TY - GEN
T1 - The Karger-Stein algorithm is optimal for k-cut
AU - Gupta, Anupam
AU - Lee, Euiwoong
AU - Li, Jason
N1 - Publisher Copyright:
© 2020 ACM.
PY - 2020/6/8
Y1 - 2020/6/8
N2 - In the k-cut problem, we are given an edge-weighted graph and want to find the least-weight set of edges whose deletion breaks the graph into k connected components. Algorithms due to Karger-Stein and Thorup showed how to find such a minimum k-cut in time approximately O(n2k-2). The best lower bounds come from conjectures about the solvability of the k-clique problem and a reduction from k-clique to k-cut, and show that solving k-cut is likely to require time ω(nk). Our recent results have given special-purpose algorithms that solve the problem in time n1.98k + O(1), and ones that have better performance for special classes of graphs (e.g., for small integer weights). In this work, we resolve the problem for general graphs, by showing that for any fixed k ≥ 2, the Karger-Stein algorithm outputs any fixed minimum k-cut with probability at least O(n-k), where O(·) hides a 2O(lnlnn)2 factor. This also gives an extremal bound of O(nk) on the number of minimum k-cuts in an n-vertex graph and an algorithm to compute a minimum k-cut in similar runtime. Both are tight up to O(1) factors. The first main ingredient in our result is a fine-grained analysis of how the graph shrinks - and how the average degree evolves - under the Karger-Stein process. The second ingredient is an extremal result bounding the number of cuts of size at most (2-) OPT/k, using the Sunflower lemma.
AB - In the k-cut problem, we are given an edge-weighted graph and want to find the least-weight set of edges whose deletion breaks the graph into k connected components. Algorithms due to Karger-Stein and Thorup showed how to find such a minimum k-cut in time approximately O(n2k-2). The best lower bounds come from conjectures about the solvability of the k-clique problem and a reduction from k-clique to k-cut, and show that solving k-cut is likely to require time ω(nk). Our recent results have given special-purpose algorithms that solve the problem in time n1.98k + O(1), and ones that have better performance for special classes of graphs (e.g., for small integer weights). In this work, we resolve the problem for general graphs, by showing that for any fixed k ≥ 2, the Karger-Stein algorithm outputs any fixed minimum k-cut with probability at least O(n-k), where O(·) hides a 2O(lnlnn)2 factor. This also gives an extremal bound of O(nk) on the number of minimum k-cuts in an n-vertex graph and an algorithm to compute a minimum k-cut in similar runtime. Both are tight up to O(1) factors. The first main ingredient in our result is a fine-grained analysis of how the graph shrinks - and how the average degree evolves - under the Karger-Stein process. The second ingredient is an extremal result bounding the number of cuts of size at most (2-) OPT/k, using the Sunflower lemma.
KW - Graph Algorithms
KW - Minimum Cut
KW - Randomized Algorithms
UR - http://www.scopus.com/inward/record.url?scp=85086755987&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85086755987&partnerID=8YFLogxK
U2 - 10.1145/3357713.3384285
DO - 10.1145/3357713.3384285
M3 - Conference contribution
AN - SCOPUS:85086755987
T3 - Proceedings of the Annual ACM Symposium on Theory of Computing
SP - 473
EP - 484
BT - STOC 2020 - Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing
A2 - Makarychev, Konstantin
A2 - Makarychev, Yury
A2 - Tulsiani, Madhur
A2 - Kamath, Gautam
A2 - Chuzhoy, Julia
PB - Association for Computing Machinery
T2 - 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020
Y2 - 22 June 2020 through 26 June 2020
ER -