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 $k+4+\varepsilon$ 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.