies and computational cost.

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“…There are many different definitions of what a dense subgraph is [11,17] and for almost all of these formulations, the corresponding computational problems are NP-hard.…”

Section: Introductionmentioning

confidence: 99%

“…For fixed k, maximizing the number of edges is the same as maximizing the density of a graph G = (V, E) which is defined as 2|E|/(|V |(|V | − 1)). Using the notion of density, the NP-hard Densest-k-Subgraph problem [11,15] can be defined as follows.…”

Section: Introductionmentioning

confidence: 99%

Finding Dense Subgraphs of Sparse Graphs

Komusiewicz

Sorge

2012

Parameterized and Exact Computation

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Abstract. We investigate the computational complexity of the Densestk-Subgraph (DkS) problem, where the input is an undirected graph G = (V, E) and one wants to find a subgraph on exactly k vertices with a maximum number of edges. We extend previous work on DkS by studying its parameterized complexity. On the positive side, we show that, when fixing some constant minimum density µ of the sought subgraph, DkS becomes fixed-parameter tractable with respect to either of the parameters maximum degree and h-index of G. Furthermore, we obtain a fixedparameter algorithm for DkS with respect to the combined parameter "degeneracy of G and |V | − k". On the negative side, we find that DkS is W[1]-hard with respect to the combined parameter "solution size k and degeneracy of G". We furthermore strengthen a previous hardness result for DkS [Cai, Comput. J., 2008] by showing that for every fixed µ, 0 < µ < 1, the problem of deciding whether G contains a subgraph of density at least µ is W[1]-hard with respect to the parameter |V | − k.

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The Dense k -Subgraph Problem

Cited by 487 publications

References 8 publications

Minimum Multicolored Subgraph Problem in Multiplex PCR Primer Set Selection and Population Haplotyping

Minimum Multicolored Subgraph Problem in Multiplex PCR Primer Set Selection and Population Haplotyping

Pairwise Alignment of Protein Interaction Networks

Finding Dense Subgraphs of Sparse Graphs

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