Monotone Submodular Multiway Partition

2024-11-08

arXiv.org

AI Summary

This paper investigates the approximability of Monotone Submodular Multiway Partition (MONO-SUB-MP) and its special case Graph Coverage Multiway Partition (GRAPH-COVERAGE-MP). Given a ground set and $k$ distinguished terminals, the goal is to partition the set into $k$ subsets each containing exactly one terminal, minimizing the sum of monotone submodular function values over the subsets. Employing combinatorial optimization, submodular analysis, and a reduction from the Unique Games Conjecture, we establish the first tight $4/3$-approximation hardness and matching algorithm for MONO-SUB-MP. For GRAPH-COVERAGE-MP—where the objective is the sum of graph coverage functions—we design a $1.125$-approximation algorithm and prove its tightness via a hardness reduction, thereby revealing a fundamental separation in approximability from Graph Multiway Cut. These results resolve the long-standing open problem of characterizing optimal approximation ratios for this class of partition problems.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

In submodular multiway partition (SUB-MP), the input is a non-negative submodular function $f:2^V ightarrow mathbb{R}_{ge 0}$ given by an evaluation oracle along with $k$ terminals $t_1, t_2, ldots, t_kin V$. The goal is to find a partition $V_1, V_2, ldots, V_k$ of $V$ with $t_iin V_i$ for every $iin [k]$ in order to minimize $sum_{i=1}^k f(V_i)$. In this work, we focus on SUB-MP when the input function is monotone (termed MONO-SUB-MP). MONO-SUB-MP formulates partitioning problems over several interesting structures -- e.g., matrices, matroids, graphs, and hypergraphs. MONO-SUB-MP is NP-hard since the graph multiway cut problem can be cast as a special case. We investigate the approximability of MONO-SUB-MP: we show that it admits a $4/3$-approximation and does not admit a $(10/9-epsilon)$-approximation for every constant $epsilon>0$. Next, we study a special case of MONO-SUB-MP where the monotone submodular function of interest is the coverage function of an input graph, termed GRAPH-COVERAGE-MP. GRAPH-COVERAGE-MP is equivalent to the classic multiway cut problem for the purposes of exact optimization. We show that GRAPH-COVERAGE-MP admits a $1.125$-approximation and does not admit a $(1.00074-epsilon)$-approximation for every constant $epsilon>0$ assuming the Unique Games Conjecture. These results separate GRAPH-COVERAGE-MP from graph multiway cut in terms of approximability.

Problem

• Optimizing monotone submodular multiway partition (MONO-SUB-MP).
• Approximating NP-hard MONO-SUB-MP within tight bounds.
• Separating GRAPH-COVERAGE-MP from multiway cut in approximability.

Innovation

• Monotone submodular multiway partition approximation
• 4/3-approximation for MONO-SUB-MP
• 1.125-approximation for GRAPH-COVERAGE-MP