On Budget-Feasible Mechanism Design for Symmetric Submodular Objectives

Abstract

We study a class of procurement auctions with a budget constraint, where an auctioneer is interested in buying resources or services from a set of agents. Ideally, the auctioneer would like to select a subset of the resources so as to maximize his valuation function, without exceeding a given budget. As the resources are owned by strategic agents however, our overall goal is to design mechanisms that are truthful, budget-feasible, and obtain a good approximation to the optimal value. Budgetfeasibility creates additional challenges, making several approaches inapplicable in this setting. Previous results on budget-feasible mechanisms have considered mostly monotone valuation functions. In this work, we mainly focus on symmetric submodular valuations, a prominent class of non-monotone submodular functions that includes cut functions. We begin first with a purely algorithmic result, obtaining a 2e e−1 -approximation for maximizing symmetric submodular functions under a budget constraint. We view this as a standalone result of independent interest, as it is the best known factor achieved by a deterministic algorithm. We then proceed to propose truthful, budget feasible mechanisms (both deterministic and randomized), paying particular attention on the Budgeted Max Cut problem. Our results significantly improve the known approximation ratios for these objectives, while establishing polynomial running time for cases where only exponential mechanisms were known. At the heart of our approach lies an appropriate combination of local search algorithms with results for monotone submodular valuations, applied to the derived local optima. * A conference version appears in WINE 2017.

Budget-Feasible Mechanism Design for Non-Monotone Submodular Objectives

“…We already discussed in the Introduction that designing deterministic budget-feasible mechanisms has been elusive. Positive results are only known for specific well-behaved objectives [1,2,17,18,29,40,41] and, even worse, beyond monotone submodular valuation functions no deterministic O(1)-approximation mechanism is known, irrespectively of time or query complexity. We consider obtaining deterministic, budget-feasible, O(1)-approximation mechanisms-or showing that they do not exist-the most intriguing related open problem.…”

“…The parameter β is later set to 9.185 in order to get the approximation factor of Corollary 3.9 but, otherwise, our analysis is independent of its value. 2 in line 9 can be any approximation algorithm for unconstrained non-monotone submodular maximization. In particular, here we may use the deterministic 2-approximation algorithm of Buchbinder and Feldman [13].…”

“…All of our mechanisms are randomized and, in fact, random sampling is an essential building block in our approach. Obtaining a good estimate of the optimal value via random sampling has been crucial in previous works on budget-feasible mechanism design for monotone objectives as well [2,5,10,35]. Designing deterministic budget-feasible mechanisms seems very challenging.…”

“…The factor for subadditive functions was later improved to O(log n/log log n) by Bei et al [10], who also gave a randomized O(1)-approximation mechanism for XOS functions, albeit in exponential time in the value query model, and further initiated the Bayesian analysis in this setting. 2 Amanatidis et al [2] suggested O(1)-approximation mechanisms for a subclass of non-monotone submodular objectives, namely symmetric submodular objectives, however their approach does not seem to generalize beyond this subclass. For settings with additional combinatorial constraints, Amanatidis et al [1] and Leonardi et al [35] gave O(1)-approximation mechanisms for additive valuation functions subject to independent system constraints.…”

“…For settings with additional combinatorial constraints, Amanatidis et al [1] and Leonardi et al [35] gave O(1)-approximation mechanisms for additive valuation functions subject to independent system constraints. There is also a line of related work under the large market assumption (where no participant can significantly 2 Bei et al [10] propose an O (1)-approximation mechanism for non-decreasing XOS objectives that runs in polynomial time in the much stronger demand query model. However, they discuss how to extend their result to general XOS functions via the use ofˆ (S ) = max T ⊆S (T ).…”