Fast approximation algorithms for the generalized survivable network design problem

In a standard f-connectivity network design problem, we are given an undirected graph G = (V, E), a cut-requirement function f : 2V → N, and non-negative costs c(e) for all e ∈ E. We are then asked to find a minimum-cost vector x ∈ ℕE such that x(δ(S)) ≥ f(S) for all S ⊆ V. We focus on the class of such problems where f is a proper function. This encodes many well-studied NP-hard problems such as the generalized survivable network design problem. In this paper we present the first strongly polynomial time FPTAS for solving the LP relaxation of the standard IP formulation of the f-connectivity problem with general proper functions f. Implementing Jain's algorithm, this yields a strongly polynomial time (2 + ε)-approximation for the generalized survivable network design problem (where we consider rounding up of rationals an arithmetic operation)

Extended formulations for combinatorial polytopes

Magdeburg, Univ., Fak. für Mathematik, Diss., 2012von Kanstantsin Pashkovic

The projected faces property and polyhedral relations

Margot (1994) in his doctoral dissertation studied extended formulations of combinatorial polytopes that arise from “smaller” polytopes via some composition rule. He introduced the “projected faces property” of a polytope and showed that this property suffices to iteratively build extended formulations of composed polytopes. For the composed polytopes, we show that an extended formulation of the type defined by Margot is always possible only if the smaller polytopes have the projected faces property. Therefore, this produces a characterization of the projected faces property. Affinely generated polyhedral relations were introduced by Kaibel and Pashkovich (Optima 85:2–7, 2011) to construct extended formulations for the convex hull of the images of a point under the action of some finite group of reflections. In this paper we prove that the projected faces property and affinely generated polyhedral relation are equivalent conditions.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Simple games versus weighted voting games: bounding the critical threshold value

A simple game (N; v) is given by a set N of n players and a partition of 2N into a set L of losing coalitions L with value v(L) = 0 that is closed under taking subsets and a set W of winning coalitions W with value v(W) = 1. We let = minp>0;p6=0 maxW2W;L2L p(L) p(W) . It is known that the subclass of simple games with 0