Optimal approximation for submodular and supermodular optimization with bounded curvature (1311.4728v3)

Abstract: We design new approximation algorithms for the problems of optimizing submodular and supermodular functions subject to a single matroid constraint. Specifically, we consider the case in which we wish to maximize a nondecreasing submodular function or minimize a nonincreasing supermodular function in the setting of bounded total curvature $c$. In the case of submodular maximization with curvature $c$, we obtain a $(1-c/e)$-approximation --- the first improvement over the greedy $(1-e^{-c})/c$-approximation of Conforti and Cornuejols from 1984, which holds for a cardinality constraint, as well as recent approaches that hold for an arbitrary matroid constraint. Our approach is based on modifications of the continuous greedy algorithm and non-oblivious local search, and allows us to approximately maximize the sum of a nonnegative, nondecreasing submodular function and a (possibly negative) linear function. We show how to reduce both submodular maximization and supermodular minimization to this general problem when the objective function has bounded total curvature. We prove that the approximation results we obtain are the best possible in the value oracle model, even in the case of a cardinality constraint. We define an extension of the notion of curvature to general monotone set functions and show $(1-c)$-approximation for maximization and $1/(1-c)$-approximation for minimization cases. Finally, we give two concrete applications of our results in the settings of maximum entropy sampling, and the column-subset selection problem.

• The paper introduces a (1 - c/e)-approximation for maximizing nondecreasing submodular functions under bounded curvature, marking a key improvement over traditional greedy algorithms.
• The study extends its methods to any matroid constraint, demonstrating broad applicability in complex optimization problems like sensor placement and feature selection.
• Using a dual approach with continuous greedy and local search, the research also tackles supermodular minimization by reducing it to submodular maximization for enhanced approximation guarantees.

An Examination of Approximation Algorithms for Submodular and Supermodular Optimization Under Bounded Curvature

This paper presents novel approximation algorithms aimed at optimizing submodular and supermodular functions when constrained by a matroid. The primary focus is on both maximizing a nondecreasing submodular function and minimizing a nonincreasing supermodular function under limited curvature constraints, specifically characterized by a curvature parameter $c$. This research succeeds in delivering a significant breakthrough in terms of approximation guarantees compared to previous methodologies, particularly those building on the greedy algorithm.

Key Contributions

1. Improved Approximation for Submodular Maximization: The paper introduces a $(1 - c/e)$-approximation for maximizing nondecreasing submodular functions. This improvement builds upon the classical Conforti and Cornuejols $(1 - e^{-c})/c$-approximation approach. Thus, this advancement marks the first substantial enhancement in this domain since the 1984 greedy algorithm foundation.
2. Generalization to Arbitrary Matroid Constraints: The improvements are neither confined to cardinality constraints nor reliant on specific assumptions about the input structure. Instead, they robustly extend to any matroid constraint, leading to widespread applicability in real-world scenarios involving complex hierarchical constraints.
3. Dual Approach using Continuous Greedy Algorithm and Non-Oblivious Local Search: The continuous greedy algorithm has been meticulously adjusted to accommodate issues of curvature, along with a variant of local search strategies. This dual-faceted approach allows a broader range of submodular functions, particularly those coupled with linear elements, to be tackled efficiently.
4. Supermodular Minimization Progress: The research also effectively reduces the problem of supermodular minimization to submodular maximization, showcasing similar improvements in approximation ratios through nuanced adjustments of the methods used for the submodular maximization scenarios.
5. Curvature Extension for General Set Functions: By extending the notion of curvature to general monotonic set functions, the work delivers a $(1-c)$-approximation for maximization and a $1/(1-c)$-approximation for minimization, which outlines the impact of curvature on the function performance metrics.

Practical Implications and Applications

The implications of this work are far-reaching, significantly influencing areas that traditionally exploit submodular optimization, such as sensor placement in machine learning, maximizing welfare in combinatorial auctions, and network influence maximization. Two illustrative applications detailed are:

• Maximum Entropy Sampling: The paper offers improvements when evaluating the determinant sampling problem, drawing on the eigenvalues of matrices to enhance approximation potential under bounded curvature.
• Column-Subset Selection Problem: This well-known problem in machine learning contexts, particularly those focusing on feature selection, benefits directly from the demonstrated capability of the proposed algorithms to effectively handle complex data matrix reductions while adhering to Frobenius norm constraints.

Future Work and Theoretical Implications

The determined optimality in the value oracle model implies a stagnated horizon for traditional algorithmic discomforts overriding the presented bounds. However, potential future exploration lies in exceeding these theoretical benchmarks with practical models or adapting these insights into nonpolynomial time contexts.

Moreover, by systematically abstracting curvature into the generalized function domain, the research paves the way for exploring uncharted territories in non-submodular function optimization, offering a preliminary yet promising scaffold for deriving novel algorithms for broader optimization challenges beyond the current state of the art.

In conclusion, the research stands to incrementally elevate the standards and efficacy of available algorithms in the domain of function optimization within bounded configurations. This progression is evident not only through the improvement in approximation ratios but also through an enriched understanding of how mathematical properties such as curvature interact with foundational optimization methodologies.