Multiway cut, pairwise realizable distributions, and descending thresholds

Abstract

We design new approximation algorithms for the Multiway Cut problem, improving the previously known factor of 1.32388 [Buchbinder et al., 2013].We proceed in three steps. First, we analyze the rounding scheme of Buchbinder et al. [2013] and design a modification that improves the approximation to 3+ √ 5 4 ≈ 1.309017. We also present a tight example showing that this is the best approximation one can achieve with the type of cuts considered by Buchbinder et al. [2013]: (1) partitioning by exponential clocks, and (2) single-coordinate cuts with equal thresholds.Then, we prove that this factor can be improved by introducing a new rounding scheme: (3) singlecoordinate cuts with descending thresholds. By combining these three schemes, we design an algorithm that achieves a factor of 10+4 √ 3 13 ≈ 1.30217. This is the best approximation factor that we are able to verify by hand.Finally, we show that by combining these three rounding schemes with the scheme of independent thresholds from Karger et al. [2004], the approximation factor can be further improved to 1.2965. This approximation factor has been verified only by computer.

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

“…As we have already mentioned, the Graph Multiway Cut problem and its generalizations to hypergraphs and submodular functions have been studied extensively over the past two decades. We omit a detailed discussion of these results and we refer the reader to [10,5,4] for additional pointers and references.…”

“…In the following, we let x be a fractional solution to the LP relaxation given in Section 2. The rounding strategies that we use have been studied in previous work for the Graph Multiway Cut and Uniform Metric Labeling problems [2,10,8], and they are given in Figure 2. We devote the rest of this section to the analysis of Algorithm 3.…”

“…In a breakthrough result, Calinescu, Karloff, and Rabani [3] obtained an (1.5 − 1/k) approximation via a novel geometric relaxation. Since the work of [3], the approximation factor has been improved in a series of papers [7,2,10], culminating with the 1.30217 approximation of [10]. All of these improvements use the CKR relaxation as a starting point and they combine the single-threshold rounding strategy of [3] with other rounding schemes.…”

Simplex Transformations and the Multiway Cut Problem

“…Using this framework, Sharma and Vondrák [30] presented an improved algorithm that achieves an approximation of 5 -approximation. To further improve the approximation factor, Sharma and Vondrák [30] introduced a third ingredient to the mix of algorithms: the descending threshold algorithm.…”

“…To further improve the approximation factor, Sharma and Vondrák [30] introduced a third ingredient to the mix of algorithms: the descending threshold algorithm. This resulted in an improved approximation of 3) ≈ 1.30217.…”