TY - GEN

T1 - Hypergraph-cut for fixed in deterministic polynomial time

AU - Chandrasekaran, Karthekeyan

AU - Chekuri, Chandra Sekhar

N1 - Publisher Copyright:
© 2020 IEEE.
Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2020/11

Y1 - 2020/11

N2 - We consider the Hypergraph-k-Cut problem. The input consists of a hypergraph G= (V, E) with nonnegative hyperedge-costs c:E rightarrow mathbb{R}+ and a positive integer k. The objective is to find a least-cost subset F subseteq E such that the number of connected components in G-F is at least k. An alternative formulation of the objective is to find a partition of V into k non-empty sets V {1}, V {2}, ldots, V {k} so as to minimize the cost of the hyperedges that cross the partition. Graph-k-Cut, the special case of Hypergraph-k-Cut obtained by restricting to graph inputs, has received considerable attention. Several different approaches lead to a polynomial-time algorithm for Graph-k-Cut when k is fixed, starting with the work of Goldschmidt and Hochbaum (1988) [1], [2]. In contrast, it is only recently that a randomized polynomial time algorithm for Hypergraph-k-Cut was developed [3] via a subtle generalization of Karger's random contraction approach for graphs. In this work, we develop the first deterministic polynomial time algorithm for Hypergraph-k-Cut for all fixed k. We describe two algorithms both of which are based on a divide and conquer approach. The first algorithm is simpler and runs in n{O(k{2})} time while the second one runs in n{O(k)} time. Our proof relies on new structural results that allow for efficient recovery of the parts of an optimum k-partition by solving minimum (S, T)-terminal cuts. Our techniques give new insights even for Graph-k-Cut.

AB - We consider the Hypergraph-k-Cut problem. The input consists of a hypergraph G= (V, E) with nonnegative hyperedge-costs c:E rightarrow mathbb{R}+ and a positive integer k. The objective is to find a least-cost subset F subseteq E such that the number of connected components in G-F is at least k. An alternative formulation of the objective is to find a partition of V into k non-empty sets V {1}, V {2}, ldots, V {k} so as to minimize the cost of the hyperedges that cross the partition. Graph-k-Cut, the special case of Hypergraph-k-Cut obtained by restricting to graph inputs, has received considerable attention. Several different approaches lead to a polynomial-time algorithm for Graph-k-Cut when k is fixed, starting with the work of Goldschmidt and Hochbaum (1988) [1], [2]. In contrast, it is only recently that a randomized polynomial time algorithm for Hypergraph-k-Cut was developed [3] via a subtle generalization of Karger's random contraction approach for graphs. In this work, we develop the first deterministic polynomial time algorithm for Hypergraph-k-Cut for all fixed k. We describe two algorithms both of which are based on a divide and conquer approach. The first algorithm is simpler and runs in n{O(k{2})} time while the second one runs in n{O(k)} time. Our proof relies on new structural results that allow for efficient recovery of the parts of an optimum k-partition by solving minimum (S, T)-terminal cuts. Our techniques give new insights even for Graph-k-Cut.

KW - algorithms

KW - hypergraphs

KW - partition

UR - http://www.scopus.com/inward/record.url?scp=85100339118&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85100339118&partnerID=8YFLogxK

U2 - 10.1109/FOCS46700.2020.00080

DO - 10.1109/FOCS46700.2020.00080

M3 - Conference contribution

AN - SCOPUS:85100339118

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 810

EP - 821

BT - Proceedings - 2020 IEEE 61st Annual Symposium on Foundations of Computer Science, FOCS 2020

PB - IEEE Computer Society

T2 - 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020

Y2 - 16 November 2020 through 19 November 2020

ER -