Surjective H-Colouring over Reflexive Digraphs

Larose, Benoît; Martin, Barnaby; Paulusma, Daniël

doi:10.1145/3282431

Cited by 7 publications

(20 citation statements)

References 34 publications

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“…Let us mention that, for the decision problem of checking existence of a surjective homomorphism, a complexity classification of templates seems to be currently elusive, although there is work in this direction (see for example [2,10] and the references therein).…”

Section: Complexity Resultsmentioning

confidence: 99%

Homomorphisms are indeed a good basis for counting: Three fixed-template dichotomy theorems, for the price of one

Chen

2017

Preprint

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Many natural combinatorial quantities can be expressed by counting the number of homomorphisms to a fixed relational structure. For example, the number of 3-colorings of an undirected graph G is equal to the number of homomorphisms from G to the 3-clique. In this setup, the structure receiving the homomorphisms is often referred to as a template; we use the term template function to refer to a function, from structures to natural numbers, that is definable as the number of homomorphisms to a fixed template. There is a literature that studies the complexity of template functions. The present work is concerned with relating template functions to the problems of counting, with respect to various fixed templates, the number of two particular types of homomorphisms: surjective homomorphisms and what we term condensations. A surjective homomorphism is a homomorphism that maps the universe of the first structure surjectively onto the universe of the second structure; a condensation is a homomorphism that, in addition, maps each relation of the first structure surjectively onto the corresponding relation of the second structure. In this article, we explain how any problem of counting surjective homomorphisms to a fixed template is polynomial-time equivalent to computing a linear combination of template functions; we also show this for any problem of counting condensations to a fixed template. Via a theorem that characterizes the complexity of computing such a linear combination, we show how a known dichotomy for template functions can be used to infer a dichotomy for counting surjective homomorphisms on fixed templates, and likewise a dichotomy for counting condensations on fixed templates. Our study is strongly inspired by, based on, and can be viewed as a dual of the graph motif framework of Curticapean, Dell, and Marx (STOC 2017); that framework is in turn based on work of Lovász (2012). PreliminariesWhen f : A → B is a map and A ′ ⊆ A, we use f (A ′ ) to denote the set {f (a) | a ∈ A ′ }.

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Section: Complexity Resultsmentioning

confidence: 99%

Homomorphisms are indeed a good basis for counting: Three fixed-template dichotomy theorems, for the price of one

Chen

2017

Preprint

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“…From the logical perspective this translates to the stipulation that all elements of B be used as witnesses to the (existential) variables of the primitive positive input 𝜑. The surjective CSP appears in the literature under a variety of names, including surjective homomorphism [2], surjective colouring [12,15] and vertex compaction [19,20]. CSP(B) and SCSP(B) have various other cousins: see the survey [2] or, in the specific context of reflexive tournaments, [15].…”

Section: Introductionmentioning

confidence: 99%

“…The surjective CSP appears in the literature under a variety of names, including surjective homomorphism [2], surjective colouring [12,15] and vertex compaction [19,20]. CSP(B) and SCSP(B) have various other cousins: see the survey [2] or, in the specific context of reflexive tournaments, [15]. The only one we will dwell on here is the retraction problem CSP 𝑐 (B) which can be defined in various ways but, in keeping with the present narrative, we could define logically as allowing atoms of the form 𝑣 = 𝑏 in the input sentence 𝜑 where 𝑏 is some element of B (the superscript 𝑐 indicates that constants are allowed).…”

Section: Introductionmentioning

confidence: 99%

“…Tournaments, both irreflexive and reflexive (and sometimes in between), have played a strong role as a testbed for conjectures and a habitat for classifications, for relatives of the CSP both complexity-theoretic [1,10,15] and algebraic [14,21]. Looking at Table 1 one can see the last unresolved case is precisely QCSP on reflexive tournaments.…”

Section: Introductionmentioning

confidence: 99%

“…When H is a reflexive tournament, we prove that QCSP(H) is in NL if H has both initial and final strongly connected components trivial, and is NP-hard otherwise. In contrast to the proof from [10] and like the proof of [15], we will henceforth work largely combinatorially rather than algebraically. Note that we do not investigate beyond NP-hard, so our dichotomy cannot be compared directly to the trichotomy of [10] for irreflexive tournaments which distinguishes between P, NP-complete and Pspace-complete.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

QCSP on Reflexive Tournaments

Larose

Marković

Martin

et al. 2022

ACM Trans. Comput. Logic

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We give a complexity dichotomy for the Quantified Constraint Satisfaction Problem \( \mathrm{QCSP}(\mathrm{H}) \) when \( \mathrm{H} \) is a reflexive tournament. It is well known that reflexive tournaments can be split into a sequence of strongly connected components \( \mathrm{H}_1,\ldots ,\mathrm{H}_n \) so that there exists an edge from every vertex of \( \mathrm{H}_i \) to every vertex of \( \mathrm{H}_j \) if and only if \( i\lt j \) . We prove that if \( \mathrm{H} \) has both its initial and final strongly connected component (possibly equal) of size 1, then \( \mathrm{QCSP}(\mathrm{H}) \) is in \( \mathsf {NL} \) and otherwise \( \mathrm{QCSP}(\mathrm{H}) \) is \( \mathsf {NP} \) -hard.

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Algebraic Global Gadgetry for Surjective Constraint Satisfaction

Chen

2024

comput. complex.

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The constraint satisfaction problem (CSP) on a finite relational structure B is to decide, given a set of constraints on variables where the relations come from B, whether or not there is an assignment to the variables satisfying all of the constraints; the surjective CSP is the variant where one decides the existence of a surjective satisfying assignment onto the universe of B. We present an algebraic framework for proving hardness results on surjective CSPs; essentially, this framework computes global gadgetry that permits one to present a reduction from a classical CSP to a surjective CSP. We show how to derive a number of hardness results for surjective CSP in this framework, including the hardness of the disconnected cut problem, of the no-rainbow three-coloring problem, and of the surjective CSP on all two-element structures known to be intractable (in this setting). Our framework thus allows us to unify these hardness results and reveal common structure among them; we believe that our hardness proof for the disconnected cut problem is more succinct than the original. In our view, the framework also makes very transparent a way in which classical CSPs can be reduced to surjective CSPs.

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Surjective H-Colouring over Reflexive Digraphs

Cited by 7 publications

References 34 publications

Homomorphisms are indeed a good basis for counting: Three fixed-template dichotomy theorems, for the price of one

Homomorphisms are indeed a good basis for counting: Three fixed-template dichotomy theorems, for the price of one

QCSP on Reflexive Tournaments

Algebraic Global Gadgetry for Surjective Constraint Satisfaction

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