Data aggregation for <Emphasis Type="Italic">p</Emphasis>-median problems

In this paper, we use a pseudo-Boolean formulation of the p-median problem and using data aggregation, provide a compact representation of p-median problem instances. We provide computational results to demonstrate this compactification in benchmark instances. We then use our representation to explain why some p-median problem instances are more difficult to solve to optimality than other instances of the same size. We also derive a preprocessing rule based on our formulation, and describe equivalent p-median problem instances, which are identical sized instances which are guaranteed to have identical optimal solutions.     

Bounds of Relative Regret Limit in p‐Robust Supply Chain Network Design

This research studies the p‐robust supply chain network design with uncertain demand and cost scenarios. The optimal design integrates the supplier selection together with the facility location and capacity problem. We provide a new framework to obtain the relative regret limit, which is critical in the robust supply chain design but is assumed to be a known value in the existing literature. We obtain lower and upper bounds for relative regret limit and obtain a sequence of optimal solutions for series relative regret limits between the upper and lower bounds. An algorithm for p‐robust supply chain network design is provided. A series of numerical examples are designed to find the properties of the bottleneck scenarios. A scenario with low probability and a low optimal objective function value for the scenario has a greater chance of being a bottleneck. To focus only on the influence from the relative regret, we also introduce three separate new objective functions in p‐robust design. The proposed new theories and approaches provide a sequence of options for decision makers to reduce the marketing risks effectively in supply chain network design.     

Median problems with positive and negative weights on cycles and cacti

This paper deals with facility location problems on graphs with positive and negative vertex weights. We consider two different objective functions: In the first one (MWD) vertices with positive weight are assigned to the closest facility, whereas vertices with negative weight are assigned to the farthest facility. In the second one (WMD) all the vertices are assigned to the nearest facility. For the MWD model it is shown that there exists a finite set of points in the graph which contains the locations of facilities in an optimal solution. Furthermore, algorithms for both models for the 2-median problem on a cycle are developed. The algorithm for the MWD model runs in linear time, whereas the algorithm for the WMD model has a time complexity of  O(n²)\mathcal{O}(n^{2}) .     

Algorithms for connected <Emphasis Type="Italic">p</Emphasis>-centdian problem on block graphs

We consider the facility location problem of locating a set \(X_p\) of p facilities (resources) on a network (or a graph) such that the subnetwork (or subgraph) induced by the selected set \(X_p\) is connected. Two problems on a block graph G are proposed: one problem is to minimizes the sum of its weighted distances from all vertices of G to \(X_p\), another problem is to minimize the maximum distance from each vertex that is not in \(X_p\) to \(X_p\) and, at the same time, to minimize the sum of its distances from all vertices of G to \(X_p\). We prove that the first problem is linearly solvable on block graphs with unit edge length. For the second problem, it is shown that the set of Pareto-optimal solutions of the two criteria has cardinality not greater than n, and can be obtained in \(O(n^2)\) time, where n is the number of vertices of the block graph G.     

The <Emphasis Type="Italic">p</Emphasis>-maxian problem on block graphs

This paper deals with the p-maxian problem on block graphs with unit edge length. It is shown that the two points with maximum distance provide an optimal solution for the 2-maxian problem of block graphs except for K ₃. It can easily be extended to the p-maxian problem of block graphs. So we solve the p-maxian problem on block graphs in linear time.     

A local search approximation algorithm for the uniform capacitated <Emphasis Type="Italic">k</Emphasis>-facility location problem

In the uniform capacitated k-facility location problem (UC-k-FLP), we are given a set of facilities and a set of clients. Every client has a demand. Every facility have an opening cost and an uniform capacity. For each client–facility pair, there is an unit service cost to serve the client with unit demand by the facility. The total demands served by a facility cannot exceed the uniform capacity. We want to open at most k facilities to serve all the demands of the clients without violating the capacity constraint such that the total opening and serving cost is minimized. The main contribution of this work is to present the first combinatorial bi-criteria approximation algorithm for the UC-k-FLP by violating the cardinality constraint.     

New algorithms for k-center and extensions

The problem of interest is covering a given point set with homothetic copies of several convex containers C ₁,…,C _k , while the objective is to minimize the maximum over the dilatation factors. Such k-containment problems arise in various applications, e.g. in facility location, shape fitting, data classification or clustering. So far most attention has been paid to the special case of the Euclidean k-center problem, where all containers C _i are Euclidean unit balls. Recent developments based on so-called core-sets enable not only better theoretical bounds in the running time of approximation algorithms but also improvements in practically solvable input sizes. Here, we present some new geometric inequalities and a Mixed-Integer-Convex-Programming formulation. Both are used in a very effective branch-and-bound routine which not only improves on best known running times in the Euclidean case but also handles general and even different containers among the C _i .     

A Bicriterion Maximal Covering Location Formulation Which Considers the Satisfaction of Uncovered Demand

There have been many applications of the maximal covering location problem (MCLP). An underlying assumption of the MCLP is that demand not covered (i.e., not within a prespecified maximal distance of a facility) is not served. This may be an unrealistic assumption in many location planning scenarios, especially in the public sector. For example, in cases such as fire protection or ambulance service, calls not technically covered will still be serviced. The MCLP, however, does not consider the distances or travel times necessary to service such demand. This paper presents a bicriterion locational covering model which explicitly considers the travel distance or time necessary to service demand not within the maximal covering distance of a facility. The model may be used to generate noninferior (Pareto optimal) siting configurations which demonstrate the inherent trade-offs between a siting scheme designed to maximize total coverage and one designed to minimize total travel time for uncovered demand to reach its nearest facility. In addition, it is shown that for any particular weighting scheme on the two objectives, the problem can be solved as a p-median problem; a problem for which several efficient solution methods exist.     

Optimal Randomized Scheduling by Replacement

In the replacement scheduling problem, a system consists of n processors drawn from a pool of p,all initially alive. At any time some processor can die. The scheduler is immediately informed of the fault butnot of its location. It must then choose another set of n processors. If this new set contains a dead processor, the system crashes and halts. The performance of a scheduling protocol is measured by the expected number of deaths the system tolerates before it crashes. We provide an optimal randomized scheduling protocol for this problem.The framework of this work combines an absolute performance measure for protocols and so-called adaptive online adversaries. This framework is rarely addressed because of the complexity of the interaction between protocols and adversaries. A major contribution of ourwork is to provide a theoretical foundation for the analysis of this interaction. In particular we make explicit how the protocol and the adversary affect the probability distribution of the analysis—a very general problem. We carefully analyze the exchange of sinformation between the two players, and reveal how they use their information optimally. The optimality of the protocol is established by using of a saddle point method for protocols and adversaries.     

A modified power spectral density test applied to?weighing matrices with small weight

The power spectral density test has been used for at least a decade in the search for many kinds of combinatorial matrices, such as weighing matrices for instance. In this paper we establish a modified power spectral density test that we apply to the search for weighing matrices of small weights constructed from two circulants. The main novelty of our approach is to define the Discrete Fourier Transform on the support of the first rows of the two circulants, thus exploiting the inherent sparsity of the problem. This new formalism turns out to be very efficient for small weights 9,18,36 and we find 10 new weighing matrices W(2⋅p,18) for prime p∈{37,47,53,59,61,67,73,79,83,97}. These matrices are given here for the first time. We also discuss briefly a connection with Combinatorial Optimization.     

On Split-Coloring Problems

We study a new coloring concept which generalizes the classical vertex coloring problem in a graph by extending the notion of stable sets to split graphs. First of all, we propose the packing problem of finding the split graph of maximum size where a split graph is a graph G = (V,E) in which the vertex set V can be partitioned into a clique K and a stable set S. No condition is imposed on the edges linking vertices in S to the vertices in K. This maximum split graph problem gives rise to an associated partitioning problem that we call the split-coloring problem. Given a graph, the objective is to cover all his vertices by a least number of split graphs. Definitions related to this new problem are introduced. We mention some polynomially solvable cases and describe open questions on this area. An erratum to this article is available at .     

Approximating soft-capacitated facility location problem with uncertainty

In this paper we devise the stochastic and robust approaches to study the soft-capacitated facility location problem with uncertainty. We first present a new stochastic soft-capacitated model called The 2-Stage Soft Capacitated Facility Location Problem and solve it via an approximation algorithm by reducing it to linear-cost version of 2-stage facility location problem and dynamic facility location problem. We then present a novel robust model of soft-capacitated facility location, The Robust Soft Capacitated Facility Location Problem. To solve it, we improve the approximation algorithm proposed by Byrka et al. (LP-rounding algorithms for facility-location problems. CoRR, 2010a) for RFTFL and then treat it similarly as in the stochastic case. The improvement results in an approximation factor of \(\alpha + 4\) for the robust fault-tolerant facility location problem, which is best so far.     

The Polymatroid Steiner Problems

The Steiner tree problem asks for a minimum cost tree spanning a given set of terminals S⊂eqV in a weighted graph G = (V,E,c), c:E→ R⁺. In this paper we consider a generalization of the Steiner tree problem, so called Polymatroid Steiner Problem, in which a polymatroid P = P(V) is defined on V and the Steiner tree is required to span at least one base of P (in particular, there may be a single base S⊂eqV). This formulation is motivated by the following application in sensor networks – given a set of sensors S = {s₁,…,s_k}, each sensor s_i can choose to monitor only a single target from a subset of targets X_i, find minimum cost tree spanning a set of sensors capable of monitoring the set of all targets X = X₁ ∪ … ∪ X_k. The Polymatroid Steiner Problem generalizes many known Steiner tree problem formulations including the group and covering Steiner tree problems. We show that this problem can be solved with the polylogarithmic approximation ratio by a generalization of the combinatorial algorithm of Chekuri et al. (2002).We also define the Polymatroid directed Steiner problem which asks for a minimum cost arborescence connecting a given root to a base of a polymatroid P defined on the terminal set S. We show that this problem can be approximately solved by algorithms generalizing methods of Chekuri et al. (2002).A preliminary version of this paper appeared in ISAAC 2004     

An approximation algorithm for the <Emphasis Type="Italic">k</Emphasis>-level capacitated facility location problem

We consider the k-level capacitated facility location problem (k-CFLP), which is a natural variant of the classical facility location problem and has applications in supply chain management. We obtain the first (combinatorial) approximation algorithm with a performance factor of \(k+2+\sqrt{k^{2}+2k+5}+\varepsilon\) (ε>0) for this problem.     

Cautious Expected Utility and the Certainty Effect

Many violations of the independence axiom of expected utility can be traced to subjects' attraction to risk‐free prospects. The key axiom in this paper, negative certainty independence ([Dillenberger, 2010]), formalizes this tendency. Our main result is a utility representation of all preferences over monetary lotteries that satisfy negative certainty independence together with basic rationality postulates. Such preferences can be represented as if the agent were unsure of how to evaluate a given lottery p; instead, she has in mind a set of possible utility functions over outcomes and displays a cautious behavior: she computes the certainty equivalent of p with respect to each possible function in the set and picks the smallest one. The set of utilities is unique in a well defined sense. We show that our representation can also be derived from a “cautious” completion of an incomplete preference relation.     

New Approximation Algorithms for Map Labeling with Sliding Labels

In this paper we present approximation algorithm for the following NP-hard map labeling problem: Given a set S of n distinct sites in the plane, one needs to place at each site a uniform square of maximum possible size such that all the squares are along the same direction. This generalizes the classical problem of labeling points with axis-parallel squares and restricts the most general version where the squares can have different orientations. We obtain factor-4 and factor- approximation algorithms for this problem. These algorithms also work for two generalized versions of the problem. We also revisit the problem of labeling each point with maximum uniform axis-parallel square pairs and improve the previous approximation factor of 4 to 3.     

On minimum <Emphasis Type="Italic">m</Emphasis>-connected <Emphasis Type="Italic">k</Emphasis>-dominating set problem in unit disc graphs

Minimum m-connected k-dominating set problem is as follows: Given a graph G=(V,E) and two natural numbers m and k, find a subset S⊆V of minimal size such that every vertex in V∖S is adjacent to at least k vertices in S and the induced graph of S is m-connected. In this paper we study this problem with unit disc graphs and small m, which is motivated by the design of fault-tolerant virtual backbone for wireless sensor networks. We propose two approximation algorithms with constant performance ratios for m≤2. We also discuss how to design approximation algorithms for the problem with arbitrarily large m. This work was supported in part by the Research Grants Council of Hong Kong under Grant No. CityU 1165/04E, the National Natural Science Foundation of China under Grant No. 70221001, 10531070 and 10771209.     

INSIGHTS ON SERVICE SYSTEM DESIGN FROM A NORMAL APPROXIMATION TO ERLANG'S DELAY FORMULA

We show how a simple normal approximation to Erlang's delay formula can be used to analyze capacity and staffing problems in service systems that can be modeled as M/M/s queues. The numbers of servers, s, needed in an M/M/s queueing system to assure a probability of delay of, at most, p can be well approximated by s ≅ p + z_***I-p+, where z_1-p, is the (1 - p)th percentile of the standard normal distribution and ρ, the presented load on the system, is the ratio of Λ, the customer arrival rate, to μ, the service rate. We examine the accuracy of this approximation over a set of parameters typical of service operations ranging from police patrol, through telemarketing to automatic teller machines, and we demonstrate that it tends to slightly underestimate the number of servers actually needed to hit the delay probability target—adding one server to the number suggested by the above formula typically gives the exact result. More importantly, the structure of the approximation promotes operational insight by explicitly linking the number of servers with server utilization and the customer service level. Using a scenario based on an actual teleservicing operation, we show how operations managers and designers can quickly obtain insights about the trade-offs between system size, system utilization and customer service. We argue that this little used approach deserves a prominent role in the operations analyst's and operations manager's toolbags.     

One‐Dimensional Inference in Autoregressive Models With the Potential Presence of a Unit Root

This paper examines the problem of testing and confidence set construction for one‐dimensional functions of the coefficients in autoregressive (AR(p)) models with potentially persistent time series. The primary example concerns inference on impulse responses. A new asymptotic framework is suggested and some new theoretical properties of known procedures are demonstrated. I show that the likelihood ratio (LR) and LR^± statistics for a linear hypothesis in an AR(p) can be uniformly approximated by a weighted average of local‐to‐unity and normal distributions. The corresponding weights depend on the weight placed on the largest root in the null hypothesis. The suggested approximation is uniform over the set of all linear hypotheses. The same family of distributions approximates the LR and LR^± statistics for tests about impulse responses, and the approximation is uniform over the horizon of the impulse response. I establish the size properties of tests about impulse responses proposed by Inoue and Kilian (2002) and Gospodinov (2004), and theoretically explain some of the empirical findings of Pesavento and Rossi (2007). An adaptation of the grid bootstrap for impulse response functions is suggested and its properties are examined.     

An improved approximation algorithm for uncapacitated facility location problem with penalties

In this paper, we consider an interesting variant of the classical facility location problem called uncapacitated facility location problem with penalties (UFLWP for short) in which each client is either assigned to an opened facility or rejected by paying a penalty. The UFLWP problem has been effectively used to model the facility location problem with outliers. Three constant approximation algorithms have been obtained (Charikar et al. in Proceedings of the Symposium on Discrete Algorithms, pp. 642–651, 2001; Jain et al. in J. ACM 50(6):795–824, 2003; Xu and Xu in Inf. Process. Lett. 94(3):119–123, 2005), and the best known performance ratio is 2. The only known hardness result is a 1.463-inapproximability result inherited from the uncapacitated facility location problem (Guha and Khuller in J. Algorithms 31(1):228–248, 1999). In this paper, We present a 1.8526-approximation algorithm for the UFLWP problem. Our algorithm significantly reduces the gap between known performance ratio and the inapproximability result. Our algorithm first enhances the primal-dual method for the UFLWP problem (Charikar et al. in Proceedings of the Symposium on Discrete Algorithms, pp. 642–651, 2001) so that outliers can be recognized more efficiently, and then applies a local search heuristic (Charikar and Guha in Proceedings of the 39th IEEE Symposium on Foundations of Computer Science, pp. 378–388, 1999) to further reduce the cost for serving those non-rejected clients. Our algorithm is simple and can be easily implemented. The research of this work was supported in part by NSF through CAREER award CCF-0546509 and grant IIS-0713489. A preliminary version of this paper appeared in the Proceedings of the 11th Annual International Computing and Combinatorics Conference (COCOON’05).