{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article11052","name":"Approximating All-Pair Bounded-Leg Shortest Path and APSP-AF in Truly-Subcubic Time","abstract":"In the bounded-leg shortest path (BLSP) problem, we are given a weighted graph G with nonnegative edge lengths, and we want to answer queries of the form \"what's the shortest path from u to v, where only edges of length <=L are considered?\". A more general problem is the APSP-AF (all-pair shortest path for all flows) problem, in which each edge has two weights - a length d and a capacity f, and a query asks about the shortest path from u to v where only edges of capacity >= f are considered.\nIn this article we give an O~(n^{(omega+3)\/2}epsilon^{-3\/2}log W) time algorithm to compute a data structure that answers APSP-AF queries in O(log(epsilon^{-1}log (nW))) time and achieves (1+epsilon)-approximation, where omega < 2.373 is the exponent of time complexity of matrix multiplication, W is the upper bound of integer edge lengths, and n is the number of vertices. This is the first truly-subcubic time algorithm for these problems on dense graphs. Our algorithm utilizes the O(n^{(omega+3)\/2}) time max-min product algorithm [Duan and Pettie 2009]. Since the all-pair bottleneck path (APBP) problem, which is equivalent to max-min product, can be seen as all-pair reachability for all flow, our approach indeed shows that these problems are almost equivalent in the approximation sense.","keywords":["Graph Theory","Approximation Algorithms","Combinatorial Optimization"],"author":[{"@type":"Person","name":"Duan, Ran","givenName":"Ran","familyName":"Duan","affiliation":"Institute for Interdisciplinary Information Sciences, Tsinghua University, China","funding":"R. Duan is supported by a China Youth 1000-Talent grant."},{"@type":"Person","name":"Ren, Hanlin","givenName":"Hanlin","familyName":"Ren","affiliation":"Institute for Interdisciplinary Information Sciences, Tsinghua University, China"}],"position":42,"pageStart":"42:1","pageEnd":"42:12","dateCreated":"2018-07-04","datePublished":"2018-07-04","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Duan, Ran","givenName":"Ran","familyName":"Duan","affiliation":"Institute for Interdisciplinary Information Sciences, Tsinghua University, China","funding":"R. Duan is supported by a China Youth 1000-Talent grant."},{"@type":"Person","name":"Ren, Hanlin","givenName":"Hanlin","familyName":"Ren","affiliation":"Institute for Interdisciplinary Information Sciences, Tsinghua University, China"}],"copyrightYear":"2018","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.ICALP.2018.42","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6310","volumeNumber":107,"name":"45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)","dateCreated":"2018-07-04","datePublished":"2018-07-04","editor":[{"@type":"Person","name":"Chatzigiannakis, Ioannis","givenName":"Ioannis","familyName":"Chatzigiannakis"},{"@type":"Person","name":"Kaklamanis, Christos","givenName":"Christos","familyName":"Kaklamanis"},{"@type":"Person","name":"Marx, D\u00e1niel","givenName":"D\u00e1niel","familyName":"Marx"},{"@type":"Person","name":"Sannella, Donald","givenName":"Donald","familyName":"Sannella"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article11052","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6310"}}}