[Paper Review] Submodular Function Maximization via the Multilinear Relaxation and Contention Resolution Schemes

Chandra Chekuri, Jan Vondrák+1 more|arXiv (Cornell University)|May 23, 2011

Complexity and Algorithms in Graphs31 references68 citations

TL;DR

This paper presents a general framework for maximizing non-negative submodular functions under packing constraints using multilinear relaxation and contention resolution schemes (CRS). It achieves constant-factor approximation guarantees for non-monotone submodular maximization over down-closed polytopes defined by separation oracles, including intersections of matroid and knapsack constraints, by introducing a novel CRS construction based on correlation gap analysis and recursive decomposition of fractional solutions.

ABSTRACT

We consider the problem of maximizing a non-negative submodular set function $f:2^N ightarrow \mathbb{R}_+$ over a ground set $N$ subject to a variety of packing type constraints including (multiple) matroid constraints, knapsack constraints, and their intersections. In this paper we develop a general framework that allows us to derive a number of new results, in particular when $f$ may be a non-monotone function. Our algorithms are based on (approximately) maximizing the multilinear extension $F$ of $f$ over a polytope $P$ that represents the constraints, and then effectively rounding the fractional solution. Although this approach has been used quite successfully, it has been limited in some important ways. We overcome these limitations as follows. First, we give constant factor approximation algorithms to maximize $F$ over a down-closed polytope $P$ described by an efficient separation oracle. Previously this was known only for monotone functions. For non-monotone functions, a constant factor was known only when the polytope was either the intersection of a fixed number of knapsack constraints or a matroid polytope. Second, we show that contention resolution schemes are an effective way to round a fractional solution, even when $f$ is non-monotone. In particular, contention resolution schemes for different polytopes can be combined to handle the intersection of different constraints. Via LP duality we show that a contention resolution scheme for a constraint is related to the correlation gap of weighted rank functions of the constraint. This leads to an optimal contention resolution scheme for the matroid polytope. Our results provide a broadly applicable framework for maximizing linear and submodular functions subject to independence constraints. We give several illustrative examples. Contention resolution schemes may find other applications.

Motivation & Objective

• To develop a general, scalable framework for maximizing non-negative submodular functions under complex packing constraints.
• To overcome limitations of prior methods that were restricted to monotone functions or specific constraint types like single matroids or knapsacks.
• To unify and extend approximation algorithms for submodular maximization across diverse constraint families, including intersections of matroids and knapsacks.
• To establish a connection between contention resolution schemes and the correlation gap of weighted rank functions, enabling optimal CRS design.
• To provide a flexible, modular approach that combines CR schemes for different constraints, enabling efficient rounding of multilinear relaxation solutions.

Proposed method

• The method uses the multilinear extension F of the submodular function f to relax the discrete optimization problem into a continuous one over a polytope P.
• It formulates the problem as maximizing F(x) over a down-closed polytope P defined by a separation oracle, enabling efficient fractional optimization.
• A recursive decomposition of the fractional solution z is performed by grouping elements by demand size into sets Nh, and scaling them down by a factor of 3^h.
• For each group Nh, a (β, 1−β′)-balanced contention resolution scheme is applied to the scaled solution z^h, producing integral solutions y^h.
• A randomized rounding rule is used: with probability 1/2, output y^0; otherwise, output the union of y^h for h≥1, ensuring feasibility and expected performance.
• The framework leverages LP duality to relate CR schemes to the correlation gap of weighted rank functions, enabling optimal CRS design for matroid polytopes.

Experimental results

Research questions

• RQ1 Can constant-factor approximation algorithms be achieved for non-monotone submodular maximization over general down-closed polytopes with a separation oracle?
• RQ2 How can contention resolution schemes be effectively designed and composed for intersecting constraints such as matroids and knapsacks?
• RQ3 What is the theoretical connection between contention resolution schemes and the correlation gap of weighted rank functions?
• RQ4 Can the multilinear relaxation approach be extended beyond monotone functions to achieve constant-factor guarantees for non-monotone cases?
• RQ5 Is there a general, modular framework that unifies rounding techniques across diverse constraint types in submodular optimization?

Key findings

• The paper achieves a constant-factor approximation for non-monotone submodular maximization over any down-closed polytope defined by a separation oracle, extending prior results limited to monotone functions or specific constraint types.
• It introduces a novel contention resolution scheme construction that achieves a (β/6, (1−β′)/2)-balanced CRS for general packing constraints, improving over prior schemes.
• For matroid polytopes, the framework yields an optimal contention resolution scheme by exploiting the correlation gap of weighted rank functions.
• The method ensures feasibility of the final integral solution through a recursive decomposition and randomized rounding strategy that maintains constraint satisfaction across all constraint types.
• The approach generalizes and unifies previous results, providing a single framework that handles intersections of matroid and knapsack constraints with constant-factor approximation guarantees.
• The framework is shown to be effective even for non-monotone submodular functions, where prior methods had limited success.