TY - JOUR

T1 - Lower bounds for on-line graph problems with application to on-line circuit and optical routing

AU - Bartal, Yair

AU - Fiat, Amos

AU - Leonardi, Stefano

PY - 2006

Y1 - 2006

N2 - We present lower bounds on the competitive ratio of randomized algorithms for a wide class of on-line graph optimization problems, and we apply such results to on-line virtual circuit and optical routing problems. Lund and Yannakakis [The approximation of maximum subgraph problems, in Proceedings of the 20th International Colloquium on Automata, Languages and Programming, 1993, pp. 40-51] give inapproximability results for the problem of finding the largest vertex induced subgraph satisfying any nontrivial, hereditary property π - e.g., independent set, planar, acyclic, bipartite. We consider the on-line version of this family of problems, where some graph G is fixed and some subgraph H of G is presented on-line, vertex by vertex. The on-line algorithm must choose a subset of the vertices of H, choosing or rejecting a vertex when it is presented, whose vertex induced subgraph satisfies property π. Furthermore, we study the on-line version of graph coloring whose offline version has also been shown to be inapproximable [C. Lund and M. Yannakakis, On the hardness of approximating minimization problems, in Proceedings of the 25th ACM Symposium on Theory of Computing, 1993], on-line max edge-disjoint paths, and on-line path coloring problems. Irrespective of the time complexity, we show an Ω(nε) lower bound on the competitive ratio of randomized on-line algorithms for any of these problems. As a consequence, we obtain an Ω(nε) lower bound on the competitive ratio of randomized on-line algorithms for virtual circuit routing on general networks, in contrast to the known results for some specific networks. Similar lower bounds are obtained for on-line optical routing as well.

AB - We present lower bounds on the competitive ratio of randomized algorithms for a wide class of on-line graph optimization problems, and we apply such results to on-line virtual circuit and optical routing problems. Lund and Yannakakis [The approximation of maximum subgraph problems, in Proceedings of the 20th International Colloquium on Automata, Languages and Programming, 1993, pp. 40-51] give inapproximability results for the problem of finding the largest vertex induced subgraph satisfying any nontrivial, hereditary property π - e.g., independent set, planar, acyclic, bipartite. We consider the on-line version of this family of problems, where some graph G is fixed and some subgraph H of G is presented on-line, vertex by vertex. The on-line algorithm must choose a subset of the vertices of H, choosing or rejecting a vertex when it is presented, whose vertex induced subgraph satisfies property π. Furthermore, we study the on-line version of graph coloring whose offline version has also been shown to be inapproximable [C. Lund and M. Yannakakis, On the hardness of approximating minimization problems, in Proceedings of the 25th ACM Symposium on Theory of Computing, 1993], on-line max edge-disjoint paths, and on-line path coloring problems. Irrespective of the time complexity, we show an Ω(nε) lower bound on the competitive ratio of randomized on-line algorithms for any of these problems. As a consequence, we obtain an Ω(nε) lower bound on the competitive ratio of randomized on-line algorithms for virtual circuit routing on general networks, in contrast to the known results for some specific networks. Similar lower bounds are obtained for on-line optical routing as well.

KW - Competitive analysis

KW - Graph problems

KW - Lower bounds

KW - Network optimization

KW - On-line computation

KW - Randomized algorithms

UR - http://www.scopus.com/inward/record.url?scp=34247185347&partnerID=8YFLogxK

U2 - 10.1137/S009753979833965X

DO - 10.1137/S009753979833965X

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AN - SCOPUS:34247185347

SN - 0097-5397

VL - 36

SP - 354

EP - 393

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

IS - 2

ER -