Patrick Briest

Approximation techniques for utilitarian mechanism design

This paper deals with the design of efficiently computable incentive compatible, or truthful, mechanisms for combinatorial optimization problems with multi-parameter agents. We focus on approximation algorithms for NP-hard mechanism design problems. These algorithms need to satisfy certain monotonicity properties to ensure truthfulness. Since most of the known approximation techniques do not fulfill these properties, we study alternative techniques.Our first contribution is a quite general method to transform a pseudopolynomial algorithm into a monotone FPTAS. This can be applied to various problems like, e.g., knapsack, constrained shortest path, or job scheduling with deadlines. For example, the monotone FPTAS for the knapsack problem gives a very efficient, truthful mechanism for single-minded multi-unit auctions. The best previous result for such auctions was a 2-approximation. In addition, we present a monotone PTAS for the generalized assignment problem with any bounded number of parameters per agent.The most efficient way to solve packing integer programs (PIPs) is LP-based randomized rounding, which also is in general not monotone. We show that primal-dual greedy algorithms achieve almost the same approximation ratios for PIPs as randomized rounding. The advantage is that these algorithms are inherently monotone. This way, we can significantly improve the approximation ratios of truthful mechanisms for various fundamental mechanism design problems like single-minded combinatorial auctions (CAs), unsplittable flow routing and multicast routing. Our approximation algorithms can also be used for the winner determination in CAs with general bidders specifying their bids through an oracle.

Single-minded unlimited supply pricing on sparse instances

We deal with the problem of finding profit-maximizing prices for a finite number of distinct goods, assuming that of each good an unlimited number of copies is available, or that goods can be reproduced at no cost (e.g., digital goods). Consumers specify subsets of the goods and the maximum prices they are willing to pay. In the considered single-minded case every consumer is interested in precisely one such subset. If the goods are the edges of a graph and consumers are requesting to purchase paths in this graph, then we can think of the problem as pricing computer network connections or transportation links.We start by showing weak NP-hardness of the very restricted case in which the requested subsets are nested, i.e., contained inside each other or non-intersecting, thereby resolving the previously open question whether the problem remains NP-hard when the underlying graph is simply a line. Using a reduction inspired by this result we present an approximation preserving reduction that proves APX-hardness even for very sparse instances defined on general graphs, where the number of requests per edge is bounded by a constant B and no path is longer than some constant l. On the algorithmic side we first present an O(log l + log B)-approximation algorithm that (almost) matches the previously best known approximation guarantee in the general case, but is especially well suited for sparse problem instances. Using a new upper bounding technique we then give an O(l2)-approximation, which is the first algorithm for the general problem with an approximation ratio that does not depend on B.

Uniform Budgets and the Envy-Free Pricing Problem

We consider the unit-demand min-buying pricing problem, in which we want to compute revenue maximizing prices for a set of products

Stackelberg Network Pricing Games

\mathcal{P}

Approximation Techniques for Utilitarian Mechanism Design

assuming that each consumer from a set of consumer samples

Experimental Supplements to the Theoretical Analysis of EAs on Problems from Combinatorial Optimization

\mathcal{C}

Buying Cheap Is Expensive: Approximability of Combinatorial Pricing Problems

will purchase her cheapest affordable product once prices are fixed. We focus on the special uniform-budget case, in which every consumer has only a single non-zero budget for some set of products. This constitutes a special case also of the unit-demand envy-free pricing problem. We show that, assuming specific hardness of the balanced bipartite independent set problem in constant degree graphs or hardness of refuting random 3CNF formulas, the unit-demand min-buying pricing problem with uniform budgets cannot be approximated in polynomial time within

Improved hardness of approximation for stackelberg shortest-path pricing

\mathcal{O}(\log ^{\varepsilon} |\mathcal{C}|)

On Stackelberg Pricing with Computationally Bounded Consumers

for some ?> 0. This is the first result giving evidence that unit-demand envy-free pricing, as well, might be hard to approximate essentially better than within the known logarithmic ratio. We then introduce a slightly more general problem definition in which consumers are given as an explicit probability distribution and show that in this case the envy-free pricing problem can be shown to be inapproximable within

On stackelberg pricing with computationally bounded customers

\mathcal{O}(|\mathcal{P}|^{\varepsilon})