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Heggernes P.,University of Bergen | Van 'T Hof P.,University of Bergen | Van Leeuwen E.J.,MPI fur Informatik | Saei R.,University of Bergen
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) | Year: 2014

The well-known Disjoint Paths problem takes as input a graph G and a set of k pairs of terminals in G, and the task is to decide whether there exists a collection of k pairwise vertex-disjoint paths in G such that the vertices in each terminal pair are connected to each other by one of the paths. This problem is known to NP-complete, even when restricted to planar graphs or interval graphs. Moreover, although the problem is fixed-parameter tractable when parameterized by k due to a celebrated result by Robertson and Seymour, it is known not to admit a polynomial kernel unless NP † coNP/poly. We prove that Disjoint Paths remains NP-complete on split graphs, and show that the problem admits a kernel with O(k 2) vertices when restricted to this graph class. We furthermore prove that, on split graphs, the edge-disjoint variant of the problem is also NP-complete and admits a kernel with O(k 3) vertices. To the best of our knowledge, our kernelization results are the first non-trivial kernelization results for these problems on graph classes. © 2014 Springer International Publishing Switzerland.


Knauer C.,University of Bayreuth | Schlipf L.,Free University of Berlin | Schmidt J.M.,MPI fur Informatik | Tiwary H.R.,Free University of Colombia
Journal of Discrete Algorithms | Year: 2012

We consider approximation algorithms for the problem of computing an inscribed rectangle having largest area in a convex polygon on n vertices. If the order of the vertices of the polygon is given, we present a randomized algorithm that computes an inscribed rectangle with area at least (1/ε) times the optimum with probability t in time O(1εlogn) for any constant t<1. We further give a deterministic approximation algorithm that computes an inscribed rectangle of area at least (1-ε) times the optimum in running time O(1 ε2logn) and show how this running time can be slightly improved. © 2012 Elsevier B.V. All rights reserved.


Baumgartner P.,Australian National University | Waldmann U.,MPI fur Informatik
Journal of Automated Reasoning | Year: 2011

We present a new calculus for first-order theorem proving with equality, Mσ+Sup, which generalizes both the Superposition calculus and the Model Evolution calculus (with equality) by integrating their inference rules and redundancy criteria in a non-trivial way. The main motivation is to combine the advantageous features of these two rather complementary calculi in a single framework. In particular, Model Evolution, as a lifted version of the propositional DPLL procedure, contributes a non-ground splitting rule that effectively permits to split a clause into non variable disjoint subclauses. In the paper we present the calculus in detail. Our main result is its completeness under semantically justified redundancy criteria and simplification rules. We also show how under certain assumptions the model representation computed by a (finite and fair) derivation can be queried in an effective way. © 2010 Springer Science+Business Media B.V.


Klasing R.,French National Center for Scientific Research | Kosowski A.,French Institute for Research in Computer Science and Automation | Pajak D.,French Institute for Research in Computer Science and Automation | Sauerwald T.,MPI fur Informatik
Proceedings of the Annual ACM Symposium on Principles of Distributed Computing | Year: 2013

The rotor-router mechanism was introduced as a deterministic alternative to the random walk in undirected graphs. In this model, an agent is initially placed at one of the nodes of the graph. Each node maintains a cyclic ordering of its outgoing arcs, and during successive visits of the agent, propagates it along arcs chosen according to this ordering in round-robin fashion. In this work we consider the setting in which multiple, indistinguishable agents are deployed in parallel in the nodes of the graph, and move around the graph in synchronous rounds, interacting with a single rotor-router system. We propose new techniques which allow us to perform a theoretical analysis of the multi-agent rotor-router model, and to compare it to the scenario of parallel independent random walks in a graph. Our main results concern the n-node ring, and suggest a strong similarity between the performance characteristics of this deterministic model and random walks. We show that on the ring the rotor-router with k agents admits a cover time of between θ(n2/k 2) in the best case and θ(n2/log k) in the worst case, depending on the initial locations of the agents, and that both these bounds are tight. The corresponding expected value of cover time for k random walks, depending on the initial locations of the walkers, is proven to belong to a similar range, namely between θ(n2/(k2/log 2 k)) and θ(n2/log k). Finally, we study the limit behavior of the rotor-router system. We show that, once the rotor-router system has stabilized, all the nodes of the ring are always visited by some agent every θ(n/k) steps, regardless of how the system was initialized. This asymptotic bound corresponds to the expected time between successive visits to a node in the case of k random walks. All our results hold up to a polynomially large number of agents (1 ≤ k < n1/11). Copyright 2013 ACM.


Baumgartner P.,Australian National University | Waldmann U.,MPI fur Informatik
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) | Year: 2013

Many applications of automated deduction require reasoning in first-order logic modulo background theories, in particular some form of integer arithmetic. A major unsolved research challenge is to design theorem provers that are "reasonably complete" even in the presence of free function symbols ranging into a background theory sort. The hierarchic superposition calculus of Bachmair, Ganzinger, and Waldmann already supports such symbols, but, as we demonstrate, not optimally. This paper aims to rectify the situation by introducing a novel form of clause abstraction, a core component in the hierarchic superposition calculus for transforming clauses into a form needed for internal operation. We argue for the benefits of the resulting calculus and provide a new completeness result for the fragment where all background-sorted terms are ground. © 2013 Springer-Verlag.

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