[Paper Review] Non-monotone submodular maximization under matroid and knapsack constraints

Jon Lee, Vahab Mirrokni+2 more|ArXiv.org|Feb 2, 2009

Complexity and Algorithms in GraphsComputer Science25 references114 citations

TL;DR

This paper presents the first constant-factor approximation algorithms for maximizing non-monotone submodular functions under multiple matroid and knapsack constraints. It introduces a novel local search framework with generalized exchange operations and randomized rounding, achieving a $\left(\frac{1}{k+1+\frac{1}{k-1}+\epsilon}\right)$-approximation for $k$ partition matroids and a $\left(\frac{1}{5}-\epsilon\right)$-approximation for $k$ knapsack constraints, with improved bounds for monotone functions.

ABSTRACT

Submodular function maximization is a central problem in combinatorial optimization, generalizing many important problems including Max Cut in directed/undirected graphs and in hypergraphs, certain constraint satisfaction problems, maximum entropy sampling, and maximum facility location problems. Unlike submodular minimization, submodular maximization is NP-hard. For the problem of maximizing a non-monotone submodular function, Feige, Mirrokni, and Vondrák recently developed a $2\over 5$-approximation algorithm \cite{FMV07}, however, their algorithms do not handle side constraints.} In this paper, we give the first constant-factor approximation algorithm for maximizing any non-negative submodular function subject to multiple matroid or knapsack constraints. We emphasize that our results are for {\em non-monotone} submodular functions. In particular, for any constant $k$, we present a $({1\over k+2+{1\over k}+ε})$-approximation for the submodular maximization problem under $k$ matroid constraints, and a $({1\over 5}-ε)$-approximation algorithm for this problem subject to $k$ knapsack constraints ($ε>0$ is any constant). We improve the approximation guarantee of our algorithm to ${1\over k+1+{1\over k-1}+ε}$ for $k\ge 2$ partition matroid constraints. This idea also gives a $({1\over k+ε})$-approximation for maximizing a {\em monotone} submodular function subject to $k\ge 2$ partition matroids, which improves over the previously best known guarantee of $\frac{1}{k+1}$.

Motivation & Objective

• To develop the first constant-factor approximation algorithms for non-monotone submodular maximization under multiple matroid and knapsack constraints.
• To address the challenge of non-monotonicity in submodular functions, which generalizes NP-hard problems like Max Cut and maximum facility location.
• To improve approximation guarantees beyond prior work, especially for partition matroid and knapsack constraints.
• To provide provable approximation guarantees for problems such as maximum entropy sampling and optimal experimental design.

Proposed method

• Introduces a generalized local search algorithm that allows including $p$ new elements while dropping up to $(k-1)\cdot p$ elements, improving over standard single-element exchanges.
• Employs a multigraph construction based on exchange maps from multiple partition matroids to analyze structural properties of feasible solutions.
• Uses a randomized rounding procedure on a fractional solution to achieve a $\left(\frac{1}{4}-\epsilon\right)$-approximation for light elements in knapsack constraints.
• Combines enumeration over heavy elements with randomized rounding to achieve a $\left(\frac{1}{5}-\epsilon\right)$-approximation for $k$ knapsack constraints.
• Applies a strengthened exchange lemma using path/cycle decomposition in multigraphs to derive the key inequality $k\cdot f(S) \geq \left(1-\frac{1}{p}\right)\cdot f(S\cup C) + (k-1)\cdot f(S\cap C)$.
• Leverages the simplified proof of Lemma 2 by Jan Vondrák to strengthen the theoretical foundation of the local search analysis.

Experimental results

Research questions

• RQ1 Can a constant-factor approximation be achieved for non-monotone submodular maximization under $k$ matroid constraints?
• RQ2 What is the best possible approximation ratio for non-monotone submodular maximization under $k$ knapsack constraints?
• RQ3 Can the approximation guarantee be improved under the special case of partition matroid constraints?
• RQ4 How does the generalized local search with multi-element exchanges improve over standard local search in submodular optimization?

Key findings

• The paper achieves a $\left(\frac{1}{k+2+\frac{1}{k}+\epsilon}\right)$-approximation for $k$ matroid constraints, providing the first constant-factor guarantee for this setting.
• For $k$ knapsack constraints, the algorithm achieves a $\left(\frac{1}{5}-\epsilon\right)$-approximation, which is the first constant-factor result for this problem class.
• Under $k$ partition matroid constraints, the approximation ratio improves to $\left(\frac{1}{k+1+\frac{1}{k-1}+\epsilon}\right)$, representing a significant improvement over prior bounds.
• For monotone submodular functions under $k$ partition matroids, the algorithm achieves a $\left(\frac{1}{k+\epsilon}\right)$-approximation, improving upon the previous best of $\frac{1}{k+1}$.
• The method combines enumeration of heavy elements with randomized rounding on light elements, ensuring bounded approximation quality under knapsack constraints.
• Theoretical analysis relies on a novel multigraph-based exchange argument and a strengthened inequality involving $f(S\cup C)$ and $f(S\cap C)$, enabling tighter bounds.