arXiv subjects

Alexey Barsukov

Publications and source records attributed to Alexey Barsukov.

8 recordsLinked to original sources

arXiv2026

Cut-homotopies and the complexity of edge-coloring problems

We study the computational complexity of problems that ask if a given graph admits an edge-coloring that does not contain an edge-colored clique from some fixed finite family. We show that every such problem is poly-time equivalent to a Constraint Satisfaction Problem, yielding a P vs. NP-complete dichotomy. Our main contribution lies in the reduction from the CSP to the coloring problem where we apply methods from Ramsey theory and a novel notion of cut-homotopy.

math.CO↗

arXiv2026

Towards infinite PCSP: a dichotomy for monochromatic cliques

The logic MMSNP is a well-studied fragment of Existential Second-Order logic that, from a computational perspective, captures finite-domain Constraint Satisfaction Problems (CSPs) modulo polynomial-time reductions. At the same time, MMSNP contains many problems that are expressible as $ω$-categorical CSPs but not as finite-domain ones. We initiate the study of Promise MMSNP (PMMSNP), a promise analogue of MMSNP. We show that every PMMSNP problem is poly-time equivalent to a (finite-domain) Promise CSP (PCSP), thereby extending the classical MMSNP-CSP correspondence to the promise setting. We then investigate the complexity of PMMSNPs arising from forbidding monochromatic cliques, a class encompassing promise graph colouring problems. For this class, we obtain a full complexity classification conditional on the Rich 2-to-1 Conjecture, a recently proposed perfect-completeness surrogate of the Unique Games Conjecture. As a key intermediate step which may be of independent interest, we prove that it is NP-hard, under the Rich 2-to-1 Conjecture, to properly colour a uniform hypergraph even if it is promised to admit a colouring satisfying a certain technical condition called reconfigurability. This proof is an extension of the recent work of Braverman, Khot, Lifshitz and Minzer (Adv. Math. 2025). To illustrate the broad applicability of this theorem, we show that it implies most of the linearly-ordered colouring conjecture of Barto, Battistelli, and Berg (STACS 2021).

cs.CC↗

arXiv2026

Edge-coloring problems with forbidden patterns and planted colors

Edge-coloring problems with forbidden patterns are decision problems asking to find an edge-coloring of the input graph which avoids a homomorphism from a fixed forbidden family of edge-colored graphs. In the precolored version of these problems, some of the edges of the input graph are already colored, and the goal is to find an extension of this coloring which omits a homomorphism from a forbidden graph. The existence of a complexity classification for such problems is an open question of Bienvenu, ten Cate, Lutz, and Wolter (ACM TODS'14) and we answer it for certain forbidden families consisting of odd cycles and cliques. The proof consists of two main stages. First, we combine the techniques from infinite constraint satisfaction and finite Ramsey theory in order to show that the edge-coloring problem is poly-time equivalent to its precolored version. After that, we show that the precolored version is poly-time equivalent to a finite constraint satisfaction problem, which has a P vs.\ NP-complete dichotomy by the seminal results of Bulatov (FOCS'17) and Zhuk (FOCS'17).

cs.CC↗

arXiv2026

The Golden Path to Guarded Monotone Strict NP

Guarded Monotone Strict NP (GMSNP) extends Monotone Monadic Strict NP (MMSNP) by guarded existentially quantified predicates of arbitrary arities. We prove that the containment and the FO-rewritability problems for GMSNP are decidable, thereby settling an open question of Bienvenu, ten Cate, Lutz, and Wolter, later restated by Bourhis and Lutz. Our proof also comes with a 2NEXPTIME upper bound on the complexity of the two problems, which matches the lower bounds for MMSNP due to Bourhis and Lutz. To obtain these results, we significantly improve the state of knowledge of the model-theoretic properties of GMSNP. Bodirsky, Knäuer, and Starke previously showed that every GMSNP sentence defines a finite union of CSPs of $ω$-categorical structures. We show that these structures can be used to obtain a reduction from the containment problem for GMSNP to the much simpler problem of testing the existence of a recolouring; a careful analysis of this yields said upper bound for containment. The upper bound for FO-rewritability is subsequently obtained by an application of several standard techniques from the theory of infinite-domain CSPs. As our secondary contribution, we refine the construction of Bodirsky, Knäuer, and Starke by adding a restricted form of homogeneity to the properties of these structures, making the logic amenable to future complexity classifications for query evaluation using techniques developed for infinite-domain CSPs.

cs.LO↗

arXiv2026

On the complexity of Sandwich Problems for $M$-partitions

We present a structural classification of constraint satisfaction problems (CSP) described by reflexive complete $2$-edge-coloured graphs. In particular, this classification extends the structural dichotomy for graph homomorphism problems known as the Hell--Nešetřil theorem (1990). Our classification is also efficient: we can check in polynomial time whether the CSP of a reflexive complete $2$-edge-coloured graph is in P or NP-complete, whereas for arbitrary $2$-edge-coloured graphs, this task is NP-complete. We then apply our main result in the context of matrix partition problems and sandwich problems. Firstly, we obtain one of the few algorithmic solutions to general classes of matrix partition problems. And secondly, we present a P vs. NP-complete classification of sandwich problems for matrix partitions.

cs.CC↗

arXiv2025

On guarded extensions of MMSNP

Feder and Vardi showed that the class Monotone Monadic SNP without inequality (MMSNP) has a P vs NP-complete dichotomy if and only if such a dichotomy holds for finite-domain Constraint Satisfaction Problems (CSPs). Moreover, they showed that none of the three classes obtained by removing one of the defining properties of MMSNP (monotonicity, monadicity, no inequality) has a dichotomy. The overall objective of this paper is to study the gaps between MMSNP and each of these three superclasses, where the existence of a dichotomy remains unknown. For the gap between MMSNP and Monotone SNP without inequality, we study the class Guarded Monotone SNP without inequality (GMSNP) introduced by Bienvenu, ten Cate, Lutz, and Wolter, and prove that GMSNP has a dichotomy if and only if a dichotomy holds for GMSNP problems over signatures consisting of a unique relation symbol. For the gap between MMSNP and MMSNP with inequality, we introduce a new class MMSNP with guarded inequality, that lies between MMSNP and MMSNP with inequality and that is strictly more expressive than the former and still has a dichotomy. For the gap between MMSNP and Monadic SNP without inequality, we introduce a logic that extends the class of Matrix Partitions in a similar way how MMSNP extends finite-domain CSP, and pose an open question about the existence of a dichotomy for this class. Finally, we revisit the theorem of Feder and Vardi, which claims that the class NP embeds into MMSNP with inequality. We give a detailed proof of this theorem as it ensures no dichotomy for the right-hand side class of each of the three gaps.

cs.CC↗

arXiv2024

Generalisations of Matrix Partitions : Complexity and Obstructions

A trigraph is a graph where each pair of vertices is labelled either 0 (a non-arc), 1 (an arc) or $\star$ (both an arc and a non-arc). In a series of papers, Hell and co-authors proposed to study the complexity of homomorphisms from graphs to trigraphs, called Matrix Partition Problems, where arcs and non-arcs can be both mapped to $\star$-arcs, while a non-arc cannot be mapped to an arc, and vice-versa. Even though Matrix Partition Problems are generalisations of CSPs, they share with them the property of being ``intrinsically'' combinatorial. So, the question of a possible P-time vs NP-complete dichotomy is a very natural one and was raised in Hell et al.'s papers. We propose a generalisation of Matrix Partitions to relational structures and study them with respect to the question of a dichotomy. We first show that trigraph homomorphisms and Matrix Partitions are P-time equivalent, and then prove that one can also restrict (with respect to having a dichotomy) to relational structures with a single relation. Failing in proving that Matrix Partitions on directed graphs are not P-time equivalent to Matrix Partitions on relational structures, we give some evidence that it might be unlikely by formalising the reductions used in the case of CSPs and by showing that such reductions cannot work for the case of Matrix Partitions. We then turn our attention to Matrix Partitions that can be described by finite sets of (induced-subgraph) obstructions. We show, in particular, that any such problem has finitely many minimal obstructions if and only if it has finite duality. We conclude by showing that on trees (seen as trigraphs) it is NP-complete to decide whether a given tree has a homomorphism to another input trigraph. The latter shows a notable difference on tractability between CSP and Matrix Partitions as it is well-known that CSP is tractable on the class of trees.

cs.CC↗

arXiv2024

Maximum Cut on Interval Graphs of Interval Count Two is NP-complete

An interval graph has interval count $\ell$ if it has an interval model, where among every $\ell+1$ intervals there are two that have the same length. Maximum Cut on interval graphs has been found to be NP-complete recently by Adhikary et al. while deciding its complexity on unit interval graphs (graphs with interval count one) remains a longstanding open problem. More recently, de Figueiredo et al. have made an advancement by showing that the problem remains NP-complete on interval graphs of interval count four. In this paper, we show that Maximum Cut is NP-complete even on interval graphs of interval count two.

cs.CC↗