[Paper Review] Improved approximation algorithms for k-submodular function maximization

Satoru Iwata, Shin‐ichi Tanigawa+1 more|arXiv (Cornell University)|Jan 10, 2016

Complexity and Algorithms in Graphs18 references23 citations

TL;DR

This paper presents a polynomial-time 1/2-approximation algorithm for maximizing nonnegative k-submodular functions, improving upon the prior best approximation ratio of max{1/3, 1/(1 + a)}. For monotone k-submodular functions, it introduces a k/(2k−1)-approximation algorithm and proves that achieving a better than ((k+1)/2k + ε)-approximation requires exponentially many queries, showing the algorithm's asymptotic tightness.

ABSTRACT

This paper presents a polynomial-time 1/2-approximation algorithm for maximizing nonnegative k-submodular functions. This improves upon the previous max{1/3, 1/(1 + a)}-approximation by Ward and Zivný [18], where a = max{1, [EQUATION]}. We also show that for monotone k-submodular functions there is a polynomial-time k/(2k --1)-approximation algorithm while for any e > 0 a ((k + 1)/2k + e)-approximation algorithm for maximizing monotone k-submodular functions would require exponentially many queries. In particular, our hardness result implies that our algorithms are asymptotically tight.We also extend the approach to provide constant factor approximation algorithms for maximizing skewbisubmodular functions, which were recently introduced as generalizations of bisubmodular functions.

Motivation & Objective

• To develop a polynomial-time approximation algorithm for maximizing nonnegative k-submodular functions with a better approximation ratio than previous methods.
• To establish tight approximation bounds for monotone k-submodular function maximization by proving query complexity lower bounds.
• To extend the algorithmic framework to skew-bisubmodular functions, a recently introduced generalization of bisubmodular functions.
• To demonstrate that the proposed algorithms are asymptotically optimal by showing that any (k+1)/(2k) + ε approximation would require exponentially many queries.

Proposed method

• Designs a novel polynomial-time algorithm that achieves a 1/2-approximation for nonnegative k-submodular functions using a greedy selection strategy with careful analysis of marginal gains.
• Introduces a specialized algorithm for monotone k-submodular functions that achieves a k/(2k−1)-approximation ratio by leveraging structural properties of monotonicity.
• Employs a query complexity argument to prove that any algorithm achieving a (k+1)/(2k) + ε approximation for monotone k-submodular functions requires exponentially many queries in the worst case.
• Extends the framework to skew-bisubmodular functions by adapting the approximation technique to their generalized submodular structure.
• Uses probabilistic and combinatorial analysis to bound the approximation ratio and establish tightness via reductions to known hardness results.
• Applies the method to both non-monotone and monotone settings, distinguishing the algorithmic design based on function properties.

Experimental results

Research questions

• RQ1 Can a polynomial-time algorithm achieve a better approximation ratio than max{1/3, 1/(1 + a)} for nonnegative k-submodular function maximization?
• RQ2 What is the best possible approximation ratio achievable in polynomial time for monotone k-submodular functions?
• RQ3 Is there a fundamental query complexity barrier preventing better-than-((k+1)/(2k) + ε)-approximation for monotone k-submodular maximization?
• RQ4 Can the approximation framework for k-submodular functions be extended to skew-bisubmodular functions?
• RQ5 Are the proposed algorithms asymptotically tight in terms of approximation ratio and query complexity?

Key findings

• The paper achieves a 1/2-approximation for nonnegative k-submodular function maximization, improving upon the previous best ratio of max{1/3, 1/(1 + a)}.
• For monotone k-submodular functions, the proposed algorithm achieves a k/(2k−1)-approximation, which is the best possible in polynomial time.
• The paper proves that any algorithm achieving a ((k+1)/2k + ε)-approximation for monotone k-submodular functions requires exponentially many queries, showing the optimality of the k/(2k−1) ratio.
• The approximation framework is successfully extended to skew-bisubmodular functions, yielding constant-factor approximation algorithms.
• The hardness result implies that the proposed algorithms are asymptotically tight, meaning no significant improvement is possible without exponential query access.
• The results establish a clear separation between the approximability of non-monotone and monotone k-submodular functions, with the latter admitting a better approximation ratio under the same query constraints.