Related papers: Forbidden vertices
We show that the problem to decide whether two (convex) polytopes, given by their vertex-facet incidences, are combinatorially isomorphic is graph isomorphism complete, even for simple or simplicial polytopes. On the other hand, we give a…
This manuscript describes the notions of blocker and interdiction applied to well-known optimization problems. The main interest of these two concepts is the capability to analyze the existence of a combinatorial structure after some…
We consider the problem of projecting a convex set onto a subspace, or equivalently formulated, the problem of computing a set obtained by applying a linear mapping to a convex feasible set. This includes the problem of approximating convex…
We study the structure of the set of all possible affine hyperplane sections of a convex polytope. We present two different cell decompositions of this set, induced by hyperplane arrangements. Using our decomposition, we bound the number of…
We distinguish two kinds of piecewise linear functions and provide an interesting representation for a piecewise linear function between two normed spaces. Based on such a representation, we study a fully piecewise linear vector…
This is a survey on algorithmic questions about combinatorial and geometric properties of convex polytopes. We give a list of 35 problems; for each the current state of knowledege on its theoretical complexity status is reported. The…
In the List $k$-Coloring problem we are given a graph whose every vertex is equipped with a list, which is a subset of $\{1,\ldots,k\}$. We need to decide if $G$ admits a proper coloring, where every vertex receives a color from its list.…
We consider combinatorial problems that can be solved in polynomial time for graphs of bounded treewidth but where the order of the polynomial that bounds the running time is expected to depend on the treewidth bound. First we review some…
This work introduces a multidimensional generalization of the maximum bisection problem. A mixed integer linear programming formulation is proposed with the proof of its correctness. The numerical tests, made on the randomly generated…
We consider the Densest-Subgraph problem, where a graph and an integer k is given and we search for a subgraph on exactly k vertices that induces the maximum number of edges. We prove that this problem is NP-hard even when the input graph…
Interdiction problems are leader-follower games in which the leader is allowed to delete a certain number of edges from the graph in order to maximally impede the follower, who is trying to solve an optimization problem on the impeded…
We study vertex-ordering problems in loop-free digraphs subject to constraints on the left-going arcs, focusing on existence conditions and computational complexity. As an intriguing special case, we explore vertex-specific lower and upper…
The Restricted Invertibility problem is the problem of selecting the largest subset of columns of a given matrix $X$, while keeping the smallest singular value of the extracted submatrix above a certain threshold. In this paper, we address…
The degree-constrained subgraph problem asks for a subgraph of a given graph such that the degree of each vertex is within some specified bounds. We study the following reconfiguration variant of this problem: Given two solutions to a…
We classify, according to their computational complexity, integer optimization problems whose constraints and objective functions are polynomials with integer coefficients and the number of variables is fixed. For the optimization of an…
We analyze integer linear programs which we obtain after discretizing two-dimensional subproblems arising from a trust-region algorithm for mixed integer optimal control problems with total variation regularization. We discuss NP-hardness…
The learning complexity of special sets of vertices in graphs is studied in the model(s) of exact learning by (extended) equivalence and membership queries. Polynomial-time learning algorithms are described for vertex covers, independent…
In p-median location interdiction the aim is to find a subset of edges in a graph, such that the objective value of the p-median problem in the same graph without the selected edges is as large as possible. We prove that this problem is…
This document introduces a strategy to solve linear optimization problems. The strategy is based on the bounding condition each constraint produces on each one of the problem's dimension. The solution of a linear optimization problem is…
In this thesis we consider ordered graphs (that is, graphs with a fixed linear ordering on their vertices). We summarize and further investigations on the number of edges an ordered graph may have while avoiding a fixed forbidden ordered…