[Paper Review] Submodular Maximization Under A Matroid Constraint: Asking more from an old friend, the Greedy Algorithm

Nived Rajaraman, Rahul Vaze|arXiv (Cornell University)|Oct 30, 2018

Complexity and Algorithms in Graphs8 references4 citations

TL;DR

This paper introduces a new measure called the 'discriminant' to refine approximation guarantees for the greedy algorithm in submodular maximization under a matroid constraint. By quantifying the relative gain of greedy choices versus the best alternative at each step, the authors derive tighter, instance-dependent approximation ratios that subsume and improve upon curvature-based bounds, with tightness proven via constructed examples.

ABSTRACT

The classical problem of maximizing a submodular function under a matroid constraint is considered. Defining a new measure for the increments made by the greedy algorithm at each step, called the discriminant, improved approximation ratio guarantees are derived for the greedy algorithm. At each step, discriminant measures the multiplicative gap in the incremental valuation between the item chosen by the greedy algorithm and the largest potential incremental valuation for eligible items not selected by it. The new guarantee subsumes all the previous known results for the greedy algorithm, including the curvature based ones, and the derived guarantees are shown to be tight via constructing specific instances. More refined approximation guarantee is derived for a special case called the submodular welfare maximization/partition problem that is also tight, for both the offline and the online case.

Motivation & Objective

• To address the gap between theoretical worst-case guarantees and practical performance of the greedy algorithm in submodular maximization under matroid constraints.
• To develop a new instance-dependent performance measure—'discriminant'—that captures how favorably the greedy choice compares to the best alternative at each step.
• To derive improved approximation guarantees for the greedy algorithm that subsume and refine existing curvature-based bounds.
• To prove tightness of the new bounds through explicit problem constructions.
• To extend the refined analysis to special cases, including the submodular welfare/partition problem and its online variant.

Proposed method

• Define the discriminant $ d_i' $ as the ratio of the incremental valuation from the greedy choice to the best possible incremental valuation among unselected eligible items at step $ i $.
• Use the discriminant to derive a new approximation guarantee for the greedy algorithm: $ \frac{Z(G)}{Z(\Omega)} \geq \frac{1}{\frac{1}{d} + c} $, where $ d $ is the minimum discriminant and $ c $ is the curvature.
• Construct a family of problem instances with controlled curvature $ c $ and discriminant $ d $ to demonstrate tightness of the bound in the limit as $ K \to \infty $.
• Establish a valid ordering of the optimal solution $ \Omega $ using the structure of the matroid and the greedy path to enable discriminant computation.
• Apply the bound to both offline and online settings of the submodular welfare/partition problem, showing it holds in both cases.
• Use a sequential exchange argument to prove optimality of the constructed solution $ \Omega $, showing $ Z(\Omega) \geq Z(\Omega') $ for any feasible $ \Omega' $.

Experimental results

Research questions

• RQ1 Can a new instance-dependent measure improve the theoretical approximation guarantee of the greedy algorithm for submodular maximization under a matroid constraint?
• RQ2 How does the discriminant, defined as the ratio of greedy gain to best alternative gain, refine existing curvature-based bounds?
• RQ3 Is the new bound tight, and can it be matched by a constructed problem instance?
• RQ4 Does the discriminant-based analysis extend to the submodular welfare/partition problem and its online variant?
• RQ5 Can the new bound subsume and improve upon all previously known curvature-based guarantees?

Key findings

• The proposed discriminant-based bound $ \frac{1}{\frac{1}{d} + c} $ strictly improves upon the classical $ \frac{1}{1+c} $ curvature-based guarantee and subsumes all prior results.
• The bound is tight, as demonstrated by a constructed instance where the approximation ratio approaches $ \frac{1}{\frac{1}{d} + c} $ in the limit as $ K \to \infty $.
• For the submodular welfare/partition problem, the same discriminant-based guarantee holds for both offline and online settings, with the online version maintaining the $ \frac{1}{\frac{1}{d} + c} $ ratio.
• The analysis proves that the optimal solution $ \Omega $ can be constructed via a sequential exchange process that maintains non-negative valuation increments, confirming its optimality.
• The discriminant $ d_i' $ is shown to be constant at $ d $ across all iterations in the tight example, validating the bound's derivation.
• The bound reduces to $ 1-1/e $ when $ d=1 $ and $ c \to 0 $, aligning with known optimal cases, and recovers the $ 1/2 $-approximation when $ d=1 $ and $ c=1 $.