Random Order Contention Resolution Schemes

Abstract

Contention resolution schemes have proven to be an incredibly powerful concept which allows to tackle a broad class of problems. The framework has been initially designed to handle submodular optimization under various types of constraints, that is, intersections of exchange systems (including matroids), knapsacks, and unsplittable flows on trees. Later on, it turned out that this framework perfectly extends to optimization under uncertainty, like stochastic probing and online selection problems, which further can be applied to mechanism design. We add to this line of work by showing how to create contention resolution schemes for intersection of matroids and knapsacks when we work in the random order setting. More precisely, we do know the whole universe of elements in advance, but they appear in an order given by a random permutation. Upon arrival we need to irrevocably decide whether to take an element or not. We bring a novel technique for analyzing procedures in the random order setting that is based on the martingale theory. This unified approach makes it easier to combine constraints, and we do not need to rely on the monotonicity of contention resolution schemes.Our paper fills the gaps, extends, and creates connections between many previous results and techniques. The main application of our framework is a k + 4 + ε approximation ratio for the Bayesian multi-parameter unit-demand mechanism design under the constraint of k matroids intersection, which improves upon the previous bounds of 4k − 2 and e(k + 1). Other results include improved approximation ratios for stochastic k-set packing and submodular stochastic probing over arbitrary non-negative submodular objective function, whereas previous results required the objective to be monotone. * This is an extended version of a paper whose preliminary version appeared in

Submodular Stochastic Probing with Prices

“…These results are given in Section 4.3. These results outperform those of Feldman et al [15] (who study the adversarial setting) and Adamczyk and W lodarcyzk [3] (who study the random-order setting). This is also shown in Section 4.3.…”

“…These results hold in the adversarial setting. More recently, Adamczyk and W lodarcyzk [3] initiated the study of random order CRSs, and use such schemes to achieve the first approximations to non-monotone submodular SP in the random-order setting, giving a 1 (k+ℓ+1)eapproximation.…”

“…Similar to the algorithms presented in [16,2,3,15] for SP, our algorithms proceed by obtaining a fractional to a suitably chosen LP-relaxation of the problem, and then rounding according to both CRSs. The first major difficulty is that while…”

“…Using Measured Continuous Greedy [14] instead of continuous greedy and the modification of its proof used in [1] and [3], we are able to extend the approach of Sviridenko et al to non-monotone submodular functions and arbitrary constraints. The result is summarized as Theorem 3.…”

Guess Free Maximization of Submodular and Linear Sums

“…In another line of work, Chekuri et al [8] designed a framework called "contention resolution schemes" which yields a rounding procedure for every constraint that can be presented as the intersection of few simple constraints. Later works extended the contention resolution schemes framework into online and stochastic settings [1,12,13].…”