Recently Dadush, Vegh, and Zambelli (2017) has devised a polynomial submodular function minimization (SFM) algorithm based on their LP algorithm. In the present note we also show a weakly polynomial algorithm for SFM based on the recently developed linear programming feasibility algorithm of Chubanov (2017). Our algorithm is different from Dadush, Vegh, and Zambelli’s but … Read more
In this paper we investigate non-linear matroid optimization problems with polynomial objective functions where the monomials satisfy certain monotonicity properties. Indeed, we study problems where the set of non-linear monomials consists of all non-linear monomials that can be built from a given subset of the variables. Linearizing all non-linear monomials we study the respective polytope. … Read more
We study the linear extension complexity of stable set polytopes of perfect graphs. We make use of known structural results permitting to decompose perfect graphs into basic perfect graphs by means of two graph operations: 2-join and skew partitions. Exploiting the link between extension complexity and the nonnegative rank of an associated slack matrix, we … Read more
The matroid parity (MP) problem is a powerful (and NP-hard) extension of the matching problem. Whereas matching polytopes are well understood, little is known about MP polytopes. We prove that, when the matroid is laminar, the MP polytope is affinely congruent to a perfect b-matching polytope. From this we deduce that, even when the matroid … Read more
A fooling-set matrix is a square matrix with nonzero diagonal, but at least one in every pair of diagonally opposite entries is 0. Dietzfelbinger et al. ’96 proved that the rank of such a matrix is at least $\sqrt n$, for a matrix of order $n$. We ask for the typical minimum rank of a … Read more
We study the Partition Coloring Problem (PCP), a generalization of the Vertex Coloring Problem where the vertex set is partitioned. The PCP asks to select one vertex for each subset of the partition in such a way that the chromatic number of the induced graph is minimum. We propose a new Integer Linear Programming formulation … Read more
A proper edge-coloring of a given undirected graph with natural numbers identified with colors is an interval (or consecutive) coloring if the colors of edges incident to each vertex form an interval of consecutive integers. Not all graphs admit such an edge-coloring and the problem of deciding whether a graph is interval colorable is NP-complete. … Read more
Recently, Buchheim and Klein suggested to study polynomial-time solvable optimisation problems with linear objective functions combined with exactly one additional quadratic monomial. They concentrated on special quadratic spanning tree or forest problems. We extend their results to general matroid optimisation problems with a set of nested monomials in the objective function. The monomials are linearised … Read more
In this paper, we introduce an $O(nm)$ time algorithm to determine the minimum length directed cycle in a directed network with $n$ nodes and $m$ arcs and with no negative length directed cycles. This result improves upon the previous best time bound of $O(nm + n^2 \log\log n)$. Our algorithm first determines the cycle with … Read more
Local graph clustering methods aim to identify well-connected clusters around a given “seed set” of reference nodes. The main focus of prior theoretical work has been on worst-case running time properties or on implicit statistical regularization; and the focus of prior empirical work has been to identify structure in large social and information networks. Here, … Read more