Conditional Lower Bounds for All-Pairs Max-Flow
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.
Specifically, 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.
In 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.
Conditional lower bounds
Hardness in P
All-Pairs Maximum Flow
Strong Exponential Time Hypothesis
20:1-20:13
Regular Paper
Robert
Krauthgamer
Robert Krauthgamer
Ohad
Trabelsi
Ohad Trabelsi
10.4230/LIPIcs.ICALP.2017.20
Amir Abboud, Karl Bringmann, Danny Hermelin, and Dvir Shabtay. SETH-based lower bounds for subset sum and bicriteria path. CoRR, 2017. URL: http://arxiv.org/abs/1704.04546.
http://arxiv.org/abs/1704.04546
Amir Abboud, Virginia Vassilevska-Williams, and Huacheng Yu. Matching triangles and basing hardness on an extremely popular conjecture. In Proceedings of the Forty-seventh Annual ACM Symposium on Theory of Computing, STOC'15, pages 41-50. ACM, 2015. URL: http://dx.doi.org/10.1145/2746539.2746594.
http://dx.doi.org/10.1145/2746539.2746594
Ravindra K. Ahuja, Thomas L. Magnanti, and James B. Orlin. Network flows - theory, algorithms and applications. Prentice Hall, 1993.
Srinivasa R. Arikati, Shiva Chaudhuri, and Christos D. Zaroliagis. All-pairs min-cut in sparse networks. J. Algorithms, 29(1):82-110, 1998. URL: http://dx.doi.org/10.1006/jagm.1998.0961.
http://dx.doi.org/10.1006/jagm.1998.0961
Sanjeev Arora, Elad Hazan, and Satyen Kale. The multiplicative weights update method: a meta-algorithm and applications. Theory of Computing, 8(1):121-164, 2012. URL: http://dx.doi.org/10.4086/toc.2012.v008a006.
http://dx.doi.org/10.4086/toc.2012.v008a006
Mokhtar S. Bazaraa, John J. Jarvis, and Hanif D. Sherali. Linear Programming and Network Flows. John Wiley &Sons, Inc., Hoboken, NJ, fourth edition, 2010.
Anand Bhalgat, Ramesh Hariharan, Telikepalli Kavitha, and Debmalya Panigrahi. An Õ(mn) Gomory-Hu tree construction algorithm for unweighted graphs. In 39th Annual ACM Symposium on Theory of Computing, STOC'07, pages 605-614. ACM, 2007. URL: http://dx.doi.org/10.1145/1250790.1250879.
http://dx.doi.org/10.1145/1250790.1250879
Glencora Borradaile, David Eppstein, Amir Nayyeri, and Christian Wulff-Nilsen. All-pairs minimum cuts in near-linear time for surface-embedded graphs. In 32nd International Symposium on Computational Geometry (SoCG 2016), volume 51 of Leibniz International Proceedings in Informatics (LIPIcs), pages 22:1-22:16. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.SoCG.2016.22.
http://dx.doi.org/10.4230/LIPIcs.SoCG.2016.22
Glencora Borradaile and Philip Klein. An Õ(nlog n) algorithm for maximum st-flow in a directed planar graph. J. ACM, 56(2):9:1-9:30, 2009. URL: http://dx.doi.org/10.1145/1502793.1502798.
http://dx.doi.org/10.1145/1502793.1502798
Marco L. Carmosino, Jiawei Gao, Russell Impagliazzo, Ivan Mihajlin, Ramamohan Paturi, and Stefan Schneider. Nondeterministic extensions of the strong exponential time hypothesis and consequences for non-reducibility. In Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science, ITCS'16, pages 261-270. ACM, 2016. URL: http://dx.doi.org/10.1145/2840728.2840746.
http://dx.doi.org/10.1145/2840728.2840746
Ho Yee Cheung, Lap Chi Lau, and Kai Man Leung. Graph connectivities, network coding, and expander graphs. In Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS'11, pages 190-199. IEEE Computer Society, 2011. URL: http://dx.doi.org/10.1109/FOCS.2011.55.
http://dx.doi.org/10.1109/FOCS.2011.55
L. R. Ford, Jr. and D. R. Fulkerson. Maximal flow through a network. Canadian Journal of Mathematics, 8:399-404, 1956. URL: http://dx.doi.org/10.4153/CJM-1956-045-5.
http://dx.doi.org/10.4153/CJM-1956-045-5
Greg N. Frederickson. Using cellular graph embeddings in solving all pairs shortest paths problems. J. Algorithms, 19(1):45-85, 1995. URL: http://dx.doi.org/10.1006/jagm.1995.1027.
http://dx.doi.org/10.1006/jagm.1995.1027
R. E. Gomory and T. C. Hu. Multi-terminal network flows. Journal of the Society for Industrial and Applied Mathematics, 9:551-570, 1961. URL: http://dx.doi.org/10.1137/0109047.
http://dx.doi.org/10.1137/0109047
Dan Gusfield. Very simple methods for all pairs network flow analysis. SIAM J. Comput., 19(1):143-155, 1990. URL: http://dx.doi.org/10.1137/0219009.
http://dx.doi.org/10.1137/0219009
Jianxiu Hao and James B. Orlin. A faster algorithm for finding the minimum cut in a directed graph. J. Algorithms, 17(3):424-446, 1994. URL: http://dx.doi.org/10.1006/jagm.1994.1043.
http://dx.doi.org/10.1006/jagm.1994.1043
Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-SAT. Journal of Computer and System Sciences, 62(2):367-375, 2001. URL: http://dx.doi.org/10.1006/jcss.2000.1727.
http://dx.doi.org/10.1006/jcss.2000.1727
Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? Journal of Computer and System Sciences, 63(4):512-530, 2001. URL: http://dx.doi.org/10.1006/jcss.2001.1774.
http://dx.doi.org/10.1006/jcss.2001.1774
Jakub Lacki, Yahav Nussbaum, Piotr Sankowski, and Christian Wulff-Nilsen. Single source - all sinks max flows in planar digraphs. In Proceedings of the 53rd IEEE Annual Symposium on Foundations of Computer Science, FOCS'12, pages 599-608. IEEE Computer Society, 2012. URL: http://dx.doi.org/10.1109/FOCS.2012.66.
http://dx.doi.org/10.1109/FOCS.2012.66
Yin Tat Lee and Aaron Sidford. Path finding methods for linear programming: Solving linear programs in Õ(√rank) iterations and faster algorithms for maximum flow. In Proceedings of the 2014 IEEE 55th Annual Symposium on Foundations of Computer Science, FOCS'14, pages 424-433. IEEE Computer Society, 2014. URL: http://dx.doi.org/10.1109/FOCS.2014.52.
http://dx.doi.org/10.1109/FOCS.2014.52
Aleksander Mądry. Computing maximum flow with augmenting electrical flows. In Proceedings of the 57th IEEE Annual Symposium on Foundations of Computer Science, FOCS'16, pages 593-602. IEEE Computer Society, 2016. URL: http://dx.doi.org/10.1109/FOCS.2016.70.
http://dx.doi.org/10.1109/FOCS.2016.70
Alexander Schrijver. On the history of the transportation and maximum flow problems. Math. Program., 91(3):437-445, 2002. URL: http://dx.doi.org/10.1007/s101070100259.
http://dx.doi.org/10.1007/s101070100259
Virginia Vassilevska-Williams. Hardness of Easy Problems: Basing Hardness on Popular Conjectures such as the Strong Exponential Time Hypothesis (Invited Talk). In 10th International Symposium on Parameterized and Exact Computation (IPEC 2015), volume 43 of Leibniz International Proceedings in Informatics (LIPIcs), pages 17-29. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.IPEC.2015.17.
http://dx.doi.org/10.4230/LIPIcs.IPEC.2015.17
Ryan Williams. A new algorithm for optimal 2-constraint satisfaction and its implications. Theoretical Computer Science, 348(2):357-365, 2005. URL: http://dx.doi.org/10.1016/j.tcs.2005.09.023.
http://dx.doi.org/10.1016/j.tcs.2005.09.023
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode