Bi-Arc Digraphs and Conservative Polymorphisms

Hell, Pavol; Rafiey, Akbar; Rafiey, Arash

doi:10.48550/arxiv.1608.03368

Cited by 3 publications

(8 citation statements)

References 27 publications

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“…Nevertheless, it is possible to obtain useful generalizations for digraphs that are neither reflexive nor irreflexive. This is done, for example, in [14,15], where general digraphs (that have some vertices with loops and others without) avoiding * 1 1 0 are investigated and found a useful unification of interval graphs, adjusted interval digraphs, two-dimensional orthogonal ray graphs (alias interval containment digraphs), and complements of threshold tolerance graphs. Another situation where it is fruitful to admit some vertices with loops and others without loops is the sub-ject of the next section; the class of graphs investigated there unifies reflexive strongly chordal graphs and irreflexive chordal bigraphs, and introduces a whole new class of well structured graphs.…”

Section: Background and Definitionsmentioning

confidence: 99%

Strongly chordal digraphs and $Γ$-free matrices

Hell¹,

Hernández-Cruz²,

Huang³

et al. 2019

Preprint

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We define strongly chordal digraphs, which generalize strongly chordal graphs, and chordal bipartite graphs, and are included in the class of chordal digraphs. They correspond to square 0, 1 matrices that admit a simultaneous row and column permutation avoiding the Γ matrix. In general, it is not clear if these digraphs can be recognized in polynomial time, and we focus on symmetric digraphs (i.e., graphs with possible loops), tournaments with possible loops, and balanced digraphs. In each of these cases we give a polynomial-time recognition algorithm and a forbidden induced subgraph characterization. We also discuss an algorithm for minimum general dominating set in strongly chordal graphs with possible loops, extending and unifying similar algorithms for strongly chordal graphs and chordal bipartite graphs.

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Section: Background and Definitionsmentioning

confidence: 99%

Strongly chordal digraphs and $Γ$-free matrices

Hell¹,

Hernández-Cruz²,

Huang³

et al. 2019

Preprint

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“…First, we consider digraphs that admit a min-ordering. Digraphs that admit a min-ordering have been studied under the name of bi-arc digraphs [36] and signed interval digraphs [29]. Deciding if digraph H has a min-ordering and finding a min-ordering of H is in P [36].…”

Section: Approximating Minimum Cost Homomorphism To Digraph Hmentioning

confidence: 99%

“…Digraphs that admit a min-ordering have been studied under the name of bi-arc digraphs [36] and signed interval digraphs [29]. Deciding if digraph H has a min-ordering and finding a min-ordering of H is in P [36]. We provide a constant factor approximation algorithm for MinHOM(H) where H admits a min-ordering.…”

Section: Approximating Minimum Cost Homomorphism To Digraph Hmentioning

confidence: 99%

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Toward a Dichotomy for Approximation of $H$-coloring

Rafiey¹,

Rafiey²,

Santos³

2019

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Given two (di)graphs G, H and a cost function c : V (G) × V (H) → Q ≥0 ∪ {+∞}, in the minimum cost homomorphism problem, MinHOM(H), we are interested in finding a homomorphism f :The complexity of exact minimization of this problem is well understood [34], and the class of digraphs H, for which the MinHOM(H) is polynomial time solvable is a small subset of all digraphs.In this paper, we consider the approximation of MinHOM within a constant factor. In terms of digraphs, MinHOM(H) is not approximable if H contains a digraph asteroidal triple (DAT). We take a major step toward a dichotomy classification of approximable cases. We give a dichotomy classification for approximating the MinHOM(H) when H is a graph (i.e. symmetric digraph). For digraphs, we provide constant factor approximation algorithms for two important classes of digraphs, namely bi-arc digraphs (digraphs with a conservative semi-lattice polymorphism or min-ordering), and k-arc digraphs (digraphs with an extended min-ordering). Specifically, we show that:• Dichotomy for Graphs: MinHOM(H) has a 2|V (H)|-approximation algorithm if graph H admits a conservative majority polymorphims (i.e. H is a bi-arc graph), otherwise, it is inapproximable;In conclusion, we show the importance of these results and provide insights for achieving a dichotomy classification of approximable cases. Our constant factors depend on the size of H. However, the implementation of our algorithms provides a much better approximation ratio. It leaves open to investigate a classification of digraphs H, where MinHOM(H) admits a constant factor approximation algorithm that is independent of |V (H)|.

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“…Second, by admitting intervals [a, b] that go in the negative direction (have b < a), we were able to extend the definition from reflexive graphs to graphs that have some vertices with loops and others without. Both these generalizations have proved very fruitful [12,9,12,14,26,16,20,21,32]. We now define a new class of digraphs that unifies these extensions.…”

Section: Signed Interval Digraphsmentioning

confidence: 99%

Interval-Like Graphs and Digraphs

Hell¹,

Huang²,

McConnell³

et al. 2018

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We unify several seemingly different graph and digraph classes under one umbrella. These classes are all, broadly speaking, different generalizations of interval graphs, and include, in addition to interval graphs, adjusted interval digraphs, threshold graphs, complements of threshold tolerance graphs (known as 'co-TT' graphs), bipartite interval containment graphs, bipartite co-circular arc graphs, and two-directional orthogonal ray graphs. (The last three classes coincide, but have been investigated in different contexts.) This common view is made possible by introducing reflexive relationships (loops) into the analysis. We also show that all the above classes are united by a common ordering characterization, the existence of a min ordering. We propose a common generalization of all these graph and digraph classes, namely signed-interval digraphs, and show that they are precisely the digraphs that are characterized by the existence of a min ordering. We also offer an alternative geometric characterization of these digraphs. For most of the above graph and digraph classes, we show that they are exactly those signed-interval digraphs that satisfy a suitable natural restriction on the digraph, like having a loop on every vertex, or having a symmetric edge-set, or being bipartite. For instance, co-TT graphs are precisely those signed-interval digraphs that have each edge symmetric. We also offer some discussion of future work on recognition algorithms and characterizations.

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Bi-Arc Digraphs and Conservative Polymorphisms

Cited by 3 publications

References 27 publications

Strongly chordal digraphs and $Γ$-free matrices

Strongly chordal digraphs and $Γ$-free matrices

Toward a Dichotomy for Approximation of $H$-coloring

Interval-Like Graphs and Digraphs

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