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13
An FPTAS for optimizing a class of lowrank functions over a polytope
, 2011
"... We present a fully polynomial time approximation scheme (FPTAS) for optimizing a very general class of nonlinear functions of low rank over a polytope. Our approximation scheme relies on constructing an approximate Paretooptimal front of the linear functions which constitute the given lowrank func ..."
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We present a fully polynomial time approximation scheme (FPTAS) for optimizing a very general class of nonlinear functions of low rank over a polytope. Our approximation scheme relies on constructing an approximate Paretooptimal front of the linear functions which constitute the given lowrank function. In contrast to existing results in the literature, our approximation scheme does not require the assumption of quasiconcavity on the objective function. For the special case of quasiconcave function minimization, we give an alternative FPTAS, which always returns a solution which is an extreme point of the polytope. Our technique can also be used to obtain an FPTAS for combinatorial optimization problems with nonlinear objective functions, for example when the objective is a product of a fixed number of linear functions. We also show that it is not possible to approximate the minimum of a general concave function over the unit hypercube to within any factor, unless P = NP. We prove this by showing a similar hardness of approximation result for supermodular function minimization, a result that may be of independent interest.
StierMoses. Stochastic selfish routing
 In SAGT
, 2011
"... Abstract. We embark on an agenda to investigate how stochastic delays and risk aversion transform traditional models of routing games and the corresponding equilibrium concepts. Moving from deterministic to stochastic delays with riskaverse players introduces nonconvexities that make the network ga ..."
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Cited by 6 (3 self)
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Abstract. We embark on an agenda to investigate how stochastic delays and risk aversion transform traditional models of routing games and the corresponding equilibrium concepts. Moving from deterministic to stochastic delays with riskaverse players introduces nonconvexities that make the network game more difficult to analyze even if one assumes that the variability of delays is exogenous. (For example, even computing players ’ best responses has an unknown complexity [24].) This paper focuses on equilibrium existence and characterization in the different settings of atomic vs. nonatomic players and exogenous vs. endogenous factors causing the variability of edge delays. We also show that succinct representations of equilibria always exist even though the game is nonadditive, i.e., the cost along a path is not a sum of costs over edges of the path as is typically assumed in selfish routing problems. Finally, we investigate the inefficiencies resulting from the stochastic nature of delays. We prove that under exogenous stochastic delays, the price of anarchy is exactly the same as in the corresponding game with deterministic delays. This implies that the stochastic delays and players’ risk aversion do not further degrade a system in the worstcase more than the selfishness of players. Keywords: Nonadditive nonatomic congestion game, stochastic Nash equilibrium, stochastic Wardrop equilibrium, risk aversion. 1
Risk Sensitivity of Price of Anarchy under Uncertainty
, 2013
"... In algorithmic game theory, the price of anarchy framework studies efficiency loss in decentralized environments. In optimization and decision theory, the price of robustness framework explores the tradeoffs between optimality and robustness in the case of single agent decision making under uncertai ..."
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In algorithmic game theory, the price of anarchy framework studies efficiency loss in decentralized environments. In optimization and decision theory, the price of robustness framework explores the tradeoffs between optimality and robustness in the case of single agent decision making under uncertainty. We establish a connection between the two that provides a novel analytic framework for proving tight performance guarantees for distributed systems in uncertain environments. We present applications of this framework to novel variants of atomic congestion games with uncertain costs, for which we provide tight performance bounds under a wide range of risk attitudes. Our results establish that the individual’s attitude towards uncertainty has a critical effect on system performance and should therefore be a subject of close and systematic investigation.
Maximizing expected utility for stochastic combinatorial optimization problems
 IN FOCS
, 2011
"... We study the stochastic versions of a broad class of combinatorial problems where the weights of the elements in the input dataset are uncertain. The class of problems that we study includes shortest paths, minimum weight spanning trees, and minimum weight matchings over probabilistic graphs, and ot ..."
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Cited by 5 (3 self)
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We study the stochastic versions of a broad class of combinatorial problems where the weights of the elements in the input dataset are uncertain. The class of problems that we study includes shortest paths, minimum weight spanning trees, and minimum weight matchings over probabilistic graphs, and other combinatorial problems like knapsack. We observe that the expected value is inadequate in capturing different types of riskaverse or riskprone behaviors, and instead we consider a more general objective which is to maximize the expected utility of the solution for some given utility function, rather than the expected weight (expected weight becomes a special case). We show that we can obtain a polynomial time approximation algorithm with additive error for any > 0, if there is a pseudopolynomial time algorithm for the exact version of the problem (This is true for the problems mentioned above) and the maximum value of the utility function is bounded by a constant. 1 Our result generalizes several prior results on stochastic shortest path, stochastic spanning tree, and stochastic knapsack. Our algorithm for utility maximization makes use of the separability of exponential utility and a technique to decompose a general utility function into exponential utility functions, which may be useful in other stochastic optimization problems.
Computing a Most Probable Delay Constrained Path: NPHardness and Approximation Schemes
"... Abstract — Delay constrained path selection is concerned with finding a sourcetodestination path so that the delay of the path is within a given delay bound. When the network is modeled by a directed graph where the delay of a link is a random variable with a known mean and a known variance, the p ..."
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Abstract — Delay constrained path selection is concerned with finding a sourcetodestination path so that the delay of the path is within a given delay bound. When the network is modeled by a directed graph where the delay of a link is a random variable with a known mean and a known variance, the problem becomes that of computing a most probable delay constrained path. In this paper, we present a comprehensive theoretical study of this problem. First, we prove that the problem is NPhard. Next, for the case where there exists a sourcetodestination path with a delay mean no more than the given delay bound, we present a fully polynomial time approximation scheme. In other words, for any given constant ɛ such that 0 <ɛ<1, our algorithm computes a path whose probability of satisfying the delay constraint is at least (1 − ɛ) times the probability that the optimal path satisfies the delay constraint, with a time complexity bounded by a polynomial in the number of network nodes and 1/ɛ. Finally, for the case where any sourcetodestination path has a delay mean larger than the given delay bound, we present a simple approximation algorithm with an approximation ratio bounded by the square root of the hopcount of the optimal path. Index Terms — Delay constrained path selection, computational complexity, approximation schemes. 1.
A MeanRisk Model for the Traffic Assignment Problem With Stochastic Travel Times
 FORTHCOMING IN OPERATIONS RESEARCH
, 2011
"... Heavy and uncertain traffic conditions exacerbate the commuting experience of millions of people across the globe. When planning important trips, commuters typically add an extra buffer to the expected trip duration to ensure ontime arrival. Motivated by this, we propose a new traffic assignment mo ..."
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Heavy and uncertain traffic conditions exacerbate the commuting experience of millions of people across the globe. When planning important trips, commuters typically add an extra buffer to the expected trip duration to ensure ontime arrival. Motivated by this, we propose a new traffic assignment model that takes into account the stochastic nature of travel times. Our model extends the traditional model of Wardrop competition when uncertainty is present in the network. The focus is on strategic riskaverse users who capture the tradeoff between travel times and their variability in a meanstandard deviation objective, defined as the mean travel time plus a riskaversion factor times the standard deviation of travel time along a path. We consider both infinitesimal users, leading to a nonatomic game, and atomic users, leading to a discrete finite game. We establish conditions that characterize an equilibrium traffic assignment and find when it exists. The main challenge is posed by the users’ risk aversion, since the meanstandard deviation objective is nonconvex and nonseparable, meaning that a path cannot be split as a sum of edge costs. As a result, even an individual user’s subproblem—a stochastic shortest path problem—is a nonconvex optimization problem for which no polynomial time algorithms are known. In turn, the mathematical structure of the traffic assignment model with stochastic travel times is fundamentally different from the deterministic counterpart. In particular,
Practical Route Planning Under Delay Uncertainty: Stochastic Shortest Path Queries
"... Abstract—We describe an algorithm for stochastic path planning and applications to route planning in the presence of traffic delays. We improve on the prior state of the art by designing, analyzing, implementing, and evaluating data structures that answer approximate stochastic shortestpath queries ..."
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Abstract—We describe an algorithm for stochastic path planning and applications to route planning in the presence of traffic delays. We improve on the prior state of the art by designing, analyzing, implementing, and evaluating data structures that answer approximate stochastic shortestpath queries. For example, our data structures can be used to efficiently compute paths that maximize the probability of arriving at a destination before a given time deadline. Our main theoretical result is an algorithm that, given a directed planar network with edge lengths characterized by expected travel time and variance, precomputes a data structure in quasilinear time such that approximate stochastic shortestpath queries can be answered in polylogarithmic time (actual worstcase bounds depend on the probabilistic model). Our main experimental results are twofold: (i) we provide methods to extract traveltime distributions from a large set of heterogenous GPS traces and we build a stochastic model of an entire city, and (ii) we adapt our algorithms to work for realworld road networks, we provide an efficient implementation, and we evaluate the performance of our method for the model of the aforementioned city. I.
A Fully PolynomialTime Approximation Scheme for TimingConstrained Minimum Cost Layer Assignment
"... Abstract—As VLSI technology enters the nanoscale regime, the interconnect delay becomes the bottleneck of circuit performance. Compared with gate delays, wires are becoming increasingly resistive, making it more difficult to propagate signals across the chip. However, more advanced technologies (65 ..."
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Abstract—As VLSI technology enters the nanoscale regime, the interconnect delay becomes the bottleneck of circuit performance. Compared with gate delays, wires are becoming increasingly resistive, making it more difficult to propagate signals across the chip. However, more advanced technologies (65 and 45 nm) provide relief as the number of metal layers continues to increase. The wires on the upper metal layers are much less resistive and can be used to drive further and faster than on thin metals. This provides an entirely new dimension to the traditional wiresizing problem, namely, layer assignment for efficient timing closure. Assigning all wires to thick metals improves timing; however, the routability of the design may be hurt. The challenge is to assign a minimal amount of wires to thick metals to meet timing constraints. In this brief, the minimum cost layer assignment problem is proven to be NPcomplete. As a theoretical solution for NPcomplete problems, a fully polynomialtime approximation scheme is proposed. The new algorithm can approximate the optimal layer assignment solution by a factor of 1+ɛ in O(m log log M · n 3 /ɛ 2) time for 0 <ɛ<1, wheren is the number of nodes in the tree, m is the number of routing layers, and M is the maximum cost ratio among layers. This work presents the first theoretical advance for the timingdriven minimum cost layer assignment problem. In addition to its theoretical guarantee, the new algorithm is highly practical. Our experiments on 500 industrial test cases demonstrate that the new algorithm can run 2 × faster than the optimal dynamic programming algorithm, with only 2 % additional wire. Index Terms—Fully polynomialtime approximation scheme (FPTAS), interconnect synthesis, layer assignment, NPcomplete. I.