[Paper Review] Stochastic Combinatorial Optimization under Probabilistic Constraints

Shipra Agrawal, Amin Saberi+1 more|ArXiv.org|Sep 2, 2008

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TL;DR

This paper presents approximation algorithms for stochastic combinatorial optimization under probabilistic constraints, focusing on the k-center and set cover problems with uncertain element sets. It introduces a reformulation using logarithmic penalties to transform probabilistic constraints into a partial set cover framework, achieving a $(\frac{4}{3}+\epsilon)\log n$-approximation for non-adaptive problems and establishing inapproximability results for adaptive settings.

ABSTRACT

In this paper, we present approximation algorithms for combinatorial optimization problems under probabilistic constraints. Specifically, we focus on stochastic variants of two important combinatorial optimization problems: the k-center problem and the set cover problem, with uncertainty characterized by a probability distribution over set of points or elements to be covered. We consider these problems under adaptive and non-adaptive settings, and present efficient approximation algorithms for the case when underlying distribution is a product distribution. In contrast to the expected cost model prevalent in stochastic optimization literature, our problem definitions support restrictions on the probability distributions of the total costs, via incorporating constraints that bound the probability with which the incurred costs may exceed a given threshold.

Motivation & Objective

• Address the limitations of expected cost models in stochastic optimization by incorporating probabilistic constraints that bound the likelihood of infeasibility or high cost.
• Formulate stochastic variants of the k-center and set cover problems where elements or points to be covered are randomly realized with known marginal probabilities.
• Develop efficient approximation algorithms for non-adaptive and adaptive settings under product distributions, focusing on value-at-risk (VaR) and chance constraints.
• Provide theoretical guarantees for approximation quality, particularly for the non-adaptive case, and establish inapproximability results for the adaptive case.
• Reframe probabilistic constraints into a partial set cover formulation using logarithmic transformations of failure probabilities to enable algorithmic treatment.

Proposed method

• Transform the probabilistic constraint $P(\max_{v\in\tilde{V}}d(v,C)\leq r)\geq 1-\rho$ into a constraint involving the sum of $l_j = \log\frac{1}{1-p_j}$, enabling a reformulation as a partial set cover problem.
• Introduce auxiliary binary variables $z_j$ to model uncovered elements and penalize them with weights $l_j$, leading to an integer program with a total penalty limit $l = \log\frac{1}{1-\rho}$.
• Leverage existing $(\frac{4}{3}+\epsilon)\log n$-approximation algorithms for the partial set cover problem to solve the non-adaptive stochastic set cover and k-center problems.
• Establish a reduction from the adaptive stochastic set cover problem to the problem of counting maximum independent sets in a graph.
• Use the inapproximability result from Luby (1993) to show that approximating the failure probability in the adaptive case is hard unless RP = NP.
• Derive bounds showing that approximating the failure probability $f(m)$ for edge cover cost $>m$ yields a constant-factor approximation for the number of maximum independent sets $I(G)$.

Experimental results

Research questions

• RQ1 Can we design efficient approximation algorithms for stochastic k-center and set cover problems under probabilistic constraints that bound the probability of exceeding a cost threshold?
• RQ2 How does the non-adaptive setting, where decisions are made before observing the random subset, compare to the adaptive setting in terms of approximation complexity?
• RQ3 To what extent can probabilistic constraints in stochastic optimization be reformulated as structured integer programs amenable to approximation?
• RQ4 What are the inherent computational limits of approximating the value-at-risk (VaR) in adaptive stochastic set cover problems?
• RQ5 Is there a connection between approximating failure probabilities in adaptive stochastic set cover and hard problems in counting complexity, such as counting maximum independent sets?

Key findings

• The non-adaptive stochastic k-center and set cover problems can be reformulated as a partial set cover problem with logarithmic penalties, enabling a $(\frac{4}{3}+\epsilon)\log n$-approximation algorithm.
• The probabilistic constraint $P(\max_{v\in\tilde{V}}d(v,C)\leq r)\geq 1-\rho$ is equivalent to a sum constraint on $\sum I_j(x) \log \frac{1}{1-p_j} \leq \log \frac{1}{1-\rho}$, enabling efficient optimization.
• For the adaptive stochastic set cover problem, approximating the value-at-risk $B$ is at least as hard as approximating the number of maximum independent sets in a graph.
• A reduction from the independent set counting problem to the adaptive failure probability computation shows that a polynomial-time approximation for $f(m)$ would yield a constant-factor approximation for $I(G)$, implying inapproximability unless RP = NP.
• The failure probability $f(m)$ satisfies $\frac{1}{3} \cdot \frac{f(m)}{p^m(1-p)^{n-m}} \leq I(G) \leq \frac{f(m)}{p^m(1-p)^{n-m}}$, linking the two problems tightly.
• When $p \leq 1/2^n$, the failure probability $f(m)$ is within a constant factor of $I(G) \cdot p^m(1-p)^{n-m}$, showing that good approximation of $f(m)$ implies good approximation of $I(G)$.