Mathematics of Operations Research current issue

On a Continuous-Time Game with Incomplete Information

Cardaliaguet, P., Rainer, C. — Wed, 18 Nov 2009 10:02:20 PST

For zero-sum two-player continuous-time games with integral payoff and incomplete information on one side, the authors show that the optimal strategy of the informed player can be computed through an auxiliary optimization problem over some martingale measures. The authors also characterize the optimal martingale measures and compute them explicitly in several examples.

Simplified Control Problems for Multiclass Many-Server Queueing Systems

Atar, R., Mandelbaum, A., Shaikhet, G. — Wed, 18 Nov 2009 10:02:20 PST

We consider scheduling and routing control problems for queueing models with I customer classes and J server pools, each consisting of many statistically identical, exponential servers. Customers require a single service that can be performed by a server from one of the pools; the service rate is µ_ij ≥ 0, which depends on the customer's class i and the server's pool j, and customers can abandon the system while waiting to be served. In the heavy traffic regime of Halfin and Whitt, these problems are formally equivalent to I-dimensional diffusion control problems. We analyze the diffusion control problems is two special cases. First, when the service rates depend only on the pool (µ_ij = µ_j), the diffusion control problem is shown to be similar to (but distinct from) the diffusion control problem for a single class model, which greatly reduces the complexity of the problem. Second, when the service rates depend only on the class (µ_ij = µ_i), the diffusion control problem is shown to be equivalent to a diffusion control problem for a single pool model, a problem that has previously been studied. In the first case, we also establish a rigorous relation between the queueing control problem and the diffusion control problem, showing that a policy for the queueing model, based on an ordinary differential equation of Hamilton-Jacobi-Bellman type, is asymptotically optimal.

Gear Composition of Stable Set Polytopes and G-Perfection

Galluccio, A., Gentile, C., Ventura, P. — Wed, 18 Nov 2009 10:02:20 PST

Graphs obtained by applying the gear composition to a given graph H are called geared graphs. We show how a linear description of the stable set polytope STAB(G) of a geared graph G can be obtained by extending the linear inequalities defining STAB(H) and STAB(H^e), where H^e is the graph obtained from H by subdividing the edge e. We also introduce the class of G-perfect graphs, i.e., graphs whose stable set polytope is described by nonnegativity inequalities, rank inequalities, lifted 5-wheel inequalities, and some special inequalities called geared inequalities and g-lifted inequalities. We prove that graphs obtained by repeated applications of the gear composition to a given graph H are G-perfect, provided that any graph obtained from H by subdividing a subset of its simplicial edges is G-perfect. In particular, we show that a large subclass of claw-free graphs is G-perfect.

Locating Objects in the Plane Using Global Optimization Techniques

Blanquero, R., Carrizosa, E., Hansen, P. — Wed, 18 Nov 2009 10:02:20 PST

We address the problem of locating objects in the plane such as segments, arcs of circumferences, arbitrary convex sets, their complements or their boundaries. Given a set of points, we seek the rotation and translation for such an object optimizing a very general performance measure, which includes as a particular case the classical objectives in semi-obnoxious facility location. In general, the above-mentioned model yields a global optimization problem, whose resolution is dealt with using difference of convex (DC) techniques such as outer approximation or branch and bound.

On the Maximum Quadratic Assignment Problem

Nagarajan, V., Sviridenko, M. — Wed, 18 Nov 2009 10:02:20 PST

Quadratic assignment is a basic problem in combinatorial optimization that generalizes several other problems such as traveling salesman, linear arrangement, dense k subgraph, and clustering with given sizes. The input to the quadratic assignment problem consists of two n x n symmetric nonnegative matrices $W=(w_{i, j})$ and $D=(d_{i, j})$. Given matrices W, D, and a permutation $\pi: [n] \rightarrow [n]$, the objective function is $Q(\pi):= \sum_{i, j \in [n], i \ne j} w_{i, j} \cdot d_{\pi(i), \pi(j)}$. In this paper, we study the maximum quadratic assignment problem, where the goal is to find a permutation that maximizes $Q(\pi)$. We give an $\tilde{O}(\sqrt{n})$-approximation algorithm, which is the first nontrivial approximation guarantee for this problem. The above guarantee also holds when the matrices W, D are asymmetric. An indication of the hardness of maximum quadratic assignment is that it contains as a special case the dense k subgraph problem, for which the best-known approximation ratio is $\approx n^{1/3}$ (Feige et al. [Feige, U., G. Kortsarz, D. Peleg. 2001. The dense k-subgraph problem. Algorithmica 29(3) 410–421]).

When one of the matrices W, D satisfies triangle inequality, we obtain a $2e/(e-1) \approx 3.16$-approximation algorithm. This improves over the previously best-known approximation guarantee of four (Arkin et al. [Arkin, E. M., R. Hassin, M. Sviridenko. 2001. Approximating the maximum quadratic assignment problem. Inform. Processing Lett. 77 13–16]) for this special case of maximum quadratic assignment.

The performance guarantee for maximum quadratic assignment with triangle inequality can be proved relative to an optimal solution of a natural linear programming relaxation that has been used earlier in branch-and-bound approaches (see, eg., Adams and Johnson [Adams, W. P., T. A. Johnson. 1994. Improved linear programming-based lower bounds for the quadratic assignment problem. DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 16 43–77]). It can also be shown that this linear program (LP) has an integrality gap of $\tilde{\Omega}(\sqrt{n})$ for general maximum quadratic assignment.

Equivalence of Convex Problem Geometry and Computational Complexity in the Separation Oracle Model

Freund, R. M., Vera, J. R. — Wed, 18 Nov 2009 10:02:21 PST

Consider the supposedly simple problem of computing a point in a convex set that is conveyed by a separation oracle with no further information (e.g., no domain ball containing or intersecting the set, etc.). The authors' interest in this problem stems from fundamental issues involving the interplay of (i) the computational complexity of computing a point in the set, (ii) the geometry of the set, and (iii) the stability or conditioning of the set under perturbation. Under suitable definitions of these terms, the authors show herein that problem instances with favorable geometry have favorable computational complexity, validating conventional wisdom. The authors also show a converse of this implication by showing that there exist problem instances that require more computational effort to solve in certain families characterized by unfavorable geometry. This in turn leads, under certain assumptions, to a form of equivalence among computational complexity, geometry, and the conditioning of the set. The authors' measures of the geometry, relative to a given reference point, are based on the radius of a certain domain ball whose intersection with the set contains a certain inscribed ball.

The geometry of the set is then measured by the radius of the domain ball, the radius of the inscribed ball, and the ratio between these two radii, the latter of which is called the aspect ratio. The aspect ratio arises in the analysis of many algorithms for convex problems, and its importance in convex algorithm analysis has been well-known for several decades. However, the presence in our bound of terms involving only the radius of the domain ball and only the radius of the inscribed ball are a bit counterintuitive; nevertheless, we show that the computational complexity must involve these terms in addition to the aspect ratio, even when the aspect ratio itself is small. This lower-bound complexity analysis relies on simple features of the separation oracle model; if we instead assume that the set is conveyed by a self-concordant barrier function, then it is an open challenge to prove such a complexity lower-bound.

Fluid Limits for Shortest Remaining Processing Time Queues

Down, D. G., Gromoll, H. C., Puha, A. L. — Wed, 18 Nov 2009 10:02:21 PST

We consider a single-server queue with renewal arrivals and i.i.d. service times in which the server uses the shortest remaining processing time policy. To describe the evolution of this queue, we use a measure-valued process that keeps track of the residual service times of all buffered jobs. We propose a fluid model (or formal law of large numbers approximation) for this system and, under mild assumptions, prove the existence and uniqueness of fluid model solutions. Furthermore, we prove a scaling limit theorem that justifies the fluid model as a first-order approximation of the stochastic model. The state descriptor of the fluid model is a measure-valued function whose dynamics are governed by certain inequalities in conjunction with the standard workload equation. In particular, these dynamics determine the evolution of the left edge (infimum) of the state descriptor's support, which yields conclusions about response times. We characterize the evolution of this left edge as an inverse functional of the initial condition, arrival rate, and service time distribution. This characterization reveals the manner in which the growth rate of the left edge depends on the service time distribution. By considering varying examples, the authors show that the rate can vary from logarithmic to polynomial.

Bid-Price Controls for Network Revenue Management: Martingale Characterization of Optimal Bid Prices

Akan, M., Ata, B. — Wed, 18 Nov 2009 10:02:21 PST

We consider a continuous-time, rate-based model of network revenue management. Under mild assumptions, we construct a simple -optimal bid-price control, which can be viewed as a perturbation of a bid-price control in the classical sense [Williamson, E. L. 1992. Airline network seat control. Ph.D. thesis, MIT, Cambridge, MA]. We show that the associated bid-price process forms a martingale and the corresponding booking controls converge in an appropriate sense to an optimal control as tends to 0. Moreover, we show that there exists an optimal generalized bid-price control, where the bid-price process forms a martingale and is used in conjunction with a capacity usage limit process. We also discuss its connection to the bid-price controls in the classical sense and sufficient conditions for the (near) optimality of the latter.

Law of Large Number Limits of Limited Processor-Sharing Queues

Zhang, J., Dai, J. G., Zwart, B. — Wed, 18 Nov 2009 10:02:21 PST

Motivated by applications in computer and communication systems, we consider a processor-sharing queue where the number of jobs served is not larger than K. We propose a measure-valued fluid model for this limited processor-sharing queue and show that there exists a unique associated fluid model solution. In addition, we show that this fluid model arises as the limit of a sequence of appropriately scaled processor-sharing queues.

Players' Effects Under Limited Independence

Gradwohl, R., Reingold, O., Yadin, A., Yehudayoff, A. — Wed, 18 Nov 2009 10:02:21 PST

In a function that takes its inputs from various players, the effect of a player measures the variation he can cause in the expectation of that function. In this paper we prove a tight upper bound on the number of players with large effect, a bound that holds even when the players' inputs are only known to be pairwise independent. We also study the effect of a set of players, and show that there always exists a "small" set of players that, when eliminated, leaves every small set with little effect. Finally, we ask whether there always exists a player with positive effect, and show that, in general, the answer is negative. More specifically, we show that if the function is nonmonotone or the distribution is only known to be pairwise independent, then it is possible that all players have zero effect.

On the Core and f-Nucleolus of Flow Games

Kern, W., Paulusma, D. — Wed, 18 Nov 2009 10:02:21 PST

Using the ellipsoid method, both Deng et al. [Deng, X., Q. Fang, X. Sun. 2006. Finding nucleolus of flow game. Proc. 17th ACM-SIAM Sympos. Discrete Algorithms. ACM Press, New York, 124–131] and Potters et al. [Potters, J., H. Reijnierse, A. Biswas. 2006. The nucleolus of balanced simple flow networks. Games Econom. Behav. 54 205–225] show that the nucleolus of simple flow games (where all edge capacities are equal to one) can be computed in polynomial time. Our main result is a combinatorial method based on eliminating redundant s–t path constraints such that only a polynomial number of constraints remains. This leads to efficient algorithms for computing the core, nucleolus, and nucleon of simple flow games. Deng et al. also prove that computing the nucleolus for (general) flow games is NP-hard. We generalize this by proving that computing the f-nucleolus of flow games is NP-hard for all priority functions f that satisfy f(A) > 0 for all coalitions A with worth v(A) > 0 (so, including the priority functions corresponding to the nucleolus, nucleon, and per-capita nucleolus).

A Strongly Polynomial Algorithm for Controlled Queues

Zadorojniy, A., Even, G., Shwartz, A. — Wed, 18 Nov 2009 10:02:21 PST

We consider the problem of computing optimal policies of finite-state finite-action Markov decision processes (MDPs). A reduction to a continuum of constrained MDPs (CMDPs) is presented such that the optimal policies for these CMDPs constitute a path in a graph defined over the deterministic policies. This path contains, in particular, an optimal policy of the original MDP. We present an algorithm based on this new approach that finds this path, and thus an optimal policy. In the general case, this path might be exponentially long in the number of states and actions. We prove that the length of this path is polynomial if the MDP satisfies a coupling property. Thus we obtain a strongly polynomial algorithm for MDP s that satisfies the coupling property. We prove that discrete time versions of controlled M/M/1 queues induce MDP s that satisfy the coupling property. The only previously known polynomial algorithm for controlled M/M/1 queues in the expected average cost model is based on linear programming (and is not known to be strongly polynomial). Our algorithm works both for the discounted and expected average cost models, and the running time does not depend on the discount factor.

A Low-Dimensional Semidefinite Relaxation for the Quadratic Assignment Problem

Ding, Y., Wolkowicz, H. — Wed, 18 Nov 2009 10:02:22 PST

The quadratic assignment problem (QAP) is arguably one of the hardest NP-hard discrete optimization problems. Problems of dimension greater than 25 are still considered to be large scale. Current successful solution techniques use branch-and-bound methods, which rely on obtaining strong and inexpensive bounds. In this paper, we introduce a new semidefinite programming (SDP) relaxation for generating bounds for the QAP in the trace formulation. We apply majorization to obtain a relaxation of the orthogonal similarity set of the quadratic part of the objective function. This exploits the matrix structure of QAP and results in a relaxation with much smaller dimension than other current SDP relaxations. We compare the resulting bounds with several other computationally inexpensive bounds such as the convex quadratic programming relaxation (QPB). We find that our method provides stronger bounds on average and is adaptable for branch-and-bound methods.

Author Index of Volume 34

Wed, 18 Nov 2009 10:02:22 PST

No abstract available.