Under the UGC, an optimal approximation factor for the $k$-way Cut Problem can be obtained by an adequate rounding of the LP relaxation suggested by C{\u{a}}linescu, Karloff, and Rabani [STOC 1998]. Hence, the challange is finding such a rounding procedure. Recent works constuct such a rounding by taking a convex combination (or a mixture) of several natural rounding schemes. The current work takes this approach to the extreme by considering many versions of these rounding schemes, where the versions differ by the parameters used in these schemes. In particular, the current work uses a mixture of hundreds of such versions. Candidate mixtures are found by a natural heuristics, and their performance (i.e., approximation factor) is determined based on a combination of rigorous analysis and execution of programs whose code and specification are posted publically.
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 Vondr\'ak [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, Klein, Stein, Thorup, and Young [STOC 1999]. (For $k=3$ an optimal approximation algorithm is known.)
Our main technical contributions are new insights on rounding the LP relaxation of C{\u{a}}linescu, Karloff, and Rabani [STOC 1998], whose integrality ratio matches Multiway Cut's approximability ratio, assuming the Unique Games Conjecture [Manokaran, Naor, Raghavendra, and Schwartz, 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, Naor, and Schwartz [STOC 2013] and by Sharma and Vondr\'ak. 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.
To appear in the proceedings of STOC26.