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Sparsification and subexponential approximation

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Abstract

Instance sparsification is well-known in the world of exact computation since it is very closely linked to the Exponential Time Hypothesis. In this paper, we extend the concept of sparsification in order to capture subexponential time approximation. We develop a new tool for inapproximability, called approximation preserving sparsification, and use it in order to get strong inapproximability results in subexponential time for several fundamental optimization problems such as min dominating set , min feedback vertex set , min set cover, min feedback arc set, and others.

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Notes

  1. All the problems discussed in this paper are defined in the appendix.

  2. These reductions rely on the fact that, in graphs without isolated vertices, a vertex cover is both a dominating set and a feedback vertex set.

  3. Two vertices are twins if they have the same neighborhood.

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Acknowledgments

The numerous very pertinent comments and suggestions of two anonymous reviewers are gratefully acknowledged.

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Authors and Affiliations

  1. Institute for Computer Science and Control, Hungarian Academy of Sciences (MTA SZTAKI), Budapest, Hungary

    Édouard Bonnet

  2. CNRS UMR 7243, LAMSADE, PSL Research University, Université Paris-Dauphine, 75016, Paris, France

    Vangelis Th. Paschos

Authors
  1. Édouard Bonnet
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  2. Vangelis Th. Paschos
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Corresponding author

Correspondence to Édouard Bonnet.

Appendix: Definitions of the problems discussed in the paper

Appendix: Definitions of the problems discussed in the paper

  • max independent set . Given a graph G(V, E), determine a maximum cardinality set \(V' \subseteq V\), such that any two vertices of \(V'\) are not adjacent in G.

  • min vertex cover . Given a graph G(V, E), determine a minimum cardinality set \(V' \subseteq V\), such that any edge in E has at least one of its endpoints in \(V'\).

  • min dominating set . Given a graph G(V, E), determine a minimum cardinality set \(V' \subseteq V\) such that every vertex \(v \in V {\setminus } V'\) is neighbor of some vertex in \(V'\).

  • min independent dominating set . Given a graph G(V, E), determine a minimum cardinality set \(V' \subseteq V\) that is simultaneously an independent and a dominating set.

  • min feedback vertex set . Given a graph G(V, E), determine a minimum cardinality set \(V' \subseteq V\), such that any cycle of G has at least one vertex in \(V'\).

  • max complete bipartite subgraph . Given a graph G(V, E), determine a maximum cardinality set \(V' \subseteq V\) that induces a complete bipartite graph.

  • max \(\ell \)-colorable induced subgraph . Given a graph G(V, E) and some fixed constant \(\ell \), determine a maximum cardinality set \(V' \subseteq V\) that induces an \(\ell \)-colorable graph.

  • max planar induced subgraph. Given a graph G(V, E), determine a maximum cardinality set \(V' \subseteq V\) that induces a planar graph.

  • min set cover. Given a system \(\mathcal {S}\) of subsets of a ground set C, determine a minimum cardinality subsystem \(\mathcal {S}'\) that covers C.

  • min hitting set. Given a system \(\mathcal {S}\) of subsets of a ground set C, determine a minimum cardinality subset \(C' \subseteq C\) that hits all the sets of \(\mathcal {S}'\).

  • max minimal vertex cover . Given a graph G(V, E), determine a maximum cardinality set \(V' \subseteq V\), that is a minimal (for exclusion) vertex cover of G.

  • min feedback arc set. Given a directed graph G(V, E), determine a minimum cardinality set \(E' \subseteq E\), such that any cycle of G has at least one edge in \(E'\).

  • min sat . Given a CNF-formula, find the assignment of variables that satisfies the smallest number of clauses.

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Bonnet, É., Paschos, V.T. Sparsification and subexponential approximation. Acta Informatica 55, 1–15 (2018). https://doi.org/10.1007/s00236-016-0281-2

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