Approximation Algorithms for the Feedback Vertex Set Problem with Applications to Constraint Satisfaction and Bayesian Inference

Bar-Yehuda, Reuven; Geiger, Dan; Naor, Joseph; Roth, Ron M.

doi:10.1137/s0097539796305109

Cited by 167 publications

(128 citation statements)

References 16 publications

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“…For any fixed d the distance to d-degenerate graphs is NP-hard to compute [42]. However, we can use existing linear-time constant-factor approximation algorithms for d = 0 and d = 1 [4]. Since a minimum feedback vertex set of a graph is always (and possibly much) smaller than its smallest vertex cover, it is natural to use this parameter rather than the vertex cover if comparably good results can be shown for both parameters.…”

Section: Parameters Lower-bounded By Degeneracymentioning

confidence: 99%

“…This is also known as the h-index of the current graph. 4 Complementing algorithmic results, the problems of finding and counting triangles are studied from a running time lower bounds perspective, where the lower bounds are based on popular conjectures like, for instance, the Strong Exponential-Time Hypothesis [33,32]. Accordingly, lower bounds are proven for finding (in a given edge-weighted graph) a triangle of negative weight [53] or counting (in a given vertex-colored graph) triangles with pairwise different colors on the vertices [3].…”

Section: Introductionmentioning

confidence: 99%

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Parameterized Aspects of Triangle Enumeration

Bentert

Fluschnik

Nichterlein

et al. 2017

Fundamentals of Computation Theory

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Listing all triangles in an undirected graph is a fundamental graph primitive with numerous applications. It is trivially solvable in time cubic in the number of vertices. It has seen a significant body of work contributing to both theoretical aspects (e.g., lower and upper bounds on running time, adaption to new computational models) as well as practical aspects (e.g. algorithms tuned for large graphs). Motivated by the fact that the worst-case running time is cubic, we perform a systematic parameterized complexity study of triangle enumeration, providing both positive results (new enumerative kernelizations, "subcubic" parameterized solving algorithms) as well as negative results (likely uselessness in terms of "faster" parameterized algorithms of certain parameters such as graph diameter).

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Section: Parameters Lower-bounded By Degeneracymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Parameterized Aspects of Triangle Enumeration

Bentert

Fluschnik

Nichterlein

et al. 2017

Fundamentals of Computation Theory

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“…Application to Feedback Problems: The integrality gap of the standard LP for feedback vertex set is shown to be O(log n) in [4] and O(log k) for the more general subset feedback vertex set in [13] (hence also for the feedback edge problems). The above lemma combined with the O(log k) integrality gap for the 1-route multicut problem [15] yields simple alternate proofs of these results.…”

Section: A Useful Lemmamentioning

confidence: 99%

Algorithms for 2-Route Cut Problems

Chekuri

Khanna

2008

Automata, Languages and Programming

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In this paper we study approximation algorithms for multi-route cut problems in undirected graphs. In these problems the goal is to find a minimum cost set of edges to be removed from a given graph such that the edge-connectivity (or node-connectivity) between certain pairs of nodes is reduced below a given threshold K. In the usual cut problems the edge connectivity is required to be reduced below 1 (i.e. disconnected). We consider the case of K = 2 and obtain poly-logarithmic approximation algorithms for fundamental cut problems including single-source, multiway-cut, multicut, and sparsest cut. These cut problems are dual to multi-route flows that are of interest in fault-tolerant networks flows. Our results show that the flow-cut gap between 2-route cuts and 2-route flows is poly-logarithmic in undirected graphs with arbitrary capacities. 2-route cuts are also closely related to well-studied feedback problems and we obtain results on some new variants. Multi-route cuts pose interesting algorithmic challenges. The new techniques developed here are of independent technical interest, and may have applications to other cut and partitioning problems.

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“…This problem is known as the Minimum Feedback-Vertex Set Problem [6], see [7] for a recent review. It is known to be NP-hard [8] and has applications in many diverse areas, including program verification [9] and Bayesian inference [10].…”

Section: Minimum Feedback-vertex Setsmentioning

confidence: 99%

The Footprint Sorting Problem

Fried¹,

Hordijk²,

Prohaska³

et al. 2004

J. Chem. Inf. Comput. Sci.

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Phylogenetic footprints are short pieces of non-coding DNA sequence in the vicinity of a gene that are conserved between evolutionary distant species. A seemingly simple problem is to sort footprints in their order along the genomes. It is complicated by the fact that not all footprints are co-linear: they may cross each other. The problem thus becomes the identification of the crossing footprints, the sorting of the remaining co-linear cliques, and finally the insertion of the non-colinear ones at "reasonable" positions. We show that solving the footprint sorting problem requires the solution of the "Minimum Weight Vertex Feedback Set Problem", which is known to be NP-complete and APX-hard. Nevertheless good approximations can be obtained for datasets of interest. The remaining steps of the sorting process are straight forward: computation of the transitive closure of an acyclic graph, linear extension of the resulting partial order, and finally sorting w.r.t. the linear extension. Alternatively, the footprint sorting problem can be rephrased as a combinatorial optimization problem for which approximate solutions can be obtained by means of general purpose heuristics. Footprint sortings obtained with different methods can be compared using a version of multiple sequence alignment that allows the identification of unambiguously ordered sub-lists.

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Approximation Algorithms for the Feedback Vertex Set Problem with Applications to Constraint Satisfaction and Bayesian Inference

Cited by 167 publications

References 16 publications

Parameterized Aspects of Triangle Enumeration

Parameterized Aspects of Triangle Enumeration

Algorithms for 2-Route Cut Problems

The Footprint Sorting Problem

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