Friedrich Eisenbrand

On the membership problem for the elementary closure of a polyhedron

A, an integral vector b and some rational vector , decide whether is outside the elementary closure , is NP-complete. This result is achieved by an extension of a result by Caprara and Fischetti.

On the complexity of fixed parameter clique and dominating set

We provide simple, faster algorithms for the detection of cliques and dominating sets of fixed order. Our algorithms are based on reductions to rectangular matrix multiplication. We also describe an improved algorithm for diamonds detection.

Carathéodory bounds for integer cones

We provide analogues of Caratheodorys theorem for integer cones and apply our bounds to integer programming and to the cutting stock problem. In particular, we provide an NP certificate for the latter, whose existence has not been known so far.

The stable set polytope of quasi-line graphs

It is a long standing open problem to find an explicit description of the stable set polytope of claw-free graphs. Yet more than 20 years after the discovery of a polynomial algorithm for the maximum stable set problem for claw-free graphs, there is even no conjecture at hand today.Such a conjecture exists for the class of quasi-line graphs. This class of graphs is a proper superclass of line graphs and a proper subclass of claw-free graphs for which it is known that not all facets have 0/1 normal vectors. The Ben Rebea conjecture states that the stable set polytope of a quasi-line graph is completely described by clique-family inequalities. Chudnovsky and Seymour recently provided a decomposition result for claw-free graphs and proved that the Ben Rebea conjecture holds, if the quasi-line graph is not a fuzzy circular interval graph.In this paper, we give a proof of the Ben Rebea conjecture by showing that it also holds for fuzzy circular interval graphs. Our result builds upon an algorithm of Bartholdi, Orlin and Ratliff which is concerned with integer programs defined by circular ones matrices.

Static-Priority Real-Time Scheduling: Response Time Computation Is NP-Hard

We show that response time computation for rate-monotonic,preemptive scheduling of periodic tasks is NP-hard under Turing reductions. More precisely, we show that the response time of a task cannot be approximated within any constant factor, unless P=NP.

Connected facility location via random facility sampling and core detouring

We present a simple randomized algorithmic framework for connected facility location problems. The basic idea is as follows: We run a black-box approximation algorithm for the unconnected facility location problem, randomly sample the clients, and open the facilities serving sampled clients in the approximate solution. Via a novel analytical tool, which we term core detouring, we show that this approach significantly improves over the previously best known approximation ratios for several NP-hard network design problems. For example, we reduce the approximation ratio for the connected facility location problem from 8.55 to 4.00 and for the single-sink rent-or-buy problem from 3.55 to 2.92. The mentioned results can be derandomized at the expense of a slightly worse approximation ratio. The versatility of our framework is demonstrated by devising improved approximation algorithms also for other related problems.

On the Chvátal rank of polytopes in the 0/1 cube

Abstract Given a polytope P⊆ R n , the Chvatal–Gomory procedure computes iteratively the integer hull PI of P. The Chvatal rank of P is the minimal number of iterations needed to obtain PI. It is always finite, but already the Chvatal rank of polytopes in R 2 can be arbitrarily large. In this paper, we study polytopes in the 0/1 cube, which are of particular interest in combinatorial optimization. We show that the Chvatal rank of any polytope P⊆[0,1]n is O (n 3 log n) and prove the linear upper and lower bound n for the case P∩ Z n =∅ .

New Approaches for Virtual Private Network Design

Virtual private network design is the following NP-hard problem. We are given a communication network represented as a weighted graph with thresholds on the nodes which represent the amount of flow that a node can send to and receive from the network. The task is to reserve capacities at minimum cost and to specify paths between every ordered pair of nodes such that all valid traffic-matrices can be routed along the corresponding paths. Recently, this network design problem has received considerable attention in the literature. It is motivated by the fact that the exact amount of flow which is exchanged between terminals is not known in advance and prediction is often elusive. The main contributions of this paper are as follows: (1) Using Hus 2-commodity flow theorem, we provide a new and considerably stronger lower bound on the cost of an optimum solution. With this lower bound we reanalyze a simple routing scheme which has been described in the literature many times, and provide an improved upper bound on its approximation ratio. (2) We present a new randomized approximation algorithm. In contrast to earlier approaches from the literature, the resulting solution does not have tree structure. A combination of our new algorithm with the simple routing scheme yields an expected performance ratio of

Fast integer programming in fixed dimension

3.79

Flow Faster: Efficient Decision Algorithms for Probabilistic Simulations

for virtual private network design. This is a considerable improvement of the previously best known