Improved Approximation Algorithms for Multiway Cut by Large Mixtures of New and Old Rounding Schemes

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.

Figures (5)

• Figure 1: Analysis of KT$(f)$ - The computation of $P^{\,\mathrm{KT}(f),1}_{k,\varepsilon}(u_1,u_2,\ldots,u_k)$.
• Figure 2: The density functions of the ST, KT, IT and DT schemes used in mixture that obtains an approximation ratio of $1.2787$ for an arbitrary number of terminals. For ST, the combined density of all ST schemes is given. For each one of KT, IT and DT, the densities of the 8 rounding schemes of each family used with the largest probabilities are shown. Also shown are the probabilities with which each one of these rounding schemes is used. Plots in this paper are made with Matplotlib Hunter:2007.
• Figure 3: Heatmaps showing the densities of the various components of the algorithm, and of the whole algorithm, on simplex points of the form $(u_1,u_2,c,c,\ldots)$, $(u_1,0,u_3,c,c,\ldots)$ and $(u_1,u_2,u_3,0,\ldots,0)$. In points of the first form we have $c=(1-u_1-u_2)/(k-2)$, where $k\to\infty$, and similarly for points of the second form. The boundary value between blue and red shades is $1.278$. Although the different components have very large maximum densities, the mixture of all of them gives a final scheme in which the densities are almost constant in most the regions shown. Plots in this paper are made with Matplotlib Hunter:2007.
• Figure 4: A closer look at the densities of the whole algorithm.
• Figure 5: The LP relaxation of Multiway Cut introduced in CKR00.

Theorems & Definitions (50)

• Theorem 1.1
• Theorem 1.2
• Proposition 1.3: See Corollary \ref{['C-KT']}
• Definition 2.1: Cut probabilities
• Definition 2.2: Cut densities
• Lemma 2.3
• proof
• Corollary 2.4
• Lemma 2.5
• proof