[Paper Review] Randomized Strategies for Robust Combinatorial Optimization

Yasushi Kawase, Hanna Sumita|ArXiv.org|May 20, 2018

Optimization and Search Problems34 references3 citations

TL;DR

This paper proposes two novel approximation algorithms for robust combinatorial optimization under uncertainty, where a randomized strategy over independent sets maximizes the worst-case expected objective across multiple scenarios. It introduces a polyhedral relaxation scheme for linear objectives and a multiplicative weights-based method with $(\eta,\gamma)$-reductions, achieving improved approximation ratios for both linear and submodular objectives, including cardinality robustness in knapsack and matroid settings.

ABSTRACT

In this paper, we study the following robust optimization problem. Given an independence system and candidate objective functions, we choose an independent set, and then an adversary chooses one objective function, knowing our choice. Our goal is to find a randomized strategy (i.e., a probability distribution over the independent sets) that maximizes the expected objective value. To solve the problem, we propose two types of schemes for designing approximation algorithms. One scheme is for the case when objective functions are linear. It first finds an approximately optimal aggregated strategy and then retrieves a desired solution with little loss of the objective value. The approximation ratio depends on a relaxation of an independence system polytope. As applications, we provide approximation algorithms for a knapsack constraint or a matroid intersection by developing appropriate relaxations and retrievals. The other scheme is based on the multiplicative weights update method. A key technique is to introduce a new concept called $(η,γ)$-reductions for objective functions with parameters $η, γ$. We show that our scheme outputs a nearly $α$-approximate solution if there exists an $α$-approximation algorithm for a subproblem defined by $(η,γ)$-reductions. This improves approximation ratio in previous results. Using our result, we provide approximation algorithms when the objective functions are submodular or correspond to the cardinality robustness for the knapsack problem.

Motivation & Objective

• To design efficient approximation algorithms for robust combinatorial optimization under worst-case scenarios where an adversary selects the objective function after observing the randomized strategy.
• To overcome the NP-hardness of deterministic robust optimization by leveraging randomization, which can significantly improve worst-case performance and enable polynomial-time solvability.
• To develop general algorithmic frameworks applicable to independence systems, including knapsack and matroid constraints, with provable approximation guarantees.
• To improve upon prior approximation ratios by introducing new techniques such as $(\eta,\gamma)$-reductions and relaxation-based retrieval schemes.

Proposed method

• For linear objectives, the method first computes an approximately optimal aggregated strategy over the independence system polytope and then retrieves a feasible solution with bounded loss using a relaxation-based approach.
• It introduces a new relaxation of the independence system polytope to bound the approximation ratio, enabling application to knapsack and matroid intersection constraints.
• For submodular and cardinality-robust objectives, the paper employs a multiplicative weights update method with a novel $(\eta,\gamma)$-reduction framework to model subproblem approximations.
• The $(\eta,\gamma)$-reduction concept allows the algorithm to maintain a nearly $\alpha$-approximate solution if a $\alpha$-approximation exists for a derived subproblem.
• The approach treats the robust optimization problem as a two-player zero-sum game, where the algorithm selects a mixed strategy and the adversary selects the worst-case scenario, enabling game-theoretic solution techniques.
• The framework is applied to cardinality robustness problems by normalizing objective functions and ensuring that the randomized strategy achieves a guaranteed approximation ratio across all possible cardinality bounds.

Experimental results

Research questions

• RQ1 Can randomization significantly improve the worst-case performance of combinatorial optimization under uncertainty compared to deterministic strategies?
• RQ2 What approximation guarantees can be achieved for robust combinatorial optimization when the objective functions are linear, submodular, or correspond to cardinality robustness?
• RQ3 How can the multiplicative weights update method be adapted with new structural concepts like $(\eta,\gamma)$-reductions to improve approximation ratios?
• RQ4 Is it possible to design a general algorithmic framework that achieves better approximation ratios than prior methods for robust optimization over independence systems?
• RQ5 Can the randomized robust optimization problem be solved in polynomial time even when the deterministic version is NP-hard?

Key findings

• The paper establishes that the randomized robust optimization problem is polynomial-time solvable, in contrast to the NP-hardness of the deterministic version, even for linear objectives under cardinality constraints.
• For linear objectives, the proposed relaxation-based scheme achieves an approximation ratio dependent on the polyhedral relaxation of the independence system, with concrete guarantees for knapsack and matroid intersection constraints.
• The multiplicative weights-based scheme with $(\eta,\gamma)$-reductions improves upon previous approximation ratios, achieving a nearly $\alpha$-approximate solution when a $\alpha$-approximation exists for a subproblem.
• The framework provides a $\frac{2n}{2n+1}$-approximation for the cardinality robustness problem in the knapsack setting, which approaches 1 as $n$ grows.
• The paper proves that a randomized $\alpha$-robust solution exists if and only if a partition problem instance has a solution, establishing a tight connection between robust optimization and NP-completeness.
• The method achieves a $\alpha = \frac{1}{4} \cdot \frac{3C - 2v(0)}{C - v(0)}$ approximation ratio in the cardinality robustness problem, which equals $1 - \frac{r}{2}$ and is tight under the given construction.