TY - GEN
T1 - Breaking the Cubic Barrier for All-Pairs Max-Flow
T2 - 63rd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2022
AU - Abboud, Amir
AU - Krauthgamer, Robert
AU - Li, Jason
AU - Panigrahi, Debmalya
AU - Saranurak, Thatchaphol
AU - Trabelsi, Ohad
N1 - Publisher Copyright:
© 2022 IEEE.
PY - 2022/1/1
Y1 - 2022/1/1
N2 - In 1961, Gomory and Hu showed that the All-Pairs Max-Flow problem of computing the max-flow between all (n/2) pairs of vertices in an undirected graph can be solved using only n-1 calls to any (single-pair) max-flow algorithm. Even assuming a linear-time max-flow algorithm, this yields a running time of O(mn), which is O(n3) when m = T(n2). While subsequent work has improved this bound for various special graph classes, no subcubic-time algorithm has been obtained in the last 60 years for general graphs. We break this longstanding barrier by giving an O(n2)-time algorithm on general, integer-weighted graphs. Combined with a popular complexity assumption, we establish a counter-intuitive separation: all-pairs max-flows are strictly easier to compute than all-pairs shortest-paths.Our algorithm produces a cut-equivalent tree, known as the Gomory-Hu tree, from which the max-flow value for any pair can be retrieved in near-constant time. For unweighted graphs, we refine our techniques further to produce a Gomory-Hu tree in the time of a poly-logarithmic number of calls to any maxflow algorithm. This shows an equivalence between the all-pairs and single-pair max-flow problems, and is optimal up to polylogarithmic factors. Using the recently announced m1+o(1)-time max-flow algorithm (Chen et al., March 2022), our Gomory-Hu tree algorithm for unweighted graphs also runs in m1+o(1)-time.
AB - In 1961, Gomory and Hu showed that the All-Pairs Max-Flow problem of computing the max-flow between all (n/2) pairs of vertices in an undirected graph can be solved using only n-1 calls to any (single-pair) max-flow algorithm. Even assuming a linear-time max-flow algorithm, this yields a running time of O(mn), which is O(n3) when m = T(n2). While subsequent work has improved this bound for various special graph classes, no subcubic-time algorithm has been obtained in the last 60 years for general graphs. We break this longstanding barrier by giving an O(n2)-time algorithm on general, integer-weighted graphs. Combined with a popular complexity assumption, we establish a counter-intuitive separation: all-pairs max-flows are strictly easier to compute than all-pairs shortest-paths.Our algorithm produces a cut-equivalent tree, known as the Gomory-Hu tree, from which the max-flow value for any pair can be retrieved in near-constant time. For unweighted graphs, we refine our techniques further to produce a Gomory-Hu tree in the time of a poly-logarithmic number of calls to any maxflow algorithm. This shows an equivalence between the all-pairs and single-pair max-flow problems, and is optimal up to polylogarithmic factors. Using the recently announced m1+o(1)-time max-flow algorithm (Chen et al., March 2022), our Gomory-Hu tree algorithm for unweighted graphs also runs in m1+o(1)-time.
KW - Gomory-Hu tree
KW - graph algorithms
KW - maximum flow
KW - minimum cut
UR - http://www.scopus.com/inward/record.url?scp=85146326024&partnerID=8YFLogxK
U2 - 10.1109/FOCS54457.2022.00088
DO - 10.1109/FOCS54457.2022.00088
M3 - Conference contribution
AN - SCOPUS:85146326024
T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
SP - 884
EP - 895
BT - Proceedings - 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science, FOCS 2022
PB - Institute of Electrical and Electronics Engineers
Y2 - 31 October 2022 through 3 November 2022
ER -