742 research outputs found
On the Configuration LP for Maximum Budgeted Allocation
We study the Maximum Budgeted Allocation problem, i.e., the problem of
selling a set of indivisible goods to players, each with a separate
budget, such that we maximize the collected revenue. Since the natural
assignment LP is known to have an integrality gap of , which
matches the best known approximation algorithms, our main focus is to improve
our understanding of the stronger configuration LP relaxation. In this
direction, we prove that the integrality gap of the configuration LP is
strictly better than , and provide corresponding polynomial time
roundings, in the following restrictions of the problem: (i) the Restricted
Budgeted Allocation problem, in which all the players have the same budget and
every item has the same value for any player it can be sold to, and (ii) the
graph MBA problem, in which an item can be assigned to at most 2 players.
Finally, we improve the best known upper bound on the integrality gap for the
general case from to and also prove
hardness of approximation results for both cases.Comment: 29 pages, 4 figures. To appear in the 17th Conference on Integer
Programming and Combinatorial Optimization (IPCO), 201
Lottery pricing equilibria
We extend the notion of Combinatorial Walrasian Equilibrium, as defined by Feldman et al. [2013], to settings with budgets. When agents have budgets, the maximum social welfare as traditionally defined is not a suitable benchmark since it is overly optimistic. This motivated the liquid welfare of [Dobzinski and Paes Leme 2014] as an alternative. Observing that no combinatorial Walrasian equilibrium guarantees a non-zero fraction of the maximum liquid welfare in the absence of randomization, we instead work with randomized allocations and extend the notions of liquid welfare and Combinatorial Walrasian Equilibrium accordingly. Our generalization of the Combinatorial Walrasian Equilibrium prices lotteries over bundles of items rather than bundles, and we term it a lottery pricing equilibrium. Our results are two-fold. First, we exhibit an efficient algorithm which turns a randomized allocation with liquid expected welfare W into a lottery pricing equilibrium with liquid expected welfare 3-√5/2 W (≈ 0.3819-W). Next, given access to a demand oracle and an α-approximate oblivious rounding algorithm for the configuration linear program for the welfare maximization problem, we show how to efficiently compute a randomized allocation which is (a) supported on polynomially-many deterministic allocations and (b) obtains [nearly] an α fraction of the optimal liquid expected welfare. In the case of subadditive valuations, combining both results yields an efficient algorithm which computes a lottery pricing equilibrium obtaining a constant fraction of the optimal liquid expected welfare. © Copyright 2016 ACM
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