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14
Multiresource allocation: Fairnessefficiency tradeoffs in a unifying framework
 in Proc. IEEE INFOCOM
, 2012
"... Abstract—Quantifying the notion of fairness is underexplored when there are multiple types of resources and users request different ratios of the different resources. A typical example is datacenters processing jobs with heterogeneous resource requirements on CPU, memory, network, bandwidth, etc. T ..."
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Cited by 34 (1 self)
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Abstract—Quantifying the notion of fairness is underexplored when there are multiple types of resources and users request different ratios of the different resources. A typical example is datacenters processing jobs with heterogeneous resource requirements on CPU, memory, network, bandwidth, etc. This paper develops a unifying framework addressing the fairnessefficiency tradeoff in light of multiple types of resources. We develop two families of fairness functions that provide different tradeoffs, characterize the effect of user requests ’ heterogeneity, and prove conditions under which these fairness measures satisfy the Pareto efficiency, sharing incentive, and envyfree properties. Intuitions behind the analysis are explained in two visualizations of multiresource allocation. We also investigate people’s fairness perceptions through an online survey of allocation preferences and provide a brief overview of related work on fairness.
Combinatorial Auctions, Knapsack Problems, and Hillclimbing Search
 In Canadian Conference on AI
, 2001
"... . This paper examines the performance of hillclimbing algorithms on standard test problems for combinatorial auctions (CAs). On singleunit CAs, deterministic hillclimbers are found to perform well, and their performance can be improved significantly by randomizing them and restarting them sev ..."
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Cited by 21 (1 self)
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. This paper examines the performance of hillclimbing algorithms on standard test problems for combinatorial auctions (CAs). On singleunit CAs, deterministic hillclimbers are found to perform well, and their performance can be improved significantly by randomizing them and restarting them several times, or by using them collectively. For some problems this good performance is shown to be no better than chancel; on others it is due to a wellchosen scoring function. The paper draws attention to the fact that multiunit CAs have been studied widely under a different name: multidimensional knapsack problems (MDKP). On standard test problems for MDKP, one of the deterministic hillclimbers generates solutions that are on average 99% of the best known solutions. 1 Introduction Suppose there are three items for auction, X, Y, and Z, and three bidders, B1, B2, and B3. B1 wants any one of the items and will pay $5, B2 wants two items  X and one of Y or Z  and will pay $9, an...
Budgeted matching and budgeted matroid intersection via the gasoline puzzle
 IN PROC. IPCO 2008
, 2008
"... Many polynomialtime solvable combinatorial optimization problems become NPhard if an additional complicating constraint is added to restrict the set of feasible solutions. In this paper, we consider two such problems, namely maximumweight matching and maximumweight matroid intersection with on ..."
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Cited by 17 (4 self)
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Many polynomialtime solvable combinatorial optimization problems become NPhard if an additional complicating constraint is added to restrict the set of feasible solutions. In this paper, we consider two such problems, namely maximumweight matching and maximumweight matroid intersection with one additional budget constraint. We present the first polynomialtime approximation schemes for these problems. Similarly to other approaches for related problems, our schemes compute two solutions to the Lagrangian relaxation of the problem and patch them together to obtain a nearoptimal solution. However, due to the richer combinatorial structure of the problems considered here, standard patching techniques do not apply. To circumvent this problem, we crucially exploit the adjacency relations on the solution polytope and, surprisingly, the solution to an old combinatorial puzzle.
Fast approximation schemes for multicriteria combinatorial Optimization
, 1994
"... The solution to an instance of the standard Shortest Path problem is a single shortest route in a directed graph. Suppose, however, that each arc has both a distance and a cost, and that one would like to find a route that is both short and inexpensive. In general, no single route will be both short ..."
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Cited by 15 (1 self)
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The solution to an instance of the standard Shortest Path problem is a single shortest route in a directed graph. Suppose, however, that each arc has both a distance and a cost, and that one would like to find a route that is both short and inexpensive. In general, no single route will be both shortest and cheapest; rather, the solution to an instance of this multicriteria problem will be a set of efficient or Pareto optimal routes. The (distance, cost) pairs associated with the efficient routes define an efficient frontier or tradeoff curve. An efficient set for a multicriteria problem can be exponentially large, even when the underlying singlecriterion;oblem is in P. This work therefore considers approximate solutions to rlulticriteria discrete optimization problems and investigates when they can be found quickly. This requires generalizing the notion of a fully polynomial time approximatiofi scheme to multicriteria problems. In this paper, necessary and sufficient conditions are developed for the existence of such a fast approximation scheme for a problem. Although the focus is multicriteria problems, the conditions are of interest even in the single criterion case. In addition, an appropriate form of problem reduction is introduced to facilitate the application of these conditions to a variety of problems. A companion paper uses the results of this paper to study the existence of fast approximation schemes for several interesting network flow, knapsack, and
Approximability of sparse integer programs
 In Proc. 17th ESA
, 2009
"... The main focus of this paper is a pair of new approximation algorithms for sparse integer programs. First, for covering integer programs {min cx: Ax ≥ b,0 ≤ x ≤ d} where A has at most k nonzeroes per row, we give a kapproximation algorithm. (We assume A, b, c, d are nonnegative.) For any k ≥ 2 and ..."
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Cited by 9 (1 self)
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The main focus of this paper is a pair of new approximation algorithms for sparse integer programs. First, for covering integer programs {min cx: Ax ≥ b,0 ≤ x ≤ d} where A has at most k nonzeroes per row, we give a kapproximation algorithm. (We assume A, b, c, d are nonnegative.) For any k ≥ 2 and ǫ> 0, if P = NP this ratio cannot be improved to k − 1 − ǫ, and under the unique games conjecture this ratio cannot be improved to k − ǫ. One key idea is to replace individual constraints by others that have better rounding properties but the same nonnegative integral solutions; another critical ingredient is knapsackcover inequalities. Second, for packing integer programs {max cx: Ax ≤ b,0 ≤ x ≤ d} where A has at most k nonzeroes per column, we give a 2 k k 2approximation algorithm. This is the first polynomialtime approximation algorithm for this problem with approximation ratio depending only on k, for any k> 1. Our approach starts from iterated LP relaxation, and then uses probabilistic and greedy methods to recover a feasible solution. Note added after publication: This version includes subsequent developments: a O(k 2) approximation for the latter problem using the iterated rounding framework, and several literature reference updates including a O(k)approximation for the same problem by Bansal et al.
Vector bin packing with multiplechoice
 Discrete Appl. Math
, 2012
"... We consider a variant of bin packing calledmultiplechoice vector bin packing. In this problem we are given a set of n items, where each item can be selected in one of several Ddimensional incarnations. We are also given T bin types, each with its own cost and Ddimensional size. Our goal is to pac ..."
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Cited by 7 (0 self)
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We consider a variant of bin packing calledmultiplechoice vector bin packing. In this problem we are given a set of n items, where each item can be selected in one of several Ddimensional incarnations. We are also given T bin types, each with its own cost and Ddimensional size. Our goal is to pack the items in a set of bins of minimum overall cost. The problem is motivated by scheduling in networks with guaranteed quality of service (QoS), but due to its general formulation it has many other applications as well. We present an approximation algorithm that is guaranteed to produce a solution whose cost is about lnD times the optimum. For the running time to be polynomial we require D = O(1) and T = O(log n). This extends previous results for vector bin packing, in which each item has a single incarnation and there is only one bin type. To obtain our result we also present a PTAS for the multiplechoice version of multidimensional knapsack, where we are given only one bin and the goal is to pack a maximum weight set of (incarnations of) items in that bin.
There is No EPTAS for Twodimensional Knapsack
"... In the ddimensional knapsack problem given is a set of items, each having a ddimensional size vector and a profit, and a ddimensional bin. The goal is to select a subset of the items of maximum total profit such that the sum of all vectors is bounded by the bin capacity in each dimension. It is w ..."
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Cited by 6 (2 self)
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In the ddimensional knapsack problem given is a set of items, each having a ddimensional size vector and a profit, and a ddimensional bin. The goal is to select a subset of the items of maximum total profit such that the sum of all vectors is bounded by the bin capacity in each dimension. It is well known that, unless P = NP, there is no fully polynomial time approximation scheme for ddimensional knapsack, already for d = 2. The best known result is a polynomial time approximation scheme (PTAS) due to Frieze and Clarke (European J. of Operational Research, 100–109, 1984) for the case where d ≥ 2 is some fixed constant. A fundamental open question is whether the problem admits an efficient PTAS (EPTAS). In this paper we resolve this question by showing that there is no EPTAS for ddimensional knapsack, already for d = 2, unless W [1] = F P T. Furthermore, we show that unless all problems in SNP are solvable in subexponential time, there is no approximation scheme for twodimensional knapsack whose running time is f(1/ε)I  o( √ 1/ε), for any function f. Together, the two results suggest that a significant improvement over the running time of the scheme of Frieze and Clarke is unlikely to exist. 1
A Local Search Based Heuristic for the Demand Constrained Multidimensional Knapsack Problem
, 2001
"... We consider an extension of the 01 multidimensional knapsack problem in which there are greaterthanequal inequalities, called demand constraints, besides the standard lessthanequal constraints. Moreover the objective function coefficients are not constrained in sign. This problem is worth consi ..."
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Cited by 1 (0 self)
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We consider an extension of the 01 multidimensional knapsack problem in which there are greaterthanequal inequalities, called demand constraints, besides the standard lessthanequal constraints. Moreover the objective function coefficients are not constrained in sign. This problem is worth considering because it is embedded in the models of practical applications, it has an intriguing combinatorial structure and it appears to be a challenging problem for commercial ILP solvers. Our approach is based on a nested tabu search algorithm in which neighborhoods of different structure are exploited. A first tabu search procedure is carried on in which mainly the infeasible region is explored. Once feasibility has been gained, a second tabu search procedure, which analyses only feasible solutions, is applied. The algorithm has been tested on a wide set of instances. Computational results are discussed.
On bounds available from branchandprice decomposition of the multidimensional knapsack problem with generalized upper bound constraints
, 2007
"... This paper describes and evaluates alternative branchandprice decompositions (B&PDs) of the multidimensional knapsack problem with generalized upper bound constraints both analytically and computationally. As part of our theoretical analysis we compare the bounds available from B&PDs wi ..."
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This paper describes and evaluates alternative branchandprice decompositions (B&PDs) of the multidimensional knapsack problem with generalized upper bound constraints both analytically and computationally. As part of our theoretical analysis we compare the bounds available from B&PDs with three alternative relaxations (Lagrangean relaxation, Lagrangean decomposition, and Surrogate relaxation) and whether incorporating a surrogate constraint can make an improvement or not. Our computational tests compare alternative ways of implementing B&PD to assess the trade off between the tightness of resulting bounds and the run times required to obtain them.