Related papers: Solving Hard Computational Problems Efficiently: A…

Can we efficiently compute optimal solutions to instances of a hard problem from optimal solutions to neighboring (i.e., locally modified) instances? For example, can we efficiently compute an optimal coloring for a graph from optimal…

A graph is $H$-free if it does not contain an induced subgraph isomorphic to $H$. We denote by $P_k$ and $C_k$ the path and the cycle on $k$ vertices, respectively. In this paper, we prove that 4-COLORING is NP-complete for $P_7$-free…

Many complex systems and datasets are characterized by multiway interactions of different categories, and can be modeled as edge-colored hypergraphs. We focus on clustering such datasets using the NP-hard edge-colored clustering problem,…

The Promise Constraint Satisfaction Problem (PCSP) is a recently introduced vast generalization of the Constraint Satisfaction Problem (CSP). We investigate the computational complexity of a class of PCSPs beyond the most studied cases -…

This paper describes several new problems and ideas concerning algebraic geometry and complexity theory. It first uses the idea of coloring graphs with elements of finite fields. This procedure then shows that graph coloring problems can be…

In this paper, we consider the maximum $k$-edge-colorable subgraph problem. In this problem we are given a graph $G$ and a positive integer $k$, the goal is to take $k$ matchings of $G$ such that their union contains maximum number of…

Using the algebraic approach to promise constraint satisfaction problems, we establish complexity classifications of three natural variants of hypergraph colourings: standard nonmonochromatic colourings, conflict-free colourings, and…

We consider the Functional Orientation 2-Color problem, which was introduced by Valiant in his seminal paper on holographic algorithms [SIAM J. Comput., 37(5), 2008]. For this decision problem, Valiant gave a polynomial time holographic…

We consider the selective graph coloring problem, which is a generalization of the classical graph coloring problem. Given a graph together with a partition of its vertex set into clusters, we want to choose exactly one vertex per cluster…

An NP-complete coloring or homomorphism problem may become polynomial time solvable when restricted to graphs with degrees bounded by a small number, but remain NP-complete if the bound is higher. For instance, 3-colorability of graphs with…

Combinatorial optimization problems are pervasive across science and industry. Modern deep learning tools are poised to solve these problems at unprecedented scales, but a unifying framework that incorporates insights from statistical…

As it follows from G\"odel's incompleteness theorems, any consistent formal system of axioms and rules of inference should imply a true unprovable statement. Actually, this fundamental principle can be efficiently applicable in…

One of the driving problems in the CSP area is the Dichotomy Conjecture, formulated in 1993 by Feder and Vardi [STOC'93], stating that for any fixed relational structure G the Constraint Satisfaction Problem CSP(G) is either NP--complete or…

We present three new complexity results for classes of planning problems with simple causal graphs. First, we describe a polynomial-time algorithm that uses macros to generate plans for the class 3S of planning problems with binary state…

Restricted star colouring is a variant of star colouring introduced to design heuristic algorithms to estimate sparse Hessian matrices. For $k\in\mathbb{N}$, a $k$-restricted star colouring ($k$-rs colouring) of a graph $G$ is a function…

In an article [3] published recently in this journal, it was shown that when k >= 3, the problem of deciding whether the distinguishing chromatic number of a graph is at most k is NP-hard. We consider the problem when k = 2. In regards to…

The NP-complete problems Colouring and k-Colouring $(k\geq 3$) are well studied on $H$-free graphs, i.e., graphs that do not contain some fixed graph $H$ as an induced subgraph. We research to what extent the known polynomial-time…

We study the complexity of approximation on satisfiable instances for graph homomorphism problems. For a fixed graph $H$, the $H$-colouring problem is to decide whether a given graph has a homomorphism to $H$. By a result of Hell and…

We consider a variation of the prototype combinatorial-optimisation problem known as graph-colouring. Our optimisation goal is to colour the vertices of a graph with a fixed number of colours, in a way to maximise the number of different…

We consider the problem of coloring a 3-colorable graph in polynomial time using as few colors as possible. This is one of the most challenging problems in graph algorithms. In this paper using Blum's notion of ``progress'', we develop a…