Coverage, Matching, and Beyond: New Results on Budgeted Mechanism Design

Abstract

We study a type of reverse (procurement) auction problems in the presence of budget constraints. The general algorithmic problem is to purchase a set of resources, which come at a cost, so as not to exceed a given budget and at the same time maximize a given valuation function. This framework captures the budgeted version of several well known optimization problems, and when the resources are owned by strategic agents the goal is to design truthful and budget feasible mechanisms, i.e. elicit the true cost of the resources and ensure the payments of the mechanism do not exceed the budget. Budget feasibility introduces more challenges in mechanism design, and we study instantiations of this problem for certain classes of submodular and XOS valuation functions. We first obtain mechanisms with an improved approximation ratio for weighted coverage valuations, a special class of submodular functions that has already attracted attention in previous works. We then provide a general scheme for designing randomized and deterministic polynomial time mechanisms for a class of XOS problems. This class contains problems whose feasible set forms an independence system (a more general structure than matroids), and some representative problems include, among others, finding maximum weighted matchings, maximum weighted matroid members, and maximum weighted 3D-matchings. For most of these problems, only randomized mechanisms with very high approximation ratios were known prior to our results.

On Budget-Feasible Mechanism Design for Symmetric Submodular Objectives

“…inferred from[3] and Lemma D.2). Mech-SM-frac is deterministic, truthful, individually rational, budget-feasible, and has approximation ratio ρ + 2 + ρ 2 + 4ρ + 1.Also, it runs in polynomial time as long as the exact algorithm for the relaxed problem runs in polynomial time.Mech-SM-frac(A, v, c, B)1 Set A = {i | c i ≤ B} and i * ∈ arg max i∈A v(i) 2 if ρ + 1 + ρ 2 + 4ρ + 1 · v(i * ) ≥ opt f (A {i * }, B) Greedy-SM(A, v, c, B/2)Note that when the relaxed problem is the same as the original (ρ = 1), then Mech-SMfrac becomes Mech-SM and the two approximation ratios coincide.…”

“…Later, Chen et al [10] significantly improved Singer's results, obtaining a randomized 7.91-approximation mechanism and a deterministic 8.34-approximation mechanism for non-decreasing submodular functions. Several modifications of the mechanism by [10] have been proposed that run in polynomial time for special cases [31,19,3]. For subadditive functions, Dobzinski et al [12] suggested a randomized O(log 2 n)-approximation mechanism, and they gave the first constant factor mechanisms for non-monotone submodular objectives, specifically for cut functions.…”

“…In order to design deterministic mechanisms that run in polynomial time, we first optimize a deterministic mechanism of [3] to obtain Mech-SM-frac below, which is applicable for non-decreasing submodular functions. The difference with Mech-SM from Section 2, is that now we assume that a fractional relaxation of our problem can be solved optimally and that the fractional optimal solution is within a constant of the integral solution.…”

“…By substituting the values for p, κ we get opt(A, v, c, B) ≤ 243.2 · E(v(X M )). 3. v(i * ) ≤ 1 κ opt (A, v, c, B).…”

“…Hence, our ideal goal is to have truthful mechanisms that achieve a good approximation to the optimal value for the auctioneer, and are budget feasible, i.e., the sum of the payments to the agents does not exceed the prespecified budget. This framework of budget feasible mechanisms is motivated by recent application scenarios including crowdsourcing platforms, where agents can be viewed as workers providing tasks (e.g., [4,16]), and influence maximization in networks, where agents correspond to influential users (see e.g., [31,3], where the chosen objective is a coverage function).…”