On the configuration LP for maximum budgeted allocation

Kalaitzis, Christos; Mądry, Aleksander; Newman, Alantha; Poláček, Lukáš; Svensson, Ola

doi:10.1007/s10107-015-0928-8

Cited by 7 publications

(19 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The best known lower bound on the integrality gap of the Configuration-LP is 3/4, and thus is not better than that of the Assignment-LP. However, it is known [9] that for certain restrictions of the MBA problem, the integrality gap is larger than 3/4, and hence this is an indication that the Configuration-LP might actually be stronger than the Assignment-LP in general. This is the central question that we address.…”

Section: Preliminariesmentioning

confidence: 99%

“…This is the central question that we address. Now, it is not hard to see (see, for example, [9]) that we can transform any given solution to the Configuration-LP into a solution to the Assignment-LP of at least the same objective value (of course, the inverse is not possible); hence, we will use y to denote a solution to the Configuration-LP and x the corresponding solution to the Assignment-LP. We should note that throughout the paper, our algorithms will only consider assigning j to i if x ij is in the support of x, i.e., if x ij > 0.…”

Section: Preliminariesmentioning

confidence: 99%

“…Since, given an allocation of items to players, computing optimal prices that respect the agents' budgets is easy, it becomes clear that the most difficult aspect of the problem is finding such an allocation. The MBA problem is a generic allocation problem, which has In this direction, the Configuration-LP was employed by Kalaitzis et al [9] to provide a 3/4 + capproximation algorithm (for some small constant c > 0) for a restricted version of the MBA problem (which they call Restricted MBA), in which all the players have uniform budgets, and the items have uniform prices. The use of the Configuration-LP for the MBA problem was first proposed by Chakrabarty and Goel [5].…”

Section: Introductionmentioning

confidence: 99%

“…The use of the Configuration-LP for the MBA problem was first proposed by Chakrabarty and Goel [5]. The worst known upper bound on the integrality gap of the Configuration-LP is 2( √ 2 − 1) [9].…”

Section: Introductionmentioning

confidence: 99%

“…Our contributions Remember that, for the Restricted MBA problem, we already know an approximation algorithm with a guarantee strictly better than 3/4 [9]. Having this result at hand, a natural way to attack the MBA problem is to try to prove that designing an algorithm with a similar guarantee is possible for any family of instances that does not fall into this restriction.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

An Improved Approximation Guarantee for the Maximum Budgeted Allocation Problem

Kalaitzis

2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

Self Cite

View full text Add to dashboard Cite

We study the Maximum Budgeted Allocation problem, which is the problem of assigning indivisible items to players with budget constraints. In its most general form, an instance of the MBA problem might include many different prices for the same item among different players, and different budget constraints for every player. So far, the best approximation algorithms we know for the MBA problem achieve a 3/4-approximation ratio, and employ a natural LP relaxation, called the Assignment-LP. In this paper, we give an algorithm for MBA, and prove that it achieves a 3/4 + c-approximation ratio, for some constant c > 0. This algorithm works by rounding solutions to an LP called the Configuration-LP, therefore also showing that the Configuration-LP is strictly stronger than the Assignment-LP (for which we know that the integrality gap is 3/4) for the MBA problem. 1 received significant attention from the research community. The reason is that it arises naturally in the context of multiple practical scenarios, such as when we want to maximize the revenue in an auction with budget-constrained agents. However, it is an NP-hard problem, and therefore hard to solve optimally. Therefore, one natural approach is to take up approximation algorithms in the course of tackling it.So far, the best algorithms we know achieve a 3/4-approximation ratio and are due to Srinivasan [14] and Chakrabarty and Goel [5]. These algorithms work by rounding solutions to the Assignment-LP relaxation, a relaxation for which we know the integrality gap is no better than 3/4. Since our goal is to improve upon the 3/4 ratio, we will have to use a stronger relaxation. Configuration LP-s have often been used in the past to provide good approximation algorithms (e.g., for the Bin Packing problem [10,12]); towards such an end, Chakrabarty and Goel [5] proposed the use of the Configuration-LP for the MBA problem, and conjectured that its integrality gap should be better than 3/4. At this point, it was already observed [5] that the restriction of items having uniform prices (i.e., an item can be assigned to a subset of the players, and every player is willing to pay the same price for that item) is one of the most natural restrictions of MBA that do not make the problem too easy. This is in line, for example, with what was already observed for the Generalized Assignment Problem (GAP), i.e., the problem of finding an allocation of items to bins of size 1 which maximizes the total value, where every item has a separate size and value for each bin. Indeed, in their work on designing an algorithm with a better than 1 − 1/e-approximation ratio for GAP, Feige and Vondrák [6] show how to deal with uniform instances, and then focus on the rest; interestingly, their techniques rely on the use of the Configuration-LP. Furthermore, these facts highlight some structural differences between different allocation problems: while problems such as GAP and MBA might not get substantially harder when the item values are non-uniform, this is not necessarily the case for other al...

show abstract

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%