TY - GEN

T1 - Solving optimization problems with diseconomies of scale via decoupling

AU - Makarychev, Konstantin

AU - Sviridenko, Maxim

N1 - Publisher Copyright:
© 2014 IEEE.
Copyright:
Copyright 2015 Elsevier B.V., All rights reserved.

PY - 2014/12/7

Y1 - 2014/12/7

N2 - We present a new framework for solving optimization problems with a diseconomy of scale. In such problems, our goal is to minimize the cost of resources used to perform a certain task. The cost of resources grows superlinearly, as xq, q ≥ 1, with the amount x of resources used. We define a novel linear programming relaxation for such problems, and then show that the integrality gap of the relaxation is Aq, where Aq is the q-th moment of the Poisson random variable with parameter 1. Using our framework, we obtain approximation algorithms for the Minimum Energy Efficient Routing, Minimum Degree Balanced Spanning Tree, Load Balancing on Unrelated Parallel Machines, and Unrelated Parallel Machine Scheduling with Nonlinear Functions of Completion Times problems. Our analysis relies on the decoupling inequality for nonnegative random variables. The inequality states that ||σn i=1 Xi||q ≤ Cq ||σni=1 Yi||q, where Xi are independent nonnegative random variables, Yi are possibly dependent nonnegative random variable, and each Yi has the same distribution as Xi. The inequality was proved by de la Pe;a in 1990. However, the optimal constant Cq was not known. We show that the optimal constant is Cq = Aq1/q.

AB - We present a new framework for solving optimization problems with a diseconomy of scale. In such problems, our goal is to minimize the cost of resources used to perform a certain task. The cost of resources grows superlinearly, as xq, q ≥ 1, with the amount x of resources used. We define a novel linear programming relaxation for such problems, and then show that the integrality gap of the relaxation is Aq, where Aq is the q-th moment of the Poisson random variable with parameter 1. Using our framework, we obtain approximation algorithms for the Minimum Energy Efficient Routing, Minimum Degree Balanced Spanning Tree, Load Balancing on Unrelated Parallel Machines, and Unrelated Parallel Machine Scheduling with Nonlinear Functions of Completion Times problems. Our analysis relies on the decoupling inequality for nonnegative random variables. The inequality states that ||σn i=1 Xi||q ≤ Cq ||σni=1 Yi||q, where Xi are independent nonnegative random variables, Yi are possibly dependent nonnegative random variable, and each Yi has the same distribution as Xi. The inequality was proved by de la Pe;a in 1990. However, the optimal constant Cq was not known. We show that the optimal constant is Cq = Aq1/q.

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U2 - 10.1109/FOCS.2014.67

DO - 10.1109/FOCS.2014.67

M3 - Conference contribution

AN - SCOPUS:84920060282

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 571

EP - 580

BT - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

PB - IEEE Computer Society

T2 - 55th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2014

Y2 - 18 October 2014 through 21 October 2014

ER -