Max-Min Greedy Matching

A bipartite graph G(U,V;E) that admits a perfect matching is given. One player imposes a permutation pi over V, the other player imposes a permutation sigma over U. In the greedy matching algorithm, vertices of U arrive in order sigma and each vertex is matched to the highest (under pi) yet unmatched neighbor in V (or left unmatched, if all its neighbors are already matched). The obtained matching is maximal, thus matches at least a half of the vertices. The max-min greedy matching problem asks: suppose the first (max) player reveals pi, and the second (min) player responds with the worst possible sigma for pi, does there exist a permutation pi ensuring to match strictly more than a half of the vertices? Can such a permutation be computed in polynomial time? The main result of this paper is an affirmative answer for these questions: we show that there exists a polytime algorithm to compute pi for which for every sigma at least rho > 0.51 fraction of the vertices of V are matched. We provide additional lower and upper bounds for special families of graphs, including regular and Hamiltonian graphs. Our solution solves an open problem regarding the welfare guarantees attainable by pricing in sequential markets with binary unit-demand valuations

Market Pricing for Matroid Rank Valuations

In this paper, we study the problem of maximizing social welfare in combinatorial markets through pricing schemes. We consider the existence of prices that are capable to achieve optimal social welfare without a central tie-breaking coordinator. In the case of two buyers with rank valuations, we give polynomial-time algorithms that always find such prices when one of the matroids is a simple partition matroid or both matroids are strongly base orderable. This result partially answers a question raised by D\"uetting and V\'egh in 2017. We further formalize a weighted variant of the conjecture of D\"uetting and V\'egh, and show that the weighted variant can be reduced to the unweighted one based on the weight-splitting theorem for weighted matroid intersection by Frank. We also show that a similar reduction technique works for M${}^\natural$-concave functions, or equivalently, gross substitutes functions

Makespan Minimization via Posted Prices

We consider job scheduling settings, with multiple machines, where jobs arrive online and choose a machine selfishly so as to minimize their cost. Our objective is the classic makespan minimization objective, which corresponds to the completion time of the last job to complete. The incentives of the selfish jobs may lead to poor performance. To reconcile the differing objectives, we introduce posted machine prices. The selfish job seeks to minimize the sum of its completion time on the machine and the posted price for the machine. Prices may be static (i.e., set once and for all before any arrival) or dynamic (i.e., change over time), but they are determined only by the past, assuming nothing about upcoming events. Obviously, such schemes are inherently truthful. We consider the competitive ratio: the ratio between the makespan achievable by the pricing scheme and that of the optimal algorithm. We give tight bounds on the competitive ratio for both dynamic and static pricing schemes for identical, restricted, related, and unrelated machine settings. Our main result is a dynamic pricing scheme for related machines that gives a constant competitive ratio, essentially matching the competitive ratio of online algorithms for this setting. In contrast, dynamic pricing gives poor performance for unrelated machines. This lower bound also exhibits a gap between what can be achieved by pricing versus what can be achieved by online algorithms

Pricing Multi-Unit Markets

We study the power and limitations of posted prices in multi-unit markets, where agents arrive sequentially in an arbitrary order. We prove upper and lower bounds on the largest fraction of the optimal social welfare that can be guaranteed with posted prices, under a range of assumptions about the designer's information and agents' valuations. Our results provide insights about the relative power of uniform and non-uniform prices, the relative difficulty of different valuation classes, and the implications of different informational assumptions. Among other results, we prove constant-factor guarantees for agents with (symmetric) subadditive valuations, even in an incomplete-information setting and with uniform prices

Posted Prices, Smoothness, and Combinatorial Prophet Inequalities

We present a general framework for proving combinatorial prophet inequalities and constructing posted-price mechanisms. Our framework applies to stochastic welfare optimization problems, in which buyers arrive sequentially and make utility-maximizing purchases. Our analysis takes the form of an extension theorem: we derive sufficient conditions for achieving welfare bounds in the special case of deterministic valuations, then prove that these bounds extend directly to stochastic settings. Furthermore, our welfare bounds compose in the sense that the welfare guarantees are preserved when buyers participate in many optimization problems simultaneously. Our sufficient conditions have a natural economic interpretation, and our approach is closely connected to the smoothness framework for bounding the price of anarchy of mechanisms. We show that many smooth mechanisms can be recast as posted-price mechanisms with comparable performance guarantees. We illustrate the power of our framework in a range of applications, including combinatorial auctions, matroids, and sparse packing programs, where we unify and improve many of the previously known results