Surjective H-Colouring: New Hardness Results

Golovach, Petr A.; Johnson, Matthew; Martin, Barnaby; Paulusma, Daniël; Stewart, Anthony

doi:10.1007/978-3-319-58741-7_26

Cited by 5 publications

(7 citation statements)

References 23 publications

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“…Surj H-Colouring H-Colouring Figure 1: Relations between Surjective H-Colouring and its variants (from [15]). An arrow from one problem to another indicates that the latter problem is polynomial-time solvable for a graph H whenever the former is polynomial-time solvable for H. Reverse arrows do not hold for the leftmost and rightmost arrows, as witnessed by the reflexive 4-vertex cycle for the rightmost arrow and by any reflexive tree that is not a reflexive interval graph for the leftmost arrow (Feder, Hell and Huang [11] showed that the only reflexive bi-arc graphs are reflexive interval graphs).…”

Section: H-retraction H-compactionmentioning

confidence: 99%

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Surjective H-Colouring over Reflexive Digraphs

Larose

Martin

Paulusma

2018

ACM Trans. Comput. Theory

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The Surjective H-Colouring problem is to test if a given graph allows a vertex-surjective homomorphism to a fixed graph H. The complexity of this problem has been well studied for undirected (partially) reflexive graphs. We introduce endo-triviality, the property of a structure that all of its endomorphisms that do not have range of size 1 are automorphisms, as a means to obtain complexity-theoretic classifications of Surjective H-Colouring in the case of reflexive digraphs. Chen [2014] proved, in the setting of constraint satisfaction problems, that Surjective H-Colouring is NP-complete if H has the property that all of its polymorphisms are essentially unary. We give the first concrete application of his result by showing that every endo-trivial reflexive digraph H has this property. We then use the concept of endo-triviality to prove, as our main result, a dichotomy for Surjective H-Colouring when H is a reflexive tournament: if H is transitive, then Surjective H-Colouring is in NL, otherwise it is NP-complete.By combining this result with some known and new results we obtain a complexity classification for Surjective H-Colouring when H is a partially reflexive digraph of size at most 3.

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Section: H-retraction H-compactionmentioning

confidence: 99%

“…This paper concerns the computational complexity of the surjective homomorphism problem, also known in the literature as Surjective H-Colouring [15,16] and H-Vertex-Compaction [30]. This problem requires the homomorphism to be surjective.…”

Section: Introductionmentioning

confidence: 99%

Surjective H-Colouring over Reflexive Digraphs

Larose

Martin

Paulusma

2018

ACM Trans. Comput. Theory

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“…So a homomorphism from G to H is surjective if every vertex of H is "used" by the homomorphism. There is still no complete characterisation of the complexity of determining whether there is a surjective homomorphism from an input graph G to a graph H, despite an impressive collection of results [1,17,18,19,27]. A homomorphism from V (G) to V (H) is a compaction if it uses every vertex of H and also every non-loop edge of H (so it is surjective both on V (H) and on the non-loop edges in E(H)).…”

Section: Introductionmentioning

confidence: 99%

The Complexity of Counting Surjective Homomorphisms and Compactions

Focke¹,

Goldberg²,

Živný³

2019

SIAM J. Discrete Math.

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A homomorphism from a graph G to a graph H is a function from the vertices of G to the vertices of H that preserves edges. A homomorphism is surjective if it uses all of the vertices of H and it is a compaction if it uses all of the vertices of H and all of the nonloop edges of H. Hell and Nešetřil gave a complete characterisation of the complexity of deciding whether there is a homomorphism from an input graph G to a fixed graph H. A complete characterisation is not known for surjective homomorphisms or for compactions, though there are many interesting results. Dyer and Greenhill gave a complete characterisation of the complexity of counting homomorphisms from an input graph G to a fixed graph H. In this paper, we give a complete characterisation of the complexity of counting surjective homomorphisms from an input graph G to a fixed graph H and we also give a complete characterisation of the complexity of counting compactions from an input graph G to a fixed graph H. In an addendum we use our characterisations to point out a dichotomy for the complexity of the respective approximate counting problems (in the connected case).

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“…We give their formal definitions in Section 2. Both problems are well-studied in the decision setting [2,22,23,21,34,42,44,46,47]. All three of the problems #SHom(H), #Comp(H) and #Ret(H) can be interpreted as problems requiring one to count homomorphisms with some kind of surjectivity constraint.…”

Section: Name: #Lhom(h)mentioning

confidence: 99%

The Complexity of Approximately Counting Retractions

Focke

Goldberg

Živný

2020

ACM Trans. Comput. Theory

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Let G be a graph that contains an induced subgraph H . A retraction from G to H is a homomorphism from G to H that is the identity function on H . Retractions are very well studied: Given H , the complexity of deciding whether there is a retraction from an input graph G to H is completely classified, in the sense that it is known for which H this problem is tractable (assuming P ≠ NP). Similarly, the complexity of (exactly) counting retractions from G to H is classified (assuming FP ≠ #P). However, almost nothing is known about approximately counting retractions. Our first contribution is to give a complete trichotomy for approximately counting retractions to graphs without short cycles. The result is as follows: (1) Approximately counting retractions to a graph H of girth at least 5 is in FP if every connected component of H is a star, a single looped vertex, or an edge with two loops. (2) Otherwise, if every component is an irreflexive caterpillar or a partially bristled reflexive path, then approximately counting retractions to H is equivalent to approximately counting the independent sets of a bipartite graph—a problem that is complete in the approximate counting complexity class RH Π 1 . (3) Finally, if none of these hold, then approximately counting retractions to H is equivalent to approximately counting the satisfying assignments of a Boolean formula. Our second contribution is to locate the retraction counting problem for each H in the complexity landscape of related approximate counting problems. Interestingly, our results are in contrast to the situation in the exact counting context. We show that the problem of approximately counting retractions is separated both from the problem of approximately counting homomorphisms and from the problem of approximately counting list homomorphisms—whereas for exact counting all three of these problems are interreducible. We also show that the number of retractions is at least as hard to approximate as both the number of surjective homomorphisms and the number of compactions. In contrast, exactly counting compactions is the hardest of all of these exact counting problems.

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Surjective H-Colouring: New Hardness Results

Cited by 5 publications

References 23 publications

Surjective H-Colouring over Reflexive Digraphs

Surjective H-Colouring over Reflexive Digraphs

The Complexity of Counting Surjective Homomorphisms and Compactions

The Complexity of Approximately Counting Retractions

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