Submodular Cost Allocation Problem and Applications

Abstract

We study the Minimum Submodular-Cost Allocation problem (MSCA). In this problem we are given a finite ground set V and k non-negative submodular set functions f 1 , . . . , f k on V . The objective is to partition V into k (possibly empty) sets A 1 , · · · , A k such that the sum k i=1 f i (A i ) is minimized. Several well-studied problems such as the non-metric facility location problem, multiway-cut in graphs and hypergraphs, and uniform metric labeling and its generalizations can be shown to be special cases of MSCA. In this paper we consider a convex-programming relaxation obtained via the Lovász-extension for submodular functions. This allows us to understand several previous relaxations and rounding procedures in a unified fashion and also develop new formulations and approximation algorithms for several problems. In particular, we give a (1.5 − 1/k)-approximation for the hypergraph multiway partition problem. We also give a min{2(1−1/k), H ∆ }-approximation for the hypergraph multiway cut problem when ∆ is the maximum hyperedge size. Both problems generalize the multiway cut problem in graphs and the hypergraph cut problem is approximation equivalent to the node-weighted multiway cut problem in graphs.

From Graph to Hypergraph Multiway Partition: Is the Single Threshold the Only Route?

“…Given the obstacles mentioned above, in this paper we focus on bridging the gap between the graph setting (c = 2) and the 3-uniform hypergraph setting (c = 3). Our main result is a 4/3 approximation for the Hyper-MP problem on 3-uniform hypergraphs, which is the first improvement over the (1.5 − 1/k) approximation of [5]. We remark that the result immediately extends to the setting in which each hyperedge has at most 3 vertices (instead of exactly 3 vertices).…”

“…Our main result is a 4/3 approximation for the Hyper-MP problem on 3-uniform hypergraphs, which is the first improvement over the (1.5−1/k) approximation of [5]. The algorithm combines the single-threshold rounding strategy of Calinescu et al [3] with the rounding strategy of Kleinberg and Tardos [8], and it parallels the recent algorithm of Buchbinder et al [2] for the Graph Multiway Cut problem, which is a special case.…”

“…It was only recently that Chekuri and Ene [5] gave a (1.5 − 1/k) approximation for the Hyper-MP problem, improving on a previous (2 − 2/k) approximation [11]. The algorithm of [5] uses a single-threshold scheme to round a fractional solution to the CKR relaxation and it achieves the best approximation known for the problem.…”

“…It was only recently that Chekuri and Ene [5] gave a (1.5 − 1/k) approximation for the Hyper-MP problem, improving on a previous (2 − 2/k) approximation [11]. The algorithm of [5] uses a single-threshold scheme to round a fractional solution to the CKR relaxation and it achieves the best approximation known for the problem. An interesting open question, which is the main motivation behind this work, is whether one can improve the (1.5 − 1/k) factor by using other rounding schemes, and in particular the strategies underpinning the Graph Multiway Cut algorithms.…”

“…Unfortunately, it is unclear how to extend these simplifying assumptions to the hypergraph setting (in the graph setting, they can be easily achieved by subdividing the edges). The work of [5] provides an analysis of the single-threshold rounding scheme without any assumptions on the fractional solution, but this seems challenging for the other rounding schemes even for 3-uniform 2 hypergraphs.…”