Related papers: Expressing Combinatorial Optimization Problems by …

We study the problem of determining whether a given graph~$G=(V,E)$ admits a matching~$M$ whose removal destroys all odd cycles of~$G$ (or equivalently whether~$G-M$ is bipartite). This problem is equivalent to determine whether~$G$ admits…

Graph parameters such as the clique number, the chromatic number, and the independence number are central in many areas, ranging from computer networks to linguistics to computational neuroscience to social networks. In particular, the…

We deploy numerical semidefinite programming and conversion to exact rational inequalities to certify that for a positive semidefinite input polynomial or rational function, any representation as a fraction of sums-of-squares of polynomials…

In this article we study the Ramsey numbers $R(r,s)$ through Hilbert's Nullstellensatz and Alon's Combinatorial Nullstellensatz. We give polynomial encodings whose solutions correspond to Ramsey graphs of order $n$, those that do not…

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 study the problem of finding a maximum cardinality minimal separator of a graph. This problem is known to be NP-hard even for bipartite graphs. In this paper, we strengthen this hardness by showing that for planar bipartite graphs, the…

The main result of this paper is a coefficient formula that sharpens and generalizes Alon and Tarsi's Combinatorial Nullstellensatz, which provides some information about the polynomial map $P|_{\X_1\times...\times\X_n}$ when only…

Counting problems, determining the number of possible states of a large system under certain constraints, play an important role in many areas of science. They naturally arise for complex disordered systems in physics and chemistry, in…

The cyclability of a graph is the maximum integer $k$ for which every $k$ vertices lie on a cycle. The algorithmic version of the problem, given a graph $G$ and a non-negative integer $k,$ decide whether the cyclability of $G$ is at least…

We prove the existence of a function $f :\mathbb{N} \to \mathbb{N}$ such that the vertices of every planar graph with maximum degree $\Delta$ can be 3-colored in such a way that each monochromatic component has at most $f(\Delta)$ vertices.…

We investigate the question whether Subset Sum can be solved by a polynomial-time algorithm with access to a certificate of length poly(k) where k is the maximal number of bits in an input number. In other words, can it be solved using only…

For a graph $H$, a graph $G$ is an $H$-graph if it is an intersection graph of connected subgraphs of some subdivision of $H$. $H$-graphs naturally generalize several important graph classes like interval or circular-arc graph. This class…

The maximum stable set problem is NP-hard, even when restricted to triangle-free graphs. In particular, one cannot expect a polynomial time algorithm deciding if a bull-free graph has a stable set of size $k$, when $k$ is part of the…

The $k$-Colouring problem is to decide if the vertices of a graph can be coloured with at most $k$ colours for a fixed integer $k$ such that no two adjacent vertices are coloured alike. If each vertex u must be assigned a colour from a…

It is known that a graph isomorphism testing algorithm is polynomially equivalent to a detecting of a graph non-trivial automorphism algorithm. The polynomiality of the latter algorithm, is obtained by consideration of symmetry properties…

Motivated by the analogous questions in graphs, we study the complexity of coloring and stable set problems in hypergraphs with forbidden substructures and bounded edge size. Letting $\nu(G)$ denote the maximum size of a matching in $H$, we…

We present new generalizations of Olson's theorem and of a consequence of Alon's Combinatorial Nullstellensatz. These enable us to extend some of their combinatorial applications with conditions modulo primes to conditions modulo prime…

We consider the problem of coloring k-colorable graphs with the fewest possible colors. We present a randomized polynomial time algorithm that colors a 3-colorable graph on $n$ vertices with min O(Delta^{1/3} log^{1/2} Delta log n),…

Let $G = (V,E)$ be a graph, and for each $e \in E(G)$, let $L_e$ be a list of real numbers. Let $w:E(G) \to \cup_{e \in E(G)}L_e$ be an edge weighting function such that $w(e) \in L_e$ for each $e \in E(G)$, and let $c_w$ be the vertex…

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…