Maximizing Profits of Routing in WDM Networks

Let G = (V, E) be a ring (or chain) network representing an optical wavelength division multiplexing (WDM) network with k channels, where each edge e_j has an integer capacity c_j. A request s_i,t_i is a pair of two nodes in G. Given m requests s_i,t_i, i = 1, 2, ..., m, each with a profit value p_i, we would like to design/route a k-colorable set of paths for some (may not be all) of the m requests such that each edge e_j in G is used at most c_j times and the total profit of the set of designed paths is maximized. Here two paths cannot have the same color (channel) if they share some common edge(s).This problem arises in optical communication networks. In this paper, we present a polynomial-time algorithm to solve the problem when G is a chain. When G is a ring, however, the optimization problem is NP-hard (Wan and Liu, 1998), we present a 2-approximation algorithm based on our solution to the chain network. Similarly, some results in a bidirected chain and a bidirected ring are obtained.     

A best on-line algorithm for the single machine parallel-batch scheduling with restricted delivery times

We consider a single batch machine on-line scheduling problem with delivery times. In this paper on-line means that jobs arrive over time and the characteristics of jobs are unknown until their arrival times. Once the processing of a job is completed it is delivered to the destination. The objective is to minimize the time by which all jobs have been delivered. For each job J _j , its processing time and delivery time are denoted by p _j and q _j , respectively. We consider two restricted models: (1) the jobs have small delivery times, i.e., for each job J _j , q _j ≤p _j ; (2) the jobs have agreeable processing and delivery times, i.e., for any two jobs J _i and J _j , p _i >p _j implies q _i ≥q _j . We provide an on-line algorithm with competitive ratio for both problems, and the results are the best possible. Project supported by NSFC (10671183).     

A 6.55 factor primal-dual approximation algorithm for the connected facility location problem

In the connected facility location (ConFL) problem, we are given a graph G=(V,E) with nonnegative edge cost c _e on the edges, a set of facilities ??V, a set of demands (i.e., clients) $\mathcal{D}\subseteq VIn the connected facility location (ConFL) problem, we are given a graph G=(V,E) with nonnegative edge cost c _e on the edges, a set of facilities ℱ⊆V, a set of demands (i.e., clients) D í V\mathcal{D}\subseteq V , and a parameter M≥1. Each facility i has a nonnegative opening cost f _i and each client j has d _j units of demand. Our objective is to open some facilities, say F⊆ℱ, assign each demand j to some open facility i(j)∈F and connect all open facilities using a Steiner tree T such that the total cost, which is ?_{i ? F}f_i+?_{j ? D}d_jc_i(j)j+M?_{e ? T}c_e\sum_{i\in F}f_{i}+\sum_{j\in \mathcal{D}}d_{j}c_{i(j)j}+M\sum_{e\in T}c_{e} , is minimized. We present a primal-dual 6.55-approximation algorithm for the ConFL problem which improves the previous primal-dual 8.55-approximation algorithm given by Swamy and Kumar (Algorithmica 40:245–269, 2004).     

Estimating hybrid frequency moments of data streams

We consider the problem of estimating hybrid frequency moments of two dimensional data streams. In this model, data is viewed to be organized in a matrix form (A _i,j )_1≤i,j,≤n . The entries A _i,j are updated coordinate-wise, in arbitrary order and possibly multiple times. The updates include both increments and decrements to the current value of A _i,j . The hybrid frequency moment F _p,q (A) is defined as \(\sum_{j=1}^{n}(\sum_{i=1}^{n}{A_{i,j}}^{p})^{q}\) and is a generalization of the frequency moment of one-dimensional data streams.We present the first \(\tilde{O}(1)\) space algorithm for the problem of estimating F _p,q for p∈[0,2] and q∈[0,1] to within an approximation factor of 1±ε. The \(\tilde{O}\) notation hides poly-logarithmic factors in the size of the stream m, the matrix size n and polynomial factors of ε ^?1. We also present the first \(\tilde{O}(n^{1-1/q})\) space algorithm for estimating F _p,q for p∈[0,2] and q∈(1,2].     

The subdivision-constrained routing requests problem

We are given a digraph D=(V,A;w), a length (delay) function w:A→R ⁺, a positive integer d and a set $\mathcal{P}=\{(s_{i},t_{i};B_{i}) | i=1,2,\ldots,k\}$ of k requests, where s _i ∈V is called as the ith source node, t _i ∈V is called the ith sink node and B _i is called as the ith length constraint. For a given positive integer d, the subdivision-constrained routing requests problem (SCRR, for short) is to find a directed subgraph D′=(V′,A′) of D, satisfying the two constraints: (1) Each request (s _i ,t _i ;B _i ) has a path P _i from s _i to t _i in D′ with length $w(P_{i})=\sum_{e\in P_{i}} w(e)$ no more than B _i ; (2) Insert some nodes uniformly on each arc e∈A′ to ensure that each new arc has length no more than d. The objective is to minimize the total number of the nodes inserted on the arcs in A′. We obtain the following three main results: (1) The SCRR problem is at least as hard as the set cover problem even if each request has the same source s, i.e., s _i =s for each i=1,2,…,k; (2) For each request (s,t;B), we design a dynamic programming algorithm to find a path from s to t with length no more than B such that the number of the nodes inserted on such a path is minimized, and as a corollary, we present a k-approximation algorithm to solve the SCRR problem for any k requests; (3) We finally present an optimal algorithm for the case where $\mathcal{P}$ contains all possible requests (s _i ,t _i ) in V×V and B _i is equal to the length of the shortest path in D from s _i to t _i . To the best of our knowledge, this is the first time that the dynamic programming algorithm within polynomial time in (2) is designed for a weighted optimization problem while previous optimal algorithms run in pseudo-polynomial time.     

The orbit problem is in the GapL hierarchy

The Orbit problem is defined as follows: Given a matrix A∈ℚ ^n×n and vectors x,y∈ℚ ⁿ , does there exist a non-negative integer i such that A ⁱ x=y. This problem was shown to be in deterministic polynomial time by Kannan and Lipton (J. ACM 33(4):808–821, 1986). In this paper we place the problem in the logspace counting hierarchy GapLH. We also show that the problem is hard for C₌L with respect to logspace many-one reductions.     

Constant time approximation scheme for largest well predicted subset

The largest well predicted subset problem is formulated for comparison of two predicted 3D protein structures from the same sequence. A 3D protein structure is represented by an ordered point set A={a ₁,…,a _n }, where each a _i is a point in 3D space. Given two ordered point sets A={a ₁,…,a _n } and B={b ₁,b ₂,…b _n } containing n points, and a threshold d, the largest well predicted subset problem is to find the rigid body transformation T for a largest subset B _opt of B such that the distance between a _i and T(b _i ) is at most d for every b _i in B _opt . A meaningful prediction requires that the size of B _opt is at least αn for some constant α (Li et al. in CPM 2008, 2008). We use LWPS(A,B,d,α) to denote the largest well predicted subset problem with meaningful prediction. An (1+δ ₁,1?δ ₂)-approximation for LWPS(A,B,d,α) is to find a transformation T to bring a subset B′?B of size at least (1?δ ₂)|B _opt | such that for each b _i ∈B′, the Euclidean distance between the two points distance?(a _i ,T(b _i ))≤(1+δ ₁)d. We develop a constant time (1+δ ₁,1?δ ₂)-approximation algorithm for LWPS(A,B,d,α) for arbitrary positive constants δ ₁ and δ ₂. To our knowledge, this is the first constant time algorithm in this area. Li et al. (CPM 2008, 2008) showed an $O(n(\log n)^{2}/\delta_{1}^{5})$ time randomized (1+δ ₁)-distance approximation algorithm for the largest well predicted subset problem under meaningful prediction. We also study a closely related problem, the bottleneck distance problem, where we are given two ordered point sets A={a ₁,…,a _n } and B={b ₁,b ₂,…b _n } containing n points and the problem is to find the smallest d _opt such that there exists a rigid transformation T with distance(a _i ,T(b _i ))≤d _opt for every point b _i ∈B. A (1+δ)-approximation for the bottleneck distance problem is to find a transformation T, such that for each b _i ∈B, distance?(a _i ,T(b _i ))≤(1+δ)d _opt , where δ is a constant. For an arbitrary constant δ, we obtain a linear O(n/δ ⁶) time (1+δ)-algorithm for the bottleneck distance problem. The best known algorithms for both problems require super-linear time (Li et al. in CPM 2008, 2008).     

Estimated Rate of Fatal Automobile Accidents Attributable to Acute Solvent Exposure at Low Inhaled Concentrations

Acute solvent exposures may contribute to automobile accidents because they increase reaction time and decrease attention, in addition to impairing other behaviors. These effects resemble those of ethanol consumption, both with respect to behavioral effects and neurological mechanisms. These observations, along with the extensive data on the relationship between ethanol consumption and fatal automobile accidents, suggested a way to estimate the probability of fatal automobile accidents from solvent inhalation. The problem can be approached using the logic of the algebraic transitive postulate of equality: if A=B and B=C, then A=C. We first calculated a function describing the internal doses of solvent vapors that cause the same magnitude of behavioral impairment as ingestion of ethanol (A=B). Next, we fit a function to data from the literature describing the probability of fatal car crashes for a given internal dose of ethanol (B=C). Finally, we used these two functions to generate a third function to estimate the probability of a fatal car crash for any internal dose of organic solvent vapor (A=C). This latter function showed quantitatively (1) that the likelihood of a fatal car crash is increased by acute exposure to organic solvent vapors at concentrations less than 1.0 ppm, and (2) that this likelihood is similar in magnitude to the probability of developing leukemia from exposure to benzene. This approach could also be applied to other potentially adverse consequences of acute exposure to solvents (e.g., nonfatal car crashes, property damage, and workplace accidents), if appropriate data were available.     

Covering directed graphs by in-trees

Given a directed graph D=(V,A) with a set of d specified vertices S={s ₁,…,s _d }?V and a function f : S→? where ? denotes the set of positive integers, we consider the problem which asks whether there exist ∑ _i=1 ^d f(s _i ) in-trees denoted by \(T_{i,1},T_{i,2},\ldots,T_{i,f(s_{i})}\) for every i=1,…,d such that \(T_{i,1},\ldots,T_{i,f(s_{i})}\) are rooted at s _i , each T _i,j spans vertices from which s _i is reachable and the union of all arc sets of T _i,j for i=1,…,d and j=1,…,f(s _i ) covers A. In this paper, we prove that such set of in-trees covering A can be found by using an algorithm for the weighted matroid intersection problem in time bounded by a polynomial in ∑ _i=1 ^d f(s _i ) and the size of D. Furthermore, for the case where D is acyclic, we present another characterization of the existence of in-trees covering A, and then we prove that in-trees covering A can be computed more efficiently than the general case by finding maximum matchings in a series of bipartite graphs.     

On the equitable k *-laceability of hypercubes

Let G be a finite undirected bipartite graph. Let u, v be two vertices of G from different partite sets. A collection of k internal vertex disjoint paths joining u to v is referred as a k-container C _k (u,v). A k-container is a k ^*-container if it spans all vertices of G. We define G to be a k ^*-laceable graph if there is a k ^*-container joining any two vertices from different partite sets. A k ^*-container C _k ^*(u,v)={P ₁,…,P _k } is equitable if ||V(P _i )|−|V(P _j )||≤2 for all 1≤i,j≤k. A graph is equitably k ^*-laceable if there is an equitable k ^*-container joining any two vertices in different partite sets. Let Q _n be the n-dimensional hypercube. In this paper, we prove that the hypercube Q _n is equitably k ^*-laceable for all k≤n−4 and n≥5. Dedicated to Professor Frank K. Hwang on the occasion of his 65th birthday. The work of H.-M. Huang was supported in part by the National Science Council of the Republic of China under NSC94-2115-M008-013.     

Algorithms for storage allocation based on client preferences

We consider a packing problem arising in storage management of Video on Demand (VoD) systems. The system consists of a set of video files (movies) and several servers (disks), each having a limited storage capacity, C, and a limited bandwidth (load capacity), L. The goal in the storage allocation problem is to assign the video files to the servers and the bandwidth to the clients. The induced class-constrained packing problem was studied in the past assuming each client provides a single request for a single movie. This paper considers a more general and realistic model—in which each client ranks all the movies in the system. Specifically, for each client j and movie i, it is known how much client j is willing to pay in order to watch movie i. The goal is to maximize the system’s profit. Alternatively, the client might provide a ranking of the movies and the goal is to maximize the lexicographic profile of the solution.     

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 .     

AN EMPIRICAL EXAMINATION OF SOCIAL POWER IN TERMS OF DECISION THEORY1

The paper presents an extension of decision theory to the analysis of social power. The power of a person, A, over another person, B, is viewed in terms of the effect A has on B's decision. The analysis is based on the idea that B's decision regarding the performance of alternative behaviors is a function of 1) B's utility for the consequences of the behaviors and 2) B's subjective probabilities that the behaviors will lead to these consequences. In these terms, A's power over B lies in A's ability to mediate various consequences for B, contingent upon B's compliance or noncompliance. Subjects were asked to consider eight situations in which hypothetical individuals had to make a choice between two courses of action. In each situation another person (A) was attempting to induce the hypothetical individual (B) to choose one of the alternatives, while various situational factors were influencing B to choose the other alternative. The subjects were asked to consider B's utilities and subjective probabilities in each situation and to indicate whether or not B should comply with A and to make ratings of A's power. The decision theory analysis did well in predicting whether or not subjects would indicate that B should comply with A. Also, subjects generally were able to correctly specify whether A or the situational factors had more influence over B's decision. Finally, the subjects' ratings of A's power in the eight situations were highly related to the decision theoretic measure of power.     

Partitioning a weighted partial order

The problem of partitioning a partially ordered set into a minimum number of chains is a well-known problem. In this paper we study a generalization of this problem, where we not only assume that the chains have bounded size, but also that a weight w _i is given for each element i in the partial order such that w _i ≤w _j if i ≺ j. The problem is then to partition the partial order into a minimum-weight set of chains of bounded size, where the weight of a chain equals the weight of the heaviest element in the chain. We prove that this problem is -hard, and we propose and analyze lower bounds for this problem. Based on these lower bounds, we exhibit a 2-approximation algorithm, and show that it is tight. We report computational results for a number of real-world and randomly generated problem instances.     

Sensitivity of the Optimum to Perturbations of the Profit or Weight of an Item in the Binary Knapsack Problem

In the binary single constraint Knapsack Problem, denoted KP, we are given a knapsack of fixed capacity c and a set of n items. Each item j, j = 1,...,n, has an associated size or weight w_j and a profit p_j. The goal is to determine whether or not item j, j = 1,...,n, should be included in the knapsack. The objective is to maximize the total profit without exceeding the capacity c of the knapsack. In this paper, we study the sensitivity of the optimum of the KP to perturbations of either the profit or the weight of an item. We give approximate and exact interval limits for both cases (profit and weight) and propose several polynomial time algorithms able to reach these interval limits. The performance of the proposed algorithms are evaluated on a large number of problem instances.     

An optimal online algorithm for single machine scheduling to minimize total general completion time

We study the online problem of single machine scheduling to minimize total general completion time. General completion time is defined as C^a_j=(C_j)^aC^{\alpha}_{j}=(C_{j})^{\alpha}, where C _j denotes the completion time of job J _j and α≥1 is a constant integer. Total general completion time characterizes the feather in service that when a customer is served later in time, his dissatisfaction increases in a manner of power function. The objective function ∑(C _j ) ^α can also be viewed as a total weighted completion time, but the “weight” is no longer a constant number. Our purpose to minimize customers’ total dissatisfaction. The problem is online in the sense that all jobs arrive over time. Each job’s processing time becomes known at its arrival time. Preemption is not allowed. For this online problem, we show that a lower bound on competitive ratio is 2 ^α and prove that D-SPT (delayed shortest processing time) algorithm is optimal with a competitive ratio 2 ^α .     

Estimating Health Risks from Natural Hazards Using Risk Assessment and Epidemiology

Risk assessment is the process of estimating the likelihood that an adverse effect may result from exposure to a specific health hazard. The process traditionally involves hazard identification, dose-response assessment, exposure assessment, and risk characterization to answer “How many excess cases of disease A will occur in a population of size B due to exposure to agent C at dose level D?” For natural hazards, however, we modify the risk assessment paradigm to answer “How many excess cases of outcome Y will occur in a population of size B due to natural hazard event E of severity D?” Using a modified version involving hazard identification, risk factor characterization, exposure characterization, and risk characterization, we demonstrate that epidemiologic modeling and measures of risk can quantify the risks from natural hazard events. We further extend the paradigm to address mitigation, the equivalent of risk management, to answer “What is the risk for outcome Y in the presence of prevention intervention X relative to the risk for Y in the absence of X?” We use the preventable fraction to estimate the efficacy of mitigation, or reduction in adverse health outcomes as a result of a prevention strategy under ideal circumstances, and further estimate the effectiveness of mitigation, or reduction in adverse health outcomes under typical community-based settings. By relating socioeconomic costs of mitigation to measures of risk, we illustrate that prevention effectiveness is useful for developing cost-effective risk management options.     

L(j,k)- and circular L(j,k)-labellings for the products of complete graphs

Let j and k be two positive integers with j≥k. An L(j,k)-labelling of a graph G is an assignment of nonnegative integers to the vertices of G such that the difference between labels of any two adjacent vertices is at least j, and the difference between labels of any two vertices that are at distance two apart is at least k. The minimum range of labels over all L(j,k)-labellings of a graph G is called the λ _j,k -number of G, denoted by λ _j,k (G). A σ(j,k)-circular labelling with span m of a graph G is a function f:V(G)→{0,1,…,m−1} such that |f(u)−f(v)| _m ≥j if u and v are adjacent; and |f(u)−f(v)| _m ≥k if u and v are at distance two apart, where |x| _m =min {|x|,m−|x|}. The minimum m such that there exists a σ(j,k)-circular labelling with span m for G is called the σ _j,k -number of G and denoted by σ _j,k (G). The λ _j,k -numbers of Cartesian products of two complete graphs were determined by Georges, Mauro and Stein ((2000) SIAM J Discret Math 14:28–35). This paper determines the λ _j,k -numbers of direct products of two complete graphs and the σ _j,k -numbers of direct products and Cartesian products of two complete graphs. Dedicated to Professor Frank K. Hwang on the occasion of his 65th birthday. This work is partially supported by FRG, Hong Kong Baptist University, Hong Kong; NSFC, China, grant 10171013; and Southeast University Science Foundation grant XJ0607230.     

Approximation algorithms for connected facility location problems

We study Connected Facility Location problems. We are given a connected graph G=(V,E) with nonnegative edge cost c _e for each edge e∈E, a set of clients D⊆V such that each client j∈D has positive demand d _j and a set of facilities F⊆V each has nonnegative opening cost f _i and capacity to serve all client demands. The objective is to open a subset of facilities, say , to assign each client j∈D to exactly one open facility i(j) and to connect all open facilities by a Steiner tree T such that the cost is minimized for a given input parameter M≥1. We propose a LP-rounding based 8.29 approximation algorithm which improves the previous bound 8.55 (Swamy and Kumar in Algorithmica, 40:245–269, 2004). We also consider the problem when opening cost of all facilities are equal. In this case we give a 7.0 approximation algorithm.     

ALLIANCES BETWEEN COMPETITORS AND CONSUMER INFORMATION

Alliances between competitors in which established firms provide access to proprietary resources—for example, their distribution channels—are important business practices. We analyze a market where an established firm, firm A, produces a product of well‐known quality, and a firm with an unknown brand, firm B, has to choose to produce high or low quality. Firm A observes firm B's quality choice but consumers do not. Hence, firm B is subject to a moral hazard problem which can potentially be solved by firm A. Firm A can accept or reject to form an alliance with firm B, which is observed by consumers. If an alliance is formed, firm A implicitly certifies the rival's product. Consumers infer that firm B is a competitor with high quality, because otherwise why would the established firm accept to form an alliance? The mechanism we discover allows for an economic interpretation of several types of business practices. (JEL: L15, L13, L24, L42, M21, M31, D43)