{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article9800","name":"Conditional Lower Bounds for All-Pairs Max-Flow","abstract":"We provide evidence that computing the maximum flow value between every pair of nodes in a directed graph on n nodes, m edges, and capacities in the range [1..n], which we call the All-Pairs Max-Flow problem, cannot be solved in time that is faster significantly (i.e., by a polynomial factor) than O(n^2 m). Since a single maximum st-flow in such graphs can be solved in time \\tilde{O}(m\\sqrt{n}) [Lee and Sidford, FOCS 2014], we conclude that the all-pairs version might require time equivalent to \\tilde\\Omega(n^{3\/2}) computations of maximum st-flow, which strongly separates the directed case from the undirected one. Moreover, if maximum $st$-flow can be solved in time \\tilde{O}(m), then the runtime of \\tilde\\Omega(n^2) computations is needed. This is in contrast to a conjecture of Lacki, Nussbaum, Sankowski, and Wulf-Nilsen [FOCS 2012] that All-Pairs Max-Flow in general graphs can be solved faster than the time of O(n^2) computations of maximum st-flow.\r\n\r\nSpecifically, we show that in sparse graphs G=(V,E,w), if one can compute the maximum st-flow from every s in an input set of sources S\\subseteq V to every t in an input set of sinks T\\subseteq V in time O((|S||T|m)^{1-epsilon}), for some |S|, |T|, and a constant epsilon>0, then MAX-CNF-SAT (maximum satisfiability of conjunctive normal form formulas) with n' variables and m' clauses can be solved in time {m'}^{O(1)}2^{(1-delta)n'} for a constant delta(epsilon)>0, a problem for which not even 2^{n'}\/\\poly(n') algorithms are known. Such runtime for MAX-CNF-SAT would in particular refute the Strong Exponential Time Hypothesis (SETH). Hence, we improve the lower bound of Abboud, Vassilevska-Williams, and Yu [STOC 2015], who showed that for every fixed epsilon>0 and |S|=|T|=O(\\sqrt{n}), if the above problem can be solved in time O(n^{3\/2-epsilon}), then some incomparable (and intuitively weaker) conjecture is false. Furthermore, a larger lower bound than ours implies strictly super-linear time for maximum st-flow problem, which would be an amazing breakthrough.\r\n\r\nIn addition, we show that All-Pairs Max-Flow in uncapacitated networks with every edge-density m=m(n), cannot be computed in time significantly faster than O(mn), even for acyclic networks. The gap to the fastest known algorithm by Cheung, Lau, and Leung [FOCS 2011] is a factor of O(m^{omega-1}\/n), and for acyclic networks it is O(n^{omega-1}), where omega is the matrix multiplication exponent.","keywords":["Conditional lower bounds","Hardness in P","All-Pairs Maximum Flow","Strong Exponential Time Hypothesis"],"author":[{"@type":"Person","name":"Krauthgamer, Robert","givenName":"Robert","familyName":"Krauthgamer"},{"@type":"Person","name":"Trabelsi, Ohad","givenName":"Ohad","familyName":"Trabelsi"}],"position":20,"pageStart":"20:1","pageEnd":"20:13","dateCreated":"2017-07-07","datePublished":"2017-07-07","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Krauthgamer, Robert","givenName":"Robert","familyName":"Krauthgamer"},{"@type":"Person","name":"Trabelsi, Ohad","givenName":"Ohad","familyName":"Trabelsi"}],"copyrightYear":"2017","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.ICALP.2017.20","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["http:\/\/arxiv.org\/abs\/1704.04546","http:\/\/dx.doi.org\/10.1145\/2746539.2746594","http:\/\/dx.doi.org\/10.1006\/jagm.1998.0961","http:\/\/dx.doi.org\/10.4086\/toc.2012.v008a006","http:\/\/dx.doi.org\/10.1145\/1250790.1250879","http:\/\/dx.doi.org\/10.4230\/LIPIcs.SoCG.2016.22","http:\/\/dx.doi.org\/10.1145\/1502793.1502798","http:\/\/dx.doi.org\/10.1145\/2840728.2840746","http:\/\/dx.doi.org\/10.1109\/FOCS.2011.55","http:\/\/dx.doi.org\/10.4153\/CJM-1956-045-5","http:\/\/dx.doi.org\/10.1006\/jagm.1995.1027","http:\/\/dx.doi.org\/10.1137\/0109047","http:\/\/dx.doi.org\/10.1137\/0219009","http:\/\/dx.doi.org\/10.1006\/jagm.1994.1043","http:\/\/dx.doi.org\/10.1006\/jcss.2000.1727","http:\/\/dx.doi.org\/10.1006\/jcss.2001.1774","http:\/\/dx.doi.org\/10.1109\/FOCS.2012.66","http:\/\/dx.doi.org\/10.1109\/FOCS.2014.52","http:\/\/dx.doi.org\/10.1109\/FOCS.2016.70","http:\/\/dx.doi.org\/10.1007\/s101070100259","http:\/\/dx.doi.org\/10.4230\/LIPIcs.IPEC.2015.17","http:\/\/dx.doi.org\/10.1016\/j.tcs.2005.09.023"],"isPartOf":{"@type":"PublicationVolume","@id":"#volume6283","volumeNumber":80,"name":"44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)","dateCreated":"2017-07-07","datePublished":"2017-07-07","editor":[{"@type":"Person","name":"Chatzigiannakis, Ioannis","givenName":"Ioannis","familyName":"Chatzigiannakis"},{"@type":"Person","name":"Indyk, Piotr","givenName":"Piotr","familyName":"Indyk"},{"@type":"Person","name":"Kuhn, Fabian","givenName":"Fabian","familyName":"Kuhn"},{"@type":"Person","name":"Muscholl, Anca","givenName":"Anca","familyName":"Muscholl"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article9800","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":"#volume6283"}}}