Optimization of Chance-Constrained Submodular Functions

Abstract

Submodular optimization plays a key role in many real-world problems. In many real-world scenarios, it is also necessary to handle uncertainty, and potentially disruptive events that violate constraints in stochastic settings need to be avoided. In this paper, we investigate submodular optimization problems with chance constraints. We provide a first analysis on the approximation behavior of popular greedy algorithms for submodular problems with chance constraints. Our results show that these algorithms are highly effective when using surrogate functions that estimate constraint violations based on Chernoff bounds. Furthermore, we investigate the behavior of the algorithms on popular social network problems and show that high quality solutions can still be obtained even if there are strong restrictions imposed by the chance constraint.

Optimising Monotone Chance-Constrained Submodular Functions Using Evolutionary Multi-Objective Algorithms

“…Furthermore, we study GSEMO experimentally on the influence maximization problem in social networks and the maximum coverage problem. Our results show that GSEMO significantly outperforms the greedy approach [6] for the considered chance constrained submodular optimisation problems. Furthermore, we use the multi-objective problem formulation in a standard setting of NSGA-II.…”

“…The GSEMO algorithm has already been widely studied in the area of runtime analysis in the field of evolutionary computation [10] and more broadly in the area of artificial intelligence where the focus has been on submodular functions and Pareto optimisation [31,30,29,32]. We analyse this algorithm in the chance constrained submodular optimisation setting investigated in [6] in the context of greedy algorithms. Our analyses show that GSEMO is able to achieve the same approximation guarantee in expected polynomial time for uniform IID weights and the same approximation quality in expected pseudo-polynomial time for independent uniform weights having the same dispersion.…”

Evolutionary Bi-objective Optimization for the Dynamic Chance-Constrained Knapsack Problem Based on Tail Bound Objectives

“…They also carried out bi-objective optimization with respect to the profit and probability of constraint violation when the capacity is static. Doerr et al [9] have investigated adaptations of classical greedy algorithms for the optimization of submodular functions with chance constraints of knapsack type. They have shown that the adapted greedy algorithms maintain asymptotically almost the same approximation quality as in the deterministic setting when considering uniform distributions with the same dispersion for the knapsack weights.…”

Runtime Analysis of Single- and Multi-Objective Evolutionary Algorithms for Chance Constrained Optimization Problems with Normally Distributed Random Variables

“…Recent studies investigated the classical knapsack problem in static [31,32] and dynamic settings [1] as well as complex stockpile blending problems [33] and the optimization of submodular functions [20]. Theoretical analyses for submodular problems with chance constraints, where each stochastic component is uniformly distributed and has the same amount of uncertainty, have shown that greedy algorithms and evolutionary Pareto optimization approaches only lose a small amount in terms of approximation quality when comparing against the corresponding deterministic problems [7,20] and that evolutionary algorithms significantly outperform the greedy approaches in practice. Other recent theoretical runtime analyses of evolutionary algorithms have produced initial results for restricted classes of instances of the knapsack problem where the weights are chosen randomly [22,34].…”