Approximation algorithms for scheduling unrelated parallel machines
Mathematical Programming: Series A and B
ACM SIGecom Exchanges
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
An approximation algorithm for max-min fair allocation of indivisible goods
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
On allocations that maximize fairness
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Graph balancing: a special case of scheduling unrelated parallel machines
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
A polynomial-time approximation scheme for maximizing the minimum machine completion time
Operations Research Letters
Optimal Sherali-Adams Gaps from Pairwise Independence
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Approximating sparsest cut in graphs of bounded treewidth
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
A truthful constant approximation for maximizing the minimum load on related machines
WINE'10 Proceedings of the 6th international conference on Internet and network economics
An Approximation Algorithm for Max-Min Fair Allocation of Indivisible Goods
SIAM Journal on Computing
Santa Claus schedules jobs on unrelated machines
Proceedings of the forty-third annual ACM symposium on Theory of computing
Minimum congestion mapping in a cloud
Proceedings of the 30th annual ACM SIGACT-SIGOPS symposium on Principles of distributed computing
Integrality gaps of linear and semi-definite programming relaxations for Knapsack
IPCO'11 Proceedings of the 15th international conference on Integer programming and combinatoral optimization
On the configuration-LP for scheduling on unrelated machines
ESA'11 Proceedings of the 19th European conference on Algorithms
New Constructive Aspects of the Lovász Local Lemma
Journal of the ACM (JACM)
Quasi-polynomial local search for restricted max-min fair allocation
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
Hi-index | 0.00 |
We consider the problem of MaxMin allocation of indivisible goods. There are m items to be distributed among n players. Each player $i$ has a nonnegative valuation pij for an item j, and the goal is to allocate items to players so as to maximize the minimum total valuation received by each player. There is a large gap in our understanding of this problem. The best known positive result is an ~O(√ n)-approximation algorithm, while there is only a factor 2 hardness known. Better algorithms are known for the restricted assignment case where each item has exactly one nonzero value for the players. We study the effect of bounded degree for items: each item has a nonzero value for at most D players. We show that essentially the case D = 3 is equivalent to the general case, and give a 4-approximation algorithm for D = 2. The current algorithmic results for MaxMin Allocation are based on a complicated LP relaxation called the configuration LP. We present a much simpler LP which is equivalent in power to the configuration LP. We focus on a special case of MaxMin Allocation-a family of instances on which this LP has a polynomially large gap. The technical core of our result for this case comes from an algorithm for an interesting new optimization problem on directed graphs, MaxMinDegree Arborescence, where the goal is to produce an arborescence of large outdegree. We develop an nε-approximation for this problem that runs in nO(1/ε) time and obtain a a polylogarithmic approximation that runs in quasipolynomial time, using a lift-and-project inspired LP formulation. In fact, we show that our results imply a rounding algorithm for the relaxations obtained by t rounds of the Sherali-Adams hierarchy applied to a natural LP relaxation of the problem. Roughly speaking, the integrality gap of the relaxation obtained from t rounds of Sherali-Adams is at most n1/t. We are able to extend the latter result to a more general class of instances. Along the way, we prove a result about the existence of a perfect matching in a probabilistically pruned graph which may be of independent interest.