arXiv subjects

Rico Zenklusen

Publications and source records attributed to Rico Zenklusen.

60 records · Page 4Linked to original sources

arXiv2010

Optimization with More than One Budget

A natural way to deal with multiple, partially conflicting objectives is turning all the objectives but one into budget constraints. Some classical polynomial-time optimization problems, such as spanning tree and forest, shortest path, (perfect) matching, independent set (basis) in a matroid or in the intersection of two matroids, become NP-hard even with one budget constraint. Still, for most of these problems deterministic and randomized polynomial-time approximation schemes are known. In the case of two or more budgets, typically only multi-criteria approximation schemes are available, which return slightly infeasible solutions. Not much is known however for the case of strict budget constraints: filling this gap is the main goal of this paper. We show that shortest path, perfect matching, and spanning tree (and hence matroid basis and matroid intersection basis) are inapproximable already with two budget constraints. For the remaining problems, whose set of solutions forms an independence system, we present deterministic and randomized polynomial-time approximation schemes for a constant number k of budget constraints. Our results are based on a variety of techniques: 1. We present a simple and powerful mechanism to transform multi-criteria approximation schemes into pure approximation schemes. 2. We show that points in low dimensional faces of any matroid polytope are almost integral, an interesting result on its own. This gives a deterministic approximation scheme for k-budgeted matroid independent set. 3. We present a deterministic approximation scheme for 2-budgeted matching. The backbone of this result is a purely topological property of curves in R^2.

cs.DS↗

arXiv2009

Dependent Randomized Rounding for Matroid Polytopes and Applications

Motivated by several applications, we consider the problem of randomly rounding a fractional solution in a matroid (base) polytope to an integral one. We consider the pipage rounding technique and also present a new technique, randomized swap rounding. Our main technical results are concentration bounds for functions of random variables arising from these rounding techniques. We prove Chernoff-type concentration bounds for linear functions of random variables arising from both techniques, and also a lower-tail exponential bound for monotone submodular functions of variables arising from randomized swap rounding. The following are examples of our applications: (1) We give a (1-1/e-epsilon)-approximation algorithm for the problem of maximizing a monotone submodular function subject to 1 matroid and k linear constraints, for any constant k and epsilon>0. (2) We present a result on minimax packing problems that involve a matroid base constraint. We give an O(log m / log log m)-approximation for the general problem Min {lambda: x \in {0,1}^N, x \in B(M), Ax <= lambda b}, where m is the number of packing constraints. (3) We generalize the continuous greedy algorithm to problems involving multiple submodular functions, and use it to find a (1-1/e-epsilon)-approximate pareto set for the problem of maximizing a constant number of monotone submodular functions subject to a matroid constraint. An example is the Submodular Welfare Problem where we are looking for an approximate pareto set with respect to individual players' utilities.

cs.DS↗

arXiv2008

Matching Interdiction

In the matching interdiction problem, we are given an undirected graph with weights and interdiction costs on the edges and seek to remove a subset of the edges constrained to some budget, such that the weight of a maximum weight matching in the remaining graph is minimized. In this work we introduce the matching interdiction problem and show that it is strongly NP-complete even when the input is restricted to simple, bipartite graphs with unit edge weights and unit interdiction costs. Furthermore, we present a pseudo-polynomial algorithm for solving the matching interdiction problem on graphs with bounded treewidth. The proposed algorithm extends the approach that is typically used for the creation of efficient algorithms on graphs with bounded treewidth to interdiction problems.

cs.DM↗

arXiv2008

Extensions to Network Flow Interdiction on Planar Graphs

Network flow interdiction analysis studies by how much the value of a maximum flow in a network can be diminished by removing components of the network constrained to some budget. Although this problem is strongly NP-complete on general networks, pseudo-polynomial algorithms were found for planar networks with a single source and a single sink and without the possibility to remove vertices. In this work we introduce pseudo-polynomial algorithms which overcome some of the restrictions of previous methods. We propose a planarity-preserving transformation that allows to incorporate vertex removals and vertex capacities in pseudo-polynomial interdiction algorithms for planar graphs. Additionally, a pseudo-polynomial algorithm is introduced for the problem of determining the minimal interdiction budget which is at least needed to make it impossible to satisfy the demand of all sink nodes, on planar networks with multiple sources and sinks satisfying that the sum of the supplies at the source nodes equals the sum of the demands at the sink nodes. Furthermore we show that the k-densest subgraph problem on planar graphs can be reduced to a network flow interdiction problem on a planar graph with multiple sources and sinks and polynomially bounded input numbers. However it is still not known if either of these problems can be solved in polynomial time.

cs.DM↗

arXiv2007

Estimation of Small s-t Reliabilities in Acyclic Networks

In the classical s-t network reliability problem a fixed network G is given including two designated vertices s and t (called terminals). The edges are subject to independent random failure, and the task is to compute the probability that s and t are connected in the resulting network, which is known to be #P-complete. In this paper we are interested in approximating the s-t reliability in case of a directed acyclic original network G. We introduce and analyze a specialized version of the Monte-Carlo algorithm given by Karp and Luby. For the case of uniform edge failure probabilities, we give a worst-case bound on the number of samples that have to be drawn to obtain an epsilon-delta approximation, being sharper than the original upper bound. We also derive a variance reduction of the estimator which reduces the expected number of iterations to perform to achieve the desired accuracy when applied in conjunction with different stopping rules. Initial computational results on two types of random networks (directed acyclic Delaunay graphs and a slightly modified version of a classical random graph) with up to one million vertices are presented. These results show the advantage of the introduced Monte-Carlo approach compared to direct simulation when small reliabilities have to be estimated and demonstrate its applicability on large-scale instances.

cs.DM↗

arXiv2007

A tight bound on the collection of edges in MSTs of induced subgraphs

Let $G=(V,E)$ be a complete $n$-vertex graph with distinct positive edge weights. We prove that for $k\in\{1,2,...,n-1\}$, the set consisting of the edges of all minimum spanning trees (MSTs) over induced subgraphs of $G$ with $n-k+1$ vertices has at most $nk-\binom{k+1}{2}$ elements. This proves a conjecture of Goemans and Vondrak \cite{GV2005}. We also show that the result is a generalization of Mader's Theorem, which bounds the number of edges in any edge-minimal $k$-connected graph.

math.CO↗