Given as input a directed graph on N vertices and a set of source-destination pairs, we study the problem of routing the maximum possible number of source-destination pairs on paths, such that at most c(N) paths go through any edge. We show that the problem is hard to approximate within an N\u2126(1\/c(N)) factor even when we compare to the optimal solution that routes pairs on edge-disjoint paths, assuming NP doesn\u2019t have NO(log log N)-time randomized algorithms. Here the congestion c(N) can be any function in the range 1 6 c(N) 6 \u03b1logN\/loglogN for some absolute constant \u03b1>0. The hardness result is in the right ballpark since a factor NO(1\/c(N)) approximation algorithm is known for this problem, via rounding a natural multicommodity-\ufb02ow relaxation. We also give a simple integrality gap construction that shows that the multicommodity-\ufb02ow relaxation has an integrality gap of N\u2126(1\/c) for c ranging from 1 to \u0398( log n log log n).<\/p>\n","protected":false},"excerpt":{"rendered":"
Given as input a directed graph on N vertices and a set of source-destination pairs, we study the problem of routing the maximum possible number of source-destination pairs on paths, such that at most c(N) paths go through any edge. We show that the problem is hard to approximate within an N\u2126(1\/c(N)) factor even when […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"Association for Computing Machinery, Inc.","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"39th ACM Symposium on Theory of Computing (STOC)","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"","msr_page_range_start":"","msr_page_range_end":"","msr_series":"","msr_volume":"","msr_copyright":"Copyright \u00a9 2007 by the Association for Computing Machinery, Inc. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, or to redistribute to lists, requires prior specific permission and\/or a fee. Request permissions from Publications Dept, ACM Inc., fax +1 (212) 869-0481, or permissions@acm.org. 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