Semidefinite programming and linear equations vs. homomorphism problems

We introduce a relaxation for homomorphism problems that combines semidefinite programming with linear Diophantine equations, and propose a framework for the analysis of its power based on the spectral theory of association schemes. We use this framework to establish an unconditional lower bound against the semidefinite programming + linear equations model, by showing that the relaxation does not solve the approximate graph homomorphism problem and thus, in particular, the approximate graph colouring problem

Approximate Graph Colouring and the Hollow Shadow

We show that approximate graph colouring is not solved by constantly many levels of the lift-and-project hierarchy for the combined basic linear programming and affine integer programming relaxation. The proof involves a construction of tensors whose fixed-dimensional projections are equal up to reflection and satisfy a sparsity condition, which may be of independent interest.Comment: Generalises and subsumes results from Section 6 in arXiv:2203.02478; builds on and generalises results in arXiv:2210.0829

Perron values and classes of trees

The bottleneck matrix $M$ of a rooted tree $T$ is a combinatorial object encoding the spatial distribution of the vertices with respect to the root. The spectral radius of $M$, known as the Perron value of the rooted tree, is closely related to the theory of the algebraic connectivity. In this paper, we investigate the Perron values of various classes of rooted trees by making use of combinatorial and linear-algebraic techniques. This results in multiple bounds on the Perron values of these classes, which can be straightforwardly applied to provide information on the algebraic connectivity.publishe

Combinatorial Perron parameters for trees

The notion of combinatorial Perron value was introduced in [2]. We continue the study of this parameter and also introduce a new parameter πe(M) which gives a new lower bound on the spectral radius of the bottleneck matrix M of a rooted tree. We prove a bound on the approximation error for πe(M). Several properties of these two parameters are shown. These ideas are motivated by the concept of algebraic connectivity. A certain extension property for the combinatorial Perron value is shown and it allows us to define a new center concept for caterpillars. We also compare computationally this new center to the so-called characteristic set, i.e., the center obtained from algebraic connectivity.publishe

On Kemeny's constant for trees with fixed order and diameter

Kemeny's constant $\kappa(G)$ of a connected graph $G$ is a measure of the expected transit time for the random walk associated with $G$. In the current work, we consider the case when $G$ is a tree, and, in this setting, we provide lower and upper bounds for $\kappa(G)$ in terms of the order $n$ and diameter $\delta$ of $G$ by using two different techniques. The lower bound is given as Kemeny's constant of a particular caterpillar tree and, as a consequence, it is sharp. The upper bound is found via induction, by repeatedly removing pendent vertices from $G$. By considering a specific family of trees - the broom-stars - we show that the upper bound is asymptotically sharp.Comment: 20 pages, 5 figure

1-in-3 vs. not-all-equal: dichotomy of a broken promise

The 1-in-3 and the Not-All-Equal satisfiability problems for Boolean CNF formulas are two well-known NP-hard problems. In contrast, the promise 1-in-3 vs. Not-All-Equal problem can be solved in polynomial time. In the present work, we investigate this constraint satisfaction problem in a regime where the promise is weakened from either side by a rainbow-free structure, and establish a complexity dichotomy for the resulting class of computational problems

A Fiedler center for graphs generalizing the characteristic set

This work introduces a general concept of center for graphs, built on the model of the characteristic set ([12], [14]) of trees. We define it as the set of cycles in a specific directed graph associated with the original graph G, and we let it depend on a function μ. In the case of trees we consider particular instances of μ given as weights of rooted subtrees, thus retrieving the characteristic set and, interestingly, the eccentricity-center. We investigate when the center of a graph G is simple – i.e., consisting of a unique cycle – and quasi-simple – i.e., inducing a connected subgraph of G. In particular, we prove that the center of a caterpillar tree associated with the so-called combinatorial Perron parameter ρc (studied in [2] and [3]) is always simple. We also make use of a discrete version of concavity to generate examples of simple and quasi-simple centers for graphs