Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
158 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
45 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Constrained Non-Monotone Submodular Maximization: Offline and Secretary Algorithms (1003.1517v2)

Published 7 Mar 2010 in cs.DS and cs.GT

Abstract: Constrained submodular maximization problems have long been studied, with near-optimal results known under a variety of constraints when the submodular function is monotone. The case of non-monotone submodular maximization is less understood: the first approximation algorithms even for the unconstrainted setting were given by Feige et al. (FOCS '07). More recently, Lee et al. (STOC '09, APPROX '09) show how to approximately maximize non-monotone submodular functions when the constraints are given by the intersection of p matroid constraints; their algorithm is based on local-search procedures that consider p-swaps, and hence the running time may be nOmega(p), implying their algorithm is polynomial-time only for constantly many matroids. In this paper, we give algorithms that work for p-independence systems (which generalize constraints given by the intersection of p matroids), where the running time is poly(n,p). Our algorithm essentially reduces the non-monotone maximization problem to multiple runs of the greedy algorithm previously used in the monotone case. Our idea of using existing algorithms for monotone functions to solve the non-monotone case also works for maximizing a submodular function with respect to a knapsack constraint: we get a simple greedy-based constant-factor approximation for this problem. With these simpler algorithms, we are able to adapt our approach to constrained non-monotone submodular maximization to the (online) secretary setting, where elements arrive one at a time in random order, and the algorithm must make irrevocable decisions about whether or not to select each element as it arrives. We give constant approximations in this secretary setting when the algorithm is constrained subject to a uniform matroid or a partition matroid, and give an O(log k) approximation when it is constrained by a general matroid of rank k.

Citations (191)

Summary

  • The paper presents an O(p)-approximation algorithm for submodular maximization under p-independence constraints, expanding the range of applicable models.
  • It introduces a constant-factor greedy approach for knapsack constraints as a viable alternative to LP-rounding techniques.
  • The research extends to the secretary problem with algorithms achieving constant and O(log k) approximations for online decision-making scenarios.

Constrained Non-Monotone Submodular Maximization: Offline and Secretary Algorithms

The publication explores algorithms for maximizing submodular functions, particularly for non-monotone cases under various constraints, with an emphasis on \textit{p-independence systems} and the \textit{secretary problem}. The work targets scenarios not well-covered in prior research, especially when a submodular function is non-monotone and subject to complex constraints.

Submodular functions, often likened to diminishing returns, are critical in optimization problems across domains ranging from machine learning to combinatorial auctions. The problem discussed by the paper extends beyond classical settings to address non-monotone submodular functions, posing a unique set of challenges and opportunities for efficient approximations.

Key Contributions

  1. Algorithm for pp-Independence Systems:
    • The paper introduces an algorithm providing an O(p)O(p)-approximation for submodular maximization subject to pp-independence constraints, efficiently handling a wider class of constraints compared to previous approaches which were limited to the intersection of a constant number of matroids. This algorithm ensures a polynomial runtime even when pp is non-constant.
  2. Knapsack Constraints:
    • For problems involving knapsack constraints, the research offers a constant-factor greedy-based approximation algorithm, presenting an alternative to existing LP-rounding techniques.
  3. The Secretary Problem:
    • The transition to the online world with the secretary problem allows the models to handle inputs arriving in random sequences, involving irrevocable decision-making. The authors present algorithms that ensure constant approximations for cardinality constraints and partition matroid constraints and an O(logk)O(\log k)-approximation for general matroids of rank kk.

Practical and Theoretical Implications

  • The research bridges theoretical gaps in understanding non-monotone submodular functions, encouraging applications in real-world scenarios where constraints dynamically intersect and evolve, such as dynamic resource allocation, network analysis, and online marketing strategies.
  • The simplification of algorithms even in complex non-monotone settings marks a significant stride towards practical feasibility without sacrificing approximation quality, ensuring wider applicability.

Speculative Trajectory and Open Problems

  • The algorithms expose potential for expansion into broader domains and variations of constraints, suggesting possibilities for dynamic or probabilistic constraints in uncertain environments such as adaptive policies.
  • Basic approximations leave fertile ground for optimizing constants further, hinting at developments that could match or surpass the results for constant pp, aligning with traditional, highly optimized combinatorial optimization problems.

The research presented underscores a foundational extension of current submodular maximization paradigms, adapting efficiently to both offline and online contexts, while paving the way for future investigations into deeper theoretical improvements and wider applications in dynamic settings. The publication serves as a crucial resource for experts looking to enhance scalable decision-making processes in environments characterized by combinatorial complexity and partial observability.