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Twin-width III: Max Independent Set, Min Dominating Set, and Coloring

Authors Édouard Bonnet , Colin Geniet, Eun Jung Kim , Stéphan Thomassé, Rémi Watrigant



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Author Details

Édouard Bonnet
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Colin Geniet
  • University of Warsaw, Poland
Eun Jung Kim
  • Université Paris-Dauphine, PSL University, CNRS UMR7243, LAMSADE, Paris, France
Stéphan Thomassé
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Rémi Watrigant
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France

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Édouard Bonnet, Colin Geniet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant. Twin-width III: Max Independent Set, Min Dominating Set, and Coloring. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 35:1-35:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ICALP.2021.35

Abstract

We recently introduced the notion of twin-width, a novel graph invariant, and showed that first-order model checking can be solved in time f(d,k)n for n-vertex graphs given with a witness that the twin-width is at most d, called d-contraction sequence or d-sequence, and formulas of size k [Bonnet et al., FOCS '20]. The inevitable price to pay for such a general result is that f is a tower of exponentials of height roughly k. In this paper, we show that algorithms based on twin-width need not be impractical. We present 2^{O(k)}n-time algorithms for k-Independent Set, r-Scattered Set, k-Clique, and k-Dominating Set when an O(1)-sequence of the graph is given in input. We further show how to solve the weighted version of k-Independent Set, Subgraph Isomorphism, and Induced Subgraph Isomorphism, in the slightly worse running time 2^{O(k log k)}n. Up to logarithmic factors in the exponent, all these running times are optimal, unless the Exponential Time Hypothesis fails. Like our FO model checking algorithm, these new algorithms are based on a dynamic programming scheme following the sequence of contractions forward. We then show a second algorithmic use of the contraction sequence, by starting at its end and rewinding it. As an example of such a reverse scheme, we present a polynomial-time algorithm that properly colors the vertices of a graph with relatively few colors, thereby establishing that bounded twin-width classes are χ-bounded. This significantly extends the χ-boundedness of bounded rank-width classes, and does so with a very concise proof. It readily yields a constant approximation for Max Independent Set on K_t-free graphs of bounded twin-width, and a 2^{O(OPT)}-approximation for Min Coloring on bounded twin-width graphs. We further observe that a constant approximation for Max Independent Set on bounded twin-width graphs (but arbitrarily large clique number) would actually imply a PTAS. The third algorithmic use of twin-width builds on the second one. Playing the contraction sequence backward, we show that bounded twin-width graphs can be edge-partitioned into a linear number of bicliques, such that both sides of the bicliques are on consecutive vertices, in a fixed vertex ordering. This property is trivially shared with graphs of bounded average degree. Given that biclique edge-partition, we show how to solve the unweighted Single-Source Shortest Paths and hence All-Pairs Shortest Paths in time O(n log n) and time O(n² log n), respectively. In sharp contrast, even Diameter does not admit a truly subquadratic algorithm on bounded twin-width graphs, unless the Strong Exponential Time Hypothesis fails. The fourth algorithmic use of twin-width builds on the so-called versatile tree of contractions [Bonnet et al., SODA '21], a branching and more robust witness of low twin-width. We present constant-approximation algorithms for Min Dominating Set and related problems, on bounded twin-width graphs, by showing that the integrality gap is constant. This is done by going down the versatile tree and stopping accordingly to a problem-dependent criterion. At the reached node, a greedy approach yields the desired approximation.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Fixed parameter tractability
Keywords
  • Twin-width
  • Max Independent Set
  • Min Dominating Set
  • Coloring
  • Parameterized Algorithms
  • Approximation Algorithms
  • Exact Algorithms

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References

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