Improved Approximation Algorithms for Multiway Cut by Large Mixtures of New and Old Rounding Schemes
2026-03-30 • Data Structures and Algorithms
Data Structures and Algorithms
AI summaryⓘ
The authors study the Multiway Cut problem, where the goal is to split a graph into parts each with a specific terminal, minimizing the total edge weights crossing between parts. This problem is hard to solve exactly for three or more terminals, so approximation algorithms are used. The authors present a new algorithm that improves the best known approximation ratio for general numbers of terminals and for small numbers of terminals (k ≥ 4), using novel ways to round solutions from a linear programming relaxation. Their approach involves combining many different rounding techniques found by computational search and rigorously analyzing them with advanced mathematical tools.
Multiway Cut problemapproximation algorithmLP relaxationrounding schemeintegrality ratioUnique Games Conjecturegraph partitioningexponential clocksinterval arithmetic
Authors
Joshua Brakensiek, Neng Huang, Aaron Potechin, Uri Zwick
Abstract
The input to the Multiway Cut problem is a weighted undirected graph, with nonnegative edge weights, and $k$ designated terminals. The goal is to partition the vertices of the graph into $k$ parts, each containing exactly one of the terminals, such that the sum of weights of the edges connecting vertices in different parts of the partition is minimized. The problem is APX-hard for $k\ge3$. The currently best known approximation algorithm for the problem for arbitrary $k$, obtained by Sharma and Vondrák [STOC 2014] more than a decade ago, has an approximation ratio of 1.2965. We present an algorithm with an improved approximation ratio of 1.2787. Also, for small values of $k \ge 4$ we obtain the first improvements in 25 years over the currently best approximation ratios obtained by Karger et al. [STOC 1999]. (For $k=3$ an optimal approximation algorithm is known.) Our main technical contributions are new insights on rounding the LP relaxation of Călinescu, Karloff, and Rabani [STOC 1998], whose integrality ratio matches Multiway Cut's approximability ratio, assuming the Unique Games Conjecture [Manokaran et al., STOC 2008]. First, we introduce a generalized form of a rounding scheme suggested by Kleinberg and Tardos [FOCS 1999] and use it to replace the Exponential Clocks rounding scheme used by Buchbinder et al. [STOC 2013] and by Sharma and Vondrák. Second, while previous algorithms use a mixture of two, three, or four basic rounding schemes, each from a different family of rounding schemes, our algorithm uses a computationally-discovered mixture of hundreds of basic rounding schemes, each parametrized by a random variable with a distinct probability distribution, including in particular many different rounding schemes from the same family. We give a completely rigorous analysis of our improved algorithms using a combination of analytical techniques and interval arithmetic.