Related papers: Linear Diophantine Equations, Group CSPs, and Grap…
In this paper, we exhibit $\textsf{AC}^{3}$ isomorphism tests for coprime extensions $H \ltimes N$ where $H$ is elementary Abelian and $N$ is Abelian; and groups where $\text{Rad}(G) = Z(G)$ is elementary Abelian and $G =…
We consider the task of detecting a hidden bipartite subgraph in a given random graph. This is formulated as a hypothesis testing problem, under the null hypothesis, the graph is a realization of an Erd\H{o}s-R\'{e}nyi random graph over $n$…
In this paper we study the complexity of the following problems: Given a colored graph X=(V,E,c), compute a minimum cardinality set S of vertices such that no nontrivial automorphism of X fixes all vertices in S. A closely related problem…
In this paper we consider the problems of testing isomorphism of tensors, $p$-groups, cubic forms, algebras, and more, which arise from a variety of areas, including machine learning, group theory, and cryptography. These problems can all…
We propose a new family of combinatorial inference problems for graphical models. Unlike classical statistical inference where the main interest is point estimation or parameter testing, combinatorial inference aims at testing the global…
Symmetries occur naturally in CSP or SAT problems and are not very difficult to discover, but using them to prune the search space tends to be very challenging. Indeed, this usually requires finding specific elements in a group of…
The Euler genus of a graph is a fundamental and well-studied parameter in graph theory and topology. Computing it has been shown to be NP-hard by [Thomassen '89 & '93], and it is known to be fixed-parameter tractable. However, the…
Lexicographic depth first search (LexDFS) is a graph search protocol which has already proved to be a powerful tool on cocomparability graphs. Cocomparability graphs have been well studied by investigating their complements (comparability…
Given a graph $G$, the longest path problem asks to compute a simple path of $G$ with the largest number of vertices. This problem is the most natural optimization version of the well known and well studied Hamiltonian path problem, and…
We give polynomial-time algorithms for the exact computation of lowest-energy (ground) states, worst margin violators, log partition functions, and marginal edge probabilities in certain binary undirected graphical models. Our approach…
Recently, Man\v{c}inska and Roberson proved that two graphs $G$ and $G'$ are quantum isomorphic if and only if they admit the same number of homomorphisms from all planar graphs. We extend this result to planar #CSP with any pair of sets…
We consider a hierarchy of graph invariants that naturally extends the spectral invariants defined by F\"urer (Lin. Alg. Appl. 2010) based on the angles formed by the set of standard basis vectors and their projections onto eigenspaces of…
A classic result due to Schaefer (1978) classifies all constraint satisfaction problems (CSPs) over the Boolean domain as being either in $\mathsf{P}$ or $\mathsf{NP}$-hard. This paper considers a promise-problem variant of CSPs called…
These notes concern aspects of various graphs whose vertex set is a group $G$ and whose edges reflect group structure in some way (so that they are invariant under the action of the automorphism group of $G$). The graphs I will discuss are…
Colour refinement is a basic algorithmic routine for graph isomorphism testing, appearing as a subroutine in almost all practical isomorphism solvers. It partitions the vertices of a graph into "colour classes" in such a way that all…
In this paper we study the interactions between so-called fractional relaxations of the integer programs (IPs) which encode homomorphism and isomorphism of relational structures. We give a combinatorial characterization of a certain natural…
In this paper, we study the hardness of solving graph-structured linear systems with coefficients over a finite field $\mathbb{Z}_p$ and over a polynomial ring $\mathbb{F}[x_1,\ldots,x_t]$. We reduce solving general linear systems in…
The Induced Graph Matching problem asks to find k disjoint induced subgraphs isomorphic to a given graph H in a given graph G such that there are no edges between vertices of different subgraphs. This problem generalizes the classical…
In this paper we resolve the complexity of the isomorphism problem on all but finitely many of the graph classes characterized by two forbidden induced subgraphs. To this end we develop new techniques applicable for the structural and…
In this paper, a new information theoretic framework for graph matching is introduced. Using this framework, the graph isomorphism and seeded graph matching problems are studied. The maximum degree algorithm for graph isomorphism is…