Deciding Clique-Width for Graphs of Bounded Tree-Width

Espelage, Wolfgang; Gurski, Frank; Wanke, Egon

doi:10.7155/jgaa.00065

Cited by 28 publications

(4 citation statements)

References 12 publications

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“…Further, they construct, in polynomial time, a k-expression of the stated size. Espelage, Gurski, and Wanke [21] gave an algorithm that takes as input a tree decomposition with width k, and gives a clique-width 2 O(k) -expression [21] in linear time.…”

Section: Treewidth and Clique-widthmentioning

confidence: 99%

“…It follows from Theorem 5.1 and Theorem 5.3 that F has treewidth at most 5 and cliquewidth at most 48. We first find a tree decomposition of F with width at most 5 in linear time by Bodlaendar [3], and then feed this decomposition into the algorithm of Espelage, Gurski, and Wanke [21] which outputs in linear time a k-expression of F for some constant k (k could be larger than 48). Then we construct from F , K v (v ∈ V (F )) and U in linear time a k-expression of G [19].…”

Section: Solving Chromatic Number Using Clique-widthmentioning

confidence: 99%

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Structure and algorithms for (cap, even hole)-free graphs

Cameron

Silva

Huang

et al. 2018

Discrete Mathematics

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A graph is even-hole-free if it has no induced even cycles of length 4 or more. A cap is a cycle of length at least 5 with exactly one chord and that chord creates a triangle with the cycle. In this paper, we consider (cap, even hole)-free graphs, and more generally, (cap, 4-hole)-free odd-signable graphs. We give an explicit construction of these graphs. We prove that every such graph G has a vertex of degree at most 3 2 ω(G) − 1, and hencewhere ω(G) denotes the size of a largest clique in G and χ(G) denotes the chromatic number of G. We give an O(nm) algorithm for q-coloring these graphs for fixed q and an O(nm) algorithm for maximum weight stable set. We also give a polynomial-time algorithm for minimum coloring.Our algorithms are based on our results that triangle-free odd-signable graphs have treewidth at most 5 and thus have clique-width at most 48, and that (cap, 4-hole)-free odd-signable graphs G without clique cutsets have treewidth at most 6ω(G) − 1 and clique-width at most 48.

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Section: Treewidth and Clique-widthmentioning

confidence: 99%

Section: Solving Chromatic Number Using Clique-widthmentioning

confidence: 99%

Structure and algorithms for (cap, even hole)-free graphs

Cameron

Silva

Huang

et al. 2018

Discrete Mathematics

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“…Finally, we mention a result by Espelage, Gurski and Wanke [41], that the clique-width of a graph can be computed in linear time on graph classes of bounded tree-width.…”

Section: Definition 44mentioning

confidence: 99%

Algorithmic meta-theorems

Kreutzer

2011

Finite and Algorithmic Model Theory

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Algorithmic meta-theorems are algorithmic results that apply to a whole range of problems, instead of addressing just one specific problem. This kind of theorems are often stated relative to a certain class of graphs, so the general form of a meta theorem reads "every problem in a certain class C of problems can be solved efficiently on every graph satisfying a certain property P". A particularly well known example of a meta-theorem is Courcelle's theorem that every decision problem definable in monadic second-order logic (MSO) can be decided in linear time on any class of graphs of bounded tree-width [1].The class C of problems can be defined in a number of different ways. One option is to state combinatorial or algorithmic criteria of problems in C. For instance, Demaine, Hajiaghayi and Kawarabayashi [5] showed that every minimisation problem that can be solved efficiently on graph classes of bounded tree-width and for which approximate solutions can be computed efficiently from solutions of certain sub-instances, have a PTAS on any class of graphs excluding a fixed minor. While this gives a strong unifying explanation for PTAS of many problems on H-minor free graphs, the class of problems it defines is not very natural. In particular, it may require some work to decide if a given problem belongs to this class or not.Another approach to define meta-theorems is therefore based on definability in logical systems, e.g. to consider the class of problems that can be defined in first-order logic. For instance, related to the above example, a result by Dawar, Grohe, Kreutzer and Schweikardt [4] states that every minimisation problem definable in first-order logic has an EPTAS on every class of graphs excluding a minor. While the actual complexity bounds obtained in this way may not live up to bounds derivable for each individual problem, the class of problems described in this way is extremely natural and for many problems their mathematical formulation already shows that they are first-order definable. Consider, e.g., the definition of a dominating set.Such meta-theorems based on definability in a given logic have received much attention in the literature. For instance, for the case of decision problems, it is has been shown that every problem definable in monadic second-order logic can be decided in polynomial time on graph classes of bounded clique width (see e.g. [2]). For first-order logic (FO), Seese [14] showed that every FO-definable decision problem is solvable in linear time on graph classes of bounded degree. This has then been extended to planar graphs and, more generally, graph classes of bounded local tree-width by Frick and Grohe [7] and to H-minor free graphs by Flum and Grohe [6]. Finally, Dawar, Grohe, Kreutzer generalised these results even further to classes of graphs locally excluding a minor [3].

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“…A prominent example is the celebrated result by Bodlaender (1996), where he uses a DP approach on an approximate tree decomposition to compute the exact treewidth of a graph; here, the solutions are tree decompositions, which are complex structures that cannot easily be represented in terms of the graph. Other prominent examples include a DP approach to compute a graph's exact tree-depth (Reidl et al 2014) or clique-width (Espelage, Gurski, and Wanke 2003) using an optimal tree decomposition.…”

Section: Introductionmentioning

confidence: 99%

Learning Small Decision Trees for Data of Low Rank-Width

Dabrowski,

Eiben,

Ordyniak

et al. 2024

AAAI

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We consider the NP-hard problem of finding a smallest decision tree representing a classification instance in terms of a partially defined Boolean function. Small decision trees are desirable to provide an interpretable model for the given data. We show that the problem is fixed-parameter tractable when parameterized by the rank-width of the incidence graph of the given classification instance. Our algorithm proceeds by dynamic programming using an NLC decomposition obtained from a rank-width decomposition. The key to the algorithm is a succinct representation of partial solutions. This allows us to limit the space and time requirements for each dynamic programming step in terms of the parameter.

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Deciding Clique-Width for Graphs of Bounded Tree-Width

Abstract: We show that there exists a linear time algorithm for deciding whether a graph of bounded tree-width has clique-width k for some fixed integer k.

Cited by 28 publications

References 12 publications

Structure and algorithms for (cap, even hole)-free graphs

Structure and algorithms for (cap, even hole)-free graphs

Algorithmic meta-theorems

Learning Small Decision Trees for Data of Low Rank-Width

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