[Paper Review] A Tight Combinatorial Algorithm for Submodular Maximization Subject to a Matroid Constraint

Yuval Filmus, Justin Ward|arXiv (Cornell University)|Apr 20, 2012

Complexity and Algorithms in Graphs27 references19 citations

TL;DR

This paper presents a combinatorial, non-rounding algorithm for monotone submodular maximization subject to a matroid constraint, achieving the optimal $1 - 1/e$ approximation ratio. It uses a non-oblivious local search over a carefully constructed auxiliary potential function, generalizing prior work on coverage functions and matching the performance of the continuous greedy algorithm without requiring rounding.

ABSTRACT

We present an optimal, combinatorial 1-1/e approximation algorithm for monotone submodular optimization over a matroid constraint. Compared to the continuous greedy algorithm (Calinescu, Chekuri, Pal and Vondrak, 2008), our algorithm is extremely simple and requires no rounding. It consists of the greedy algorithm followed by local search. Both phases are run not on the actual objective function, but on a related non-oblivious potential function, which is also monotone submodular. Our algorithm runs in randomized time O(n^8u), where n is the rank of the given matroid and u is the size of its ground set. We additionally obtain a 1-1/e-eps approximation algorithm running in randomized time O (eps^-3n^4u). For matroids in which n = o(u), this improves on the runtime of the continuous greedy algorithm. The improvement is due primarily to the time required by the pipage rounding phase, which we avoid altogether. Furthermore, the independence of our algorithm from pipage rounding techniques suggests that our general approach may be helpful in contexts such as monotone submodular maximization subject to multiple matroid constraints. Our approach generalizes to the case where the monotone submodular function has restricted curvature. For any curvature c, we adapt our algorithm to produce a (1-e^-c)/c approximation. This result complements results of Vondrak (2008), who has shown that the continuous greedy algorithm produces a (1-e^-c)/c approximation when the objective function has curvature c. He has also proved that achieving any better approximation ratio is impossible in the value oracle model.

Motivation & Objective

• To develop a combinatorial algorithm that achieves the optimal $1 - 1/e$ approximation ratio for monotone submodular maximization under a matroid constraint.
• To eliminate the need for rounding procedures used in the continuous greedy algorithm by operating entirely on integral solutions.
• To generalize the non-oblivious local search framework from coverage functions to arbitrary monotone submodular functions.
• To extend the algorithm to handle functions with restricted curvature $c$, achieving the optimal $(1 - e^{-c})/c$ approximation ratio.
• To provide a combinatorial alternative to continuous methods that remains optimal and efficient in the value oracle model.

Proposed method

• The algorithm uses a two-phase approach: first, a greedy selection over the potential function, followed by local search on the same function.
• The potential function $g$ is defined as a weighted sum of $f$-values over subsets, using coefficients derived from a probability distribution $P$ to emphasize elements with higher marginal gains.
• The potential function $g$ is monotone submodular and serves as a surrogate objective to guide the local search toward better solutions.
• Local search iteratively improves the solution by swapping elements while maintaining independence in the matroid, using $g$ to evaluate moves.
• For functions with curvature $c$, the algorithm adapts by using a modified potential function that accounts for curvature, achieving the optimal $(1 - e^{-c})/c$ ratio.
• When curvature is unknown, the algorithm enumerates over candidate values of $c$, achieving a $(1 - e^{-c})/c - \epsilon$ approximation with high probability.

Experimental results

Research questions

• RQ1 Can a purely combinatorial algorithm achieve the optimal $1 - 1/e$ approximation for monotone submodular matroid maximization without requiring rounding?
• RQ2 How can non-oblivious local search be generalized from coverage functions to arbitrary monotone submodular functions?
• RQ3 What is the role of the potential function in improving the locality ratio of local search algorithms for submodular maximization?
• RQ4 Can the algorithm be adapted to handle functions with restricted curvature $c$ while maintaining optimality?
• RQ5 Is it possible to achieve the optimal approximation ratio combinatorially even when the curvature is unknown?

Key findings

• The algorithm achieves the optimal $1 - 1/e$ approximation ratio for monotone submodular matroid maximization using only combinatorial operations and no rounding.
• The algorithm runs in randomized time $\tilde{O}(r^7 n^2)$ for the exact $1 - 1/e$ ratio and $\tilde{O}(\epsilon^{-3} r^4 n)$ for a $1 - 1/e - \epsilon$ approximation.
• For functions with curvature $c$, the algorithm achieves the optimal $(1 - e^{-c})/c$ approximation ratio, matching the performance of the continuous greedy algorithm.
• The potential function $g$ is explicitly computable in polynomial time when $f$ is a coverage function, enabling efficient local search steps.
• When curvature is unknown, the algorithm uses enumeration over candidate $c$ values to achieve a $\rho(c) - \epsilon$ approximation with high probability.
• The approach generalizes prior work on coverage functions, with the potential function coefficients matching those used in earlier non-oblivious local search for maximum coverage.