In the Submodular Facility Location problem (SFL) we are given a collection of $n$ clients and $m$ facilities in a metric space. A feasible solution consists of an assignment of each client to some facility. For each client, one has to pay the distance to the associated facility. Furthermore, for each facility $f$ to which we assign the subset of clients $S^f$, one has to pay the opening cost $g(S^f)$, where $g(\cdot)$ is a monotone submodular function with $g(\emptyset)=0$. SFL is APX-hard since it includes the classical (metric uncapacitated) Facility Location problem (with uniform facility costs) as a special case. Svitkina and Tardos [SODA'06] gave the current-best $O(\log n)$ approximation algorithm for SFL. The same authors pose the open problem whether SFL admits a constant approximation and provide such an approximation for a very restricted special case of the problem. We make some progress towards the solution of the above open problem by presenting an $O(\log\log n)$ approximation. Our approach is rather flexible and can be easily extended to generalizations and variants of SFL. In more detail, we achieve the same approximation factor for the practically relevant generalizations of SFL where the opening cost of each facility $f$ is of the form $p_f+g(S^f)$ or $w_f\cdot g(S^f)$, where $p_f,w_f \geq 0$ are input values. We also obtain an improved approximation algorithm for the related Universal Stochastic Facility Location problem. In this problem one is given a classical (metric) facility location instance and has to a priori assign each client to some facility. Then a subset of active clients is sampled from some given distribution, and one has to pay (a posteriori) only the connection and opening costs induced by the active clients. The expected opening cost of each facility $f$ can be modelled with a submodular function of the set of clients assigned to $f$.

Capacitated k-median is one of the few outstanding optimization problems for which the existence of a polynomial time constant factor approximation algorithm remains an open problem. In a series of recent papers algorithms producing solutions violating either the number of facilities or the capacity by a multiplicative factor were obtained. However, to produce solutions without violations appears to be hard and potentially requires different algorithmic techniques. Notably, if parameterized by the number of facilities k, the problem is also W[2] hard, making the existence of an exact FPT algorithm unlikely. In this work we provide an FPT-time constant factor approximation algorithm preserving both cardinality and capacity of the facilities. The algorithm runs in time 2^O(k log k) n^O(1) and achieves an approximation ratio of 7+epsilon.

Contention resolution schemes have proven to be an incredibly powerful concept which allows tackling 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, as it was the case before. 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 nonnegative submodular objective function, whereas previous results required the objective to be monotone.

Consider the following variant of the set cover problem. We are given a universe U={1,...,n} and a collection of subsets C = {S_1,...,S_m} where each S_i is a subset of U. For every element u from U we need to find a set phi(u) from collection C such that u belongs to phi(u). Once we construct and fix the mapping phi from U to C a subset X from the universe U is revealed, and we need to cover all elements from X with exactly phi(X), that is {phi(u)}_{all u from X}. The goal is to find a mapping such that the cover phi(X) is as cheap as possible. This is an example of a universal problem where the solution has to be created before the actual instance to deal with is revealed. Such problems appear naturally in some settings when we need to optimize under uncertainty and it may be actually too expensive to begin finding a good solution once the input starts being revealed. A rich body of work was devoted to investigate such problems under the regime of worst case analysis, i.e., when we measure how good the solution is by looking at the worst-case ratio: universal solution for a given instance vs optimum solution for the same instance. As the universal solution is significantly more constrained, it is typical that such a worst-case ratio is actually quite big. One way to give a viewpoint on the problem that would be less vulnerable to such extreme worst-cases is to assume that the instance, for which we will have to create a solution, will be drawn randomly from some probability distribution. In this case one wants to minimize the expected value of the ratio: universal solution vs optimum solution. Here the bounds obtained are indeed smaller than when we compare to the worst-case ratio. But even in this case we still compare apples to oranges as no universal solution is able to construct the optimum solution for every possible instance. What if we would compare our approximate universal solution against an optimal universal solution that obeys the same rules as we do? We show that under this viewpoint, but still in the stochastic variant, we can indeed obtain better bounds than in the expected ratio model. For example, for the set cover problem we obtain $H_n$ approximation which matches the approximation ratio from the classic deterministic setup. Moreover, we show this for all possible probability distributions over $U$ that have a polynomially large carrier, while all previous results pertained to a model in which elements were sampled independently. Our result is based on rounding a proper configuration IP that captures the optimal universal solution, and using tools from submodular optimization. The same basic approach leads to improved approximation algorithms for other related problems, including Vertex Cover, Edge Cover, Directed Steiner Tree, Multicut, and Facility Location.

We study the revenue performance of sequential posted price mechanisms and some natural extensions, for a general setting where the valuations of the buyers are drawn from a correlated distribution. Sequential posted price mechanisms are conceptually simple mechanisms that work by proposing a “take-it-or-leave-it” offer to each buyer. We apply sequential posted price mechanisms to single-parameter multi-unit settings in which each buyer demands only one item and the mechanism can assign the service to at most k of the buyers. For standard sequential posted price mechanisms, we prove that with the valuation distribution having finite support, no sequential posted price mechanism can extract a constant fraction of the optimal expected revenue, even with unlimited supply. We extend this result to the case of a continuous valuation distribution when various standard assumptions hold simultaneously. In fact, it turns out that the best fraction of the optimal revenue that is extractable by a sequential posted price mechanism is proportional to the ratio of the highest and lowest possible valuation. We prove that for two simple generalizations of these mechanisms, a better revenue performance can be achieved: if the sequential posted price mechanism has for each buyer the option of either proposing an offer or asking the buyer for its valuation, then a $$\varOmega (1/\max \{1,d\})$$ fraction of the optimal revenue can be extracted, where d denotes the “degree of dependence” of the valuations, ranging from complete independence ( $$d=0$$ ) to arbitrary dependence ( $$d = n-1$$ ). When we generalize the sequential posted price mechanisms further, such that the mechanism has the ability to make a take-it-or-leave-it offer to the i-th buyer that depends on the valuations of all buyers except i, we prove that a constant fraction $$(2 - \sqrt{e})/4 \approx 0.088$$ of the optimal revenue can be always extracted.

In a stochastic probing problem we are given a universe E, where each element e in E is active independently with probability p in [0,1], and only a probe of e can tell us whether it is active or not. On this universe we execute a process that one by one probes elements - if a probed element is active, then we have to include it in the solution, which we gradually construct. Throughout the process we need to obey inner constraints on the set of elements taken into the solution, and outer constraints on the set of all probed elements. This abstract model was presented in [Gupta and Nagaraja, IPCO 2013], and provides a unified view of a number of problems. Thus far all the results in this general framework pertain only to the case in which we are maximizing a linear objective function of the successfully probed elements. In this paper we generalize the stochastic probing problem by considering a monotone submodular objective function. We give a (1-1/e)/(k_in+k_out+1)-approximation algorithm for the case in which we are given k_in greater than 0 matroids as inner constraints and k_out greater than 1 matroids as outer constraints. There are two main ingredients behind this result. First is a previously unpublished stronger bound on the continuous greedy algorithm due to Vondrak. Second is a rounding procedure that also allows us to obtain an improved 1/(k_in+k_out)-approximation for linear objective functions.