An algorithmic characterization of antimatroids

In an article entitled “Optimal sequencing of a single machine subject to precedence constraints” E.L. Lawler presented a now classical minmax result for job scheduling. In essence, Lawler's proof demonstrated that the properties of partially ordered sets were sufficient to solve the posed scheduling problem. These properties are, in fact, common to a more general class of combinatorial structures known as antimatroids, which have recently received considerable attention in the literature. It is demonstrated that the properties of antimatroids are not only sufficient but necessary to solve the scheduling problem posed by Lawler, thus yielding an algorithmic characterization of antimatroids. Examples of problems solvable by the general result are provided

(Near) Optimal Adaptivity Gaps for Stochastic Multi-Value Probing

Consider a kidney-exchange application where we want to find a max-matching in a random graph. To find whether an edge e exists, we need to perform an expensive test, in which case the edge e appears independently with a known probability p_e. Given a budget on the total cost of the tests, our goal is to find a testing strategy that maximizes the expected maximum matching size. The above application is an example of the stochastic probing problem. In general the optimal stochastic probing strategy is difficult to find because it is adaptive - decides on the next edge to probe based on the outcomes of the probed edges. An alternate approach is to show the adaptivity gap is small, i.e., the best non-adaptive strategy always has a value close to the best adaptive strategy. This allows us to focus on designing non-adaptive strategies that are much simpler. Previous works, however, have focused on Bernoulli random variables that can only capture whether an edge appears or not. In this work we introduce a multi-value stochastic probing problem, which can also model situations where the weight of an edge has a probability distribution over multiple values. Our main technical contribution is to obtain (near) optimal bounds for the (worst-case) adaptivity gaps for multi-value stochastic probing over prefix-closed constraints. For a monotone submodular function, we show the adaptivity gap is at most 2 and provide a matching lower bound. For a weighted rank function of a k-extendible system (a generalization of intersection of k matroids), we show the adaptivity gap is between O(k log k) and k. None of these results were known even in the Bernoulli case where both our upper and lower bounds also apply, thereby resolving an open question of Gupta et al. [Gupta et al., 2017]

Robust and MaxMin Optimization under Matroid and Knapsack Uncertainty Sets

Consider the following problem: given a set system (U,I) and an edge-weighted graph G = (U, E) on the same universe U, find the set A in I such that the Steiner tree cost with terminals A is as large as possible: "which set in I is the most difficult to connect up?" This is an example of a max-min problem: find the set A in I such that the value of some minimization (covering) problem is as large as possible. In this paper, we show that for certain covering problems which admit good deterministic online algorithms, we can give good algorithms for max-min optimization when the set system I is given by a p-system or q-knapsacks or both. This result is similar to results for constrained maximization of submodular functions. Although many natural covering problems are not even approximately submodular, we show that one can use properties of the online algorithm as a surrogate for submodularity. Moreover, we give stronger connections between max-min optimization and two-stage robust optimization, and hence give improved algorithms for robust versions of various covering problems, for cases where the uncertainty sets are given by p-systems and q-knapsacks.Comment: 17 pages. Preliminary version combining this paper and http://arxiv.org/abs/0912.1045 appeared in ICALP 201