Mathematics of Operations Research current issue

Absorbing Games with Compact Action Spaces

Mertens, J.-F., Neyman, A., Rosenberg, D. — 2009-05-14

We prove that games with absorbing states with compact action sets have a value.

Time Average Replicator and Best-Reply Dynamics

Hofbauer, J., Sorin, S., Viossat, Y. — 2009-05-14

Using an explicit representation in terms of the logit map, we show, in a unilateral framework, that the time average of the replicator dynamics is a perturbed solution of the best-reply dynamics.

Online Primal-Dual Algorithms for Covering and Packing

Buchbinder, N., Naor, J. — 2009-05-14

We study a wide range of online covering and packing optimization problems. In an online covering problem, a linear cost function is known in advance, but the linear constraints that define the feasible solution space are given one by one, in rounds. In an online packing problem, the profit function as well as the packing constraints are not known in advance. In each round additional information (i.e., a new variable) about the profit function and the constraints is revealed. An online algorithm needs to maintain a feasible solution in each round; in addition, the solutions generated over the different rounds need to satisfy a monotonicity property. We provide general deterministic primal-dual algorithms for online fractional covering and packing problems. We also provide deterministic algorithms for several integral online covering and packing problems. Our algorithms are designed via a novel online primal-dual technique and are evaluated via competitive analysis.

Partially Observed Markov Decision Process Multiarmed Bandits--Structural Results

Krishnamurthy, V., Wahlberg, B. — 2009-05-14

This paper considers multiarmed bandit problems involving partially observed Markov decision processes (POMDPs). We show how the Gittins index for the optimal scheduling policy can be computed by a value iteration algorithm on each process, thereby considerably simplifying the computational cost. A suboptimal value iteration algorithm based on Lovejoy's approximation is presented. We then show that for the case of totally positive of order 2 (TP2) transition probability matrices and monotone likelihood ratio (MLR) ordered observation probabilities, the Gittins index is MLR increasing in the information state. Algorithms that exploit this structure are then presented.

Penalty and Smoothing Methods for Convex Semi-Infinite Programming

Auslender, A., Goberna, M. A., Lopez, M. A. — 2009-05-14

In this paper we consider min-max convex semi-infinite programming. To solve these problems we introduce a unified framework concerning Remez-type algorithms and integral methods coupled with penalty and smoothing methods. This framework subsumes well-known classical algorithms, but also provides some new methods with interesting properties. Convergence of the primal and dual sequences are proved under minimal assumptions.

Credit Risk Models with Incomplete Information

Guo, X., Jarrow, R. A., Zeng, Y. — 2009-05-14

Incomplete information is at the heart of information-based credit risk models. In this paper, we rigorously define incomplete information with the notion of "delayed filtrations." We characterize two distinct types of delayed information, continuous and discrete: the first generated by a time change of filtrations and the second by finitely many marked point processes. This notion unifies the noisy information in Duffie and Lando [Duffie, D., D. Lando. 2001. Term structures and credit spreads with incomplete accounting information. Econometrica 69 633–664] and the notion of partial information in Collin-Dufresne et al. [Collin-Dufresne, P., R. Goldstein, J. Helwege. 2003. Is credit event risk priced? Modeling contagion via the updating of beliefs. Working paper, Carnegie Mellon University, Pittsburgh], under which structural models are translated into reduced-form intensity-based models. We illustrate through a simple example the importance of this notion of delayed information, as well as the potential pitfall for abusing the Laplacian approximation techniques for calculating the intensity process in an information-based model.

Stochastic Depletion Problems: Effective Myopic Policies for a Class of Dynamic Optimization Problems

Chan, C. W., Farias, V. F. — 2009-05-14

This paper presents a general class of dynamic stochastic optimization problems we refer to as stochastic depletion problems. A number of challenging dynamic optimization problems of practical interest are stochastic depletion problems. Optimal solutions for such problems are difficult to obtain, both from a pragmatic computational perspective as well as from a theoretical perspective. As such, simple heuristics are desirable. We isolate two simple properties that, if satisfied by a problem within this class, guarantee that a myopic policy incurs a performance loss of at most 50% relative to the optimal adaptive control policy for that problem. We are able to verify that these two properties are satisfied for several interesting families of stochastic depletion problems and, as a consequence, we identify computationally efficient approximations to optimal control policies for a number of interesting dynamic stochastic optimization problems.

A Weighted kt, t-Free t-Factor Algorithm for Bipartite Graphs

Takazawa, K. — 2009-05-14

For a simple bipartite graph and an integer t ≥ 2, we consider the problem of finding a minimum-weight k_{t, t}-free t-factor, which is a t-factor containing no complete bipartite graph k_{t, t} as a subgraph. When t = 2, this problem amounts to the square-free 2-factor problem in a bipartite graph. For the unweighted square-free 2-factor problem, a combinatorial algorithm is given by Hartvigsen, and the weighted version of the problem is NP-hard. For general t, Pap designed a combinatorial algorithm for the unweighted version, and Makai gave a dual integral description of k_{t, t}-free t-matchings for a certain case where the weight vector is vertex-induced on any subgraph isomorphic to k_{t, t}. For this class of weight vectors, we propose a strongly polynomial algorithm to find a minimum-weight k_{t, t}-free t-factor. The algorithm adapts the unweighted algorithms of Hartvigsen and Pap and a primal-dual approach to the minimum-cost flow problem. The algorithm is fully combinatorial and thus provides a dual integrality theorem, which is tantamount to Makai's one.

Queue-and-Idleness-Ratio Controls in Many-Server Service Systems

Gurvich, I., Whitt, W. — 2009-05-14

Motivated by call centers, we study large-scale service systems with multiple customer classes and multiple agent pools, each with many agents. We propose a family of routing rules called queue-and-idleness-ratio (QIR) rules. A newly available agent next serves the customer from the head of the queue of the class (from among those he is eligible to serve) whose queue length most exceeds a specified state-dependent proportion of the total queue length. An arriving customer is routed to the agent pool whose idleness most exceeds a specified state-dependent proportion of the total idleness. We identify regularity conditions on the network structure and system parameters under which QIR produces an important state-space collapse (SSC) result in the quality-and-efficiency-driven (QED) many-server heavy-traffic limiting regime. The SSC result is applied here to prove stochastic-process limits and in subsequent papers to solve important staffing and control problems for large-scale service systems.

An Adaptive Algorithm for Finding the Optimal Base-Stock Policy in Lost Sales Inventory Systems with Censored Demand

Huh, W. T., Janakiraman, G., Muckstadt, J. A., Rusmevichientong, P. — 2009-05-14

We consider a periodic-review, single-location, single-product inventory system with lost sales and positive replenishment lead times. It is well known that the optimal policy does not possess a simple structure. Motivated by recent results showing that base-stock policies perform well in these systems, we study the problem of finding the best base-stock policy in such a system. In contrast to the classical inventory literature, we assume that the manager does not know the demand distribution a priori but must make the replenishment decision in each period based only on the past sales (censored demand) data. We develop a nonparametric adaptive algorithm that generates a sequence of order-up-to levels whose running average of the inventory holding and lost sales penalty cost converges to the cost of the optimal base-stock policy, and we establish the cubic-root convergence rate of the algorithm. Our analysis is based on recent advances in stochastic online convex optimization and on the uniform ergodicity of Markov chains associated with bases-stock policies.

Tight Bounds for Permutation Flow Shop Scheduling

Nagarajan, V., Sviridenko, M. — 2009-05-14

In flow shop scheduling there are m machines and n jobs, such that every job has to be processed on the machines in the fixed order 1,...,m. In the permutation flow shop problem, it is also required that each machine process the set of all jobs in the same order. Formally, given n jobs along with their processing times on each machine, the goal is to compute a single permutation of the jobs : [n] -> [n] that minimizes the maximum job completion time (makespan) of the schedule resulting from . The previously best known approximation guarantee for this problem was O((m log m)^1/2) [Sviridenko, M. 2004. A note on permutation flow shop problem. Ann. Oper. Res. 129 247–252]. In this paper, we obtain an improved O(min{m^1/2,n^1/2}) approximation algorithm for the permutation flow shop scheduling problem, by finding a connection between the scheduling problem and the longest increasing subsequence problem. Our approximation ratio is relative to the lower bounds of maximum job length and maximum machine load, and is the best possible such result. This also resolves an open question from Potts et al. [Potts, C., D. Shmoys, D. Williamson. 1991. Permutation vs. nonpermutation flow shop schedules. Oper. Res. Lett. 10 281–284], by algorithmically matching the gap between permutation and nonpermutation schedules.

We also consider the weighted completion time objective for the permutation flow shop scheduling problem. Using a natural linear programming relaxation and our algorithm for the makespan objective, we obtain an O(min{m^1/2,n^1/2}) approximation algorithm for minimizing the total weighted completion time, improving on the previously best known guarantee of m for any constant > 0 [Smutnicki, C. 1998. Some results of the worst-case analysis for flow shop scheduling. Eur. J. Oper. Res. 109 66–87]. We give a matching lower bound on the integrality gap of our linear programming relaxation.

An Information-Based Approximation Scheme for Stochastic Optimization Problems in Continuous Time

Kuhn, D. — 2009-05-14

Dynamic stochastic optimization problems with a large (possibly infinite) number of decision stages and high-dimensional state vectors are inherently difficult to solve. In fact, scenario tree-based algorithms are unsuitable for problems with many stages, while dynamic programming-type techniques are unsuitable for problems with many state variables. This paper proposes a stage aggregation scheme for stochastic optimization problems in continuous time, thus having an extremely large (i.e., uncountable) number of decision stages. By perturbing the underlying data and information processes, we construct two approximate problems that provide bounds on the optimal value of the original problem. Moreover, we prove that the gap between the bounds converges to zero as the stage aggregation is refined. If massive aggregation of stages is possible without sacrificing too much accuracy, the aggregate approximate problems can be addressed by means of scenario tree-based methods. The suggested approach applies to problems that exhibit randomness in the objective and the constraints, while the constraint functions are required to be additively separable in the decision variables and random parameters.

Maximum Entropy Principle with General Deviation Measures

Grechuk, B., Molyboha, A., Zabarankin, M. — 2009-05-14

An approach to the Shannon and Rényi entropy maximization problems with constraints on the mean and law-invariant deviation measure for a random variable has been developed. The approach is based on the representation of law-invariant deviation measures through corresponding convex compact sets of nonnegative concave functions. A solution to the problem has been shown to have an alpha-concave distribution (log-concave for Shannon entropy), for which in the case of comonotone deviation measures, an explicit formula has been obtained. As an illustration, the problem has been solved for several deviation measures, including mean absolute deviation (MAD), conditional value-at-risk (CVaR) deviation, and mixed CVaR-deviation. Also, it has been shown that the maximum entropy principle establishes a one-to-one correspondence between the class of alpha-concave distributions and the class of comonotone deviation measures. This fact has been used to solve the inverse problem of finding a corresponding comonotone deviation measure for a given alpha-concave distribution.

Belief Propagation: An Asymptotically Optimal Algorithm for the Random Assignment Problem

Salez, J., Shah, D. — 2009-05-14

The random assignment problem asks for the minimum-cost perfect matching in the complete n x n bipartite graph K_nn with i.i.d. edge weights, say uniform on [0, 1]. In a remarkable work by Aldous [Aldous, D. 2001. The (2) limit in the random assignment problem. RSA 18 381–418], the optimal cost was shown to converge to (2) as n -> , as conjectured by Mézard and Parisi [Mézard, M., G. Parisi. 1987. On the solution of the random link matching problem. J. Phys. 48 1451–1459] through the so-called cavity method. The latter also suggested a nonrigorous decentralized strategy for finding the optimum, which turned out to be an instance of the belief propagation (BP) heuristic discussed by Pearl [Pearl, J. 1988. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, San Francisco]. In this paper we use the objective method to analyze the performance of BP as the size of the underlying graph becomes large. Specifically, we establish that the dynamic of BP on K_nn converges in distribution as n -> to an appropriately defined dynamic on the Poisson weighted infinite tree, and we then prove correlation decay for this limiting dynamic. As a consequence, we obtain that BP finds an asymptotically correct assignment in O(n²) time only. This contrasts with both the worst-case upper bound for convergence of BP derived by Bayati et al. [Bayati, M., D. Shah, M. Sharma. 2008. Max-product for maximum weight matching: Convergence, correctness, and LP duality. IEEE Trans. Inform. Theory 54(3) 1241–1251.] and the best-known computational cost of (n³) achieved by Edmonds and Karp's algorithm [Edmonds, J., R. Karp. 1972. Theoretical improvements in algorithmic efficiency for network flow problems. J. ACM 19 248–264].

Online Scheduling with Bounded Migration

Sanders, P., Sivadasan, N., Skutella, M. — 2009-05-14

Consider the classical online scheduling problem, in which jobs that arrive one by one are assigned to identical parallel machines with the objective of minimizing the makespan. We generalize this problem by allowing the current assignment to be changed whenever a new job arrives, subject to the constraint that the total size of moved jobs is bounded by some constant times the size of the arriving job. This constant is called the migration factor. For small migration factors, we obtain several simple online algorithms with constant competitive ratio. We also present a linear time "online approximation scheme," that is, a family of online algorithms with competitive ratio arbitrarily close to 1 and constant migration factor.

Earliest Arrival Flows with Multiple Sources

Baumann, N., Skutella, M. — 2009-05-14

Earliest arrival flows capture the essence of evacuation planning. Given a network with capacities and transit times on the arcs, a subset of source nodes with supplies and a sink node, the task is to send the given supplies from the sources to the sink "as quickly as possible." The latter requirement is made more precise by the earliest arrival property, which requires that the total amount of flow that has arrived at the sink is maximal for all points in time simultaneously. It is a classical result from the 1970s that, for the special case of a single source node, earliest arrival flows do exist and can be computed by essentially applying the successive shortest-path algorithm for min-cost flow computations. Although it has previously been observed that an earliest arrival flow still exists for multiple sources, the problem of computing one efficiently has been open for many years. We present an exact algorithm for this problem whose running time is strongly polynomial in the input plus output size of the problem.