Finding Optimal Solutions With Neighborly Help

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 colorings for all one-edge-deleted subgraphs? Studying such questions not only gives detailed insight into the structure of the problem itself, but also into the complexity of related problems; most notably graph theory\u27s core notion of critical graphs (e.g., graphs whose chromatic number decreases under deletion of an arbitrary edge) and the complexity-theoretic notion of minimality problems (also called criticality problems, e.g., recognizing graphs that become 3-colorable when an arbitrary edge is deleted). We focus on two prototypical graph problems, Colorability and Vertex Cover. For example, we show that it is NP-hard to compute an optimal coloring for a graph from optimal colorings for all its one-vertex-deleted subgraphs, and that this remains true even when optimal solutions for all one-edge-deleted subgraphs are given. In contrast, computing an optimal coloring from all (or even just two) one-edge-added supergraphs is in P. We observe that Vertex Cover exhibits a remarkably different behavior, demonstrating the power of our model to delineate problems from each other more precisely on a structural level. Moreover, we provide a number of new complexity results for minimality and criticality problems. For example, we prove that Minimal-3-UnColorability is complete for DP (differences of NP sets), which was previously known only for the more amenable case of deleting vertices rather than edges. For Vertex Cover, we show that recognizing beta-vertex-critical graphs is complete for Theta_2^p (parallel access to NP), obtaining the first completeness result for a criticality problem for this class

Support Sets in Exponential Families and Oriented Matroid Theory

The closure of a discrete exponential family is described by a finite set of equations corresponding to the circuits of an underlying oriented matroid. These equations are similar to the equations used in algebraic statistics, although they need not be polynomial in the general case. This description allows for a combinatorial study of the possible support sets in the closure of an exponential family. If two exponential families induce the same oriented matroid, then their closures have the same support sets. Furthermore, the positive cocircuits give a parameterization of the closure of the exponential family.Comment: 27 pages, extended version published in IJA

Vector Coloring the Categorical Product of Graphs

A vector $t$-coloring of a graph is an assignment of real vectors $p_1, \ldots, p_n$ to its vertices such that $p_i^Tp_i = t-1$ for all $i=1, \ldots, n$ and $p_i^Tp_j \le -1$ whenever $i$ and $j$ are adjacent. The vector chromatic number of $G$ is the smallest real number $t \ge 1$ for which a vector $t$-coloring of $G$ exists. For a graph $H$ and a vector $t$-coloring $p_1,\ldots,p_n$ of a graph $G$, the assignment $(i,\ell) \mapsto p_i$ is a vector $t$-coloring of the categorical product $G \times H$. It follows that the vector chromatic number of $G \times H$ is at most the minimum of the vector chromatic numbers of the factors. We prove that equality always holds, constituting a vector coloring analog of the famous Hedetniemi Conjecture from graph coloring. Furthermore, we prove a necessary and sufficient condition for when all of the optimal vector colorings of the product can be expressed in terms of the optimal vector colorings of the factors. The vector chromatic number is closely related to the well-known Lov\'{a}sz theta function, and both of these parameters admit formulations as semidefinite programs. This connection to semidefinite programming is crucial to our work and the tools and techniques we develop could likely be of interest to others in this field.Comment: 38 page