Tight Performance Bounds of CP-Scheduling on Out-Trees

The worst-case behavior of the critical path (CP) algorithm for multiprocessor scheduling with an out-tree task dependency structure is studied. The out-tree is not known in advance and the tasks are released on-line over time (each task is released at the completion time of its direct predecessor task in the out-tree). For each task, the processing time and the remainder (the length of the longest chain of the future tasks headed by this task) become known at its release time. The tight worst-case ratio and absolute error are derived for this strongly clairvoyant on-line model. For out-trees with a specific simple structure, essentially better worst-case ratio and absolute error are derived. Our bounds are given in terms of t _max, the length of the longest chain in the out-tree, and it is shown that the worst-case ratio asymptotically approaches 2 for large t _max when the number of processors , where is an integer close to . A non-clairvoyant on-line version (without knowledge of task processing time and remainder at the release time of the task) is also considered and is shown that the worst-case behavior of width-first search is better or the same as that of the depth-first search.     

Online LPT algorithms for parallel machines scheduling with a single server

We consider two parallel machines scheduling problems with a single server. For the general case we present an online LPT algorithm with competitive ratio 2, and give a lower bound $\frac{\sqrt{5} + 1}{2}$ . We also apply the online LPT algorithm to the special case where all the setup times are equal to 1. We show that the competitive ratio is 1.5, and no online algorithm can has a competitive ratio less than  $\sqrt{2}$ .     

A New Approximation Algorithm for the Capacitated Vehicle Routing Problem on a Tree

This paper presents a new approximation algorithm for a vehicle routing problem on a tree-shaped network with a single depot. Customers are located on vertices of the tree, and each customer has a positive demand. Demands of customers are served by a fleet of identical vehicles with limited capacity. It is assumed that the demand of a customer is splittable, i.e., it can be served by more than one vehicle. The problem we are concerned with in this paper asks to find a set of tours of the vehicles with minimum total lengths. Each tour begins at the depot, visits a subset of the customers and returns to the depot without violating the capacity constraint. We propose a 1.35078-approximation algorithm for the problem (exactly, ), which is an improvement over the existing 1.5-approximation.     

Online tradeoff scheduling on a single machine to minimize makespan and maximum lateness

In this paper, we consider the following single machine online tradeoff scheduling problem. A set of n independent jobs arrive online over time. Each job \(J_{j}\) has a release date \(r_{j}\), a processing time \(p_{j}\) and a delivery time \(q_{j}\). The characteristics of a job are unknown until it arrives. The goal is to find a schedule that minimizes the makespan \(C_{\max } = \max _{1 \le j \le n} C_{j}\) and the maximum lateness \(L_{\max } = \max _{1 \le j \le n} L_{j}\), where \(L_{j} = C_{j} + q_{j}\). For the problem, we present a nondominated \(( \rho , 1 + \displaystyle \frac{1}{\rho } )\)-competitive online algorithm for each \(\rho \) with \( 1 \le \rho \le \displaystyle \frac{\sqrt{5} + 1}{2}\).     

Inverse Polymatroidal Flow Problem

Let D = (V, A) be a directed graph, for each vertex v V, let ⁺(v) and ^– (v) denote the sets of arcs leaving and entering v, and be intersecting families on ⁺(v) and ^–(v), respectively, and and be submodular functions on intersecting pairs. A flow f : A R is feasible if

Augmenting a Submodular and Posi-modular Set Function by a Multigraph

Given a finite set V and a set function , we consider the problem of constructing an undirected multigraph G = (V,E) such that the cut function together has value at least 2 for all non-empty and proper subsets of V. If f is intersecting submodular and posi-modular, and satisfies the tripartite inequality, then we show that such a multigraph G with the minimum number of edges can be found in time, where is the time to compute the value of f(X) for a subset .     

Ordinal On-Line Scheduling for Maximizing the Minimum Machine Completion Time

This paper considers the on-line problem of scheduling nonpreemptively n independent jobs on m > 1 identical and parallel machines with the objective to maximize the minimum machine completion time. It is assumed that the values of the processing times are unknown but the order of the jobs by their processing times is known in advance. We are asked to decide the assignment of all the jobs to some machines at time zero by utilizing only ordinal data rather than the actual magnitudes of jobs. Algorithms to slove the problem are called ordinal algorithms. In this paper, we give lower bounds and ordinal algorithms. We first propose an algorithm MIN which is at most -competitive for any m machine case, while the lower bound is _i=1 ^m 1/i. Both are on the order of (ln m). Furthermore, for m = 3, we present an optimal algorithm.     

需求扰动下供应链反向保理的鲁棒决策

反向保理运用买方信誉缓解中小供应商资金困境，现实中需求扰动时有发生并影响供应链决策。探讨供应商资金约束背景下"买方驱动"反向保理对供应链运营的作用规律，运用极大极小法构建单供应商和单零售商组成的二级供应链模型，得到只知均值和方差时的供应链决策和收益。研究表明：鲁棒决策能为反向保理时需求信息缺失的供应链提供有效而稳健的决策，降低信用违约风险；基于供应链关系的反向保理增强了供应链合作；供应商信誉损失风险不会成为零售商主导反向保理实施的阻碍。丰富了金融与运营交互作用的理论研究。     

An approximation algorithm for <Emphasis Type="Italic">k</Emphasis>-facility location problem with linear penalties using local search scheme

In this paper, we consider an extension of the classical facility location problem, namely k-facility location problem with linear penalties. In contrast to the classical facility location problem, this problem opens no more than k facilities and pays a penalty cost for any non-served client. We present a local search algorithm for this problem with a similar but more technical analysis due to the extra penalty cost, compared to that in Zhang (Theoretical Computer Science 384:126–135, 2007). We show that the approximation ratio of the local search algorithm is \(2 + 1/p + \sqrt{3+ 2/p+ 1/p^2} + \epsilon \), where \(p \in {\mathbb {Z}}_+\) is a parameter of the algorithm and \(\epsilon >0\) is a positive number.     

Local Optimality and Its Application on Independent Sets for k-claw Free Graphs

Given a graph G = (V,E), we define the locally optimal independent sets asfollows. Let S be an independent set and T be a subset of V such that S T = and (S) T, where (S) is defined as the neighbor set of S. A minimum dominating set of S in T is defined as T_D(S) T such that every vertex of S is adjacent to a vertex inTD(S) and TD(S) has minimum cardinality. An independent setI is called r-locally optimal if it is maximal and there exists noindependent set S V\I with |ID (S)| r such that|S| >|I (S)|.In this paper, we demonstrate that for k-claw free graphs ther-locally optimal independent sets is found in polynomial timeand the worst case is bounded by , where I and I* are a locally optimal and an optimal independent set,respectively. This improves the best published bound by Hochbaum (1983) bynearly a factor of two. The bound is proved by LP duality and complementaryslackness. We provide an efficientO(|V|^r+3) algorithm to find an independent set which is notnecessarily r-locally optimal but is guarantteed with the above bound. Wealso present an algorithm to find a r-locally optimal independent set inO(|V|^r(k-1)+3) time.     

Two-stage proportionate flexible flow shop to minimize the makespan

We consider a two-stage flexible flow shop problem with a single machine at one stage and m identical machines at the other stage, where the processing times of each job at both stages are identical. The objective is to minimize the makespan. We describe some optimality conditions and show that the problem is NP-hard when m is fixed. Finally, we present an approximation algorithm that has a worst-case performance ratio of $\frac{5}{4}$ for m=2 and $\frac{\sqrt{1+m^{2}}+1+m}{2m}$ for m≥3.     

On Approximability of Boolean Formula Minimization

For a Boolean function given by a Boolean formula (or a binary circuit) S we discuss the problem of building a Boolean formula (binary circuit) of minimal size, which computes the function g equivalent to , or -equivalent to , i.e., . In this paper we prove that if P NP then this problem can not be approximated with a good approximation ratio by a polynomial time algorithm.     

On Shortest k-Edge-Connected Steiner Networks in Metric Spaces

Given a set of points P in a metric space, let l(P) denote the ratio of lengths between the shortest k-edge-connected Steiner network and the shortest k-edge-connected spanning network on P, and let r = inf l(P) P for k 1. In this paper, we show that in any metric space, r 3/4 for k 2, and there exists a polynomial-time -approximation for the shortest k-edge-connected Steiner network, where = 2 for even k and = 2 + 4/(3k) for odd k. In the Euclidean plane, and .     

Finding a Noncrossing Steiner Forest in Plane Graphs Under a 2-Face Condition

Let G = (V,E) be a plane graph with nonnegative edge weights, and let be a family of k vertex sets , called nets. Then a noncrossing Steiner forest for in G is a set of k trees in G such that each tree connects all vertices, called terminals, in net N _i, any two trees in do not cross each other, and the sum of edge weights of all trees is minimum. In this paper we give an algorithm to find a noncrossing Steiner forest in a plane graph G for the case where all terminals in nets lie on any two of the face boundaries of G. The algorithm takes time if G has n vertices and each net contains a bounded number of terminals.     

Simple Approximation Algorithms for MAXNAESP and Hypergraph 2-colorability

Hypergraph 2-colorability, also known as set splitting, is a widely studied problem in graph theory. In this paper we study the maximization version of the same. We recast the problem as a special type of satisfiability problem and give approximation algorithms for it. Our results are valid for hypergraph 2-colorability, set splitting and MAX-CUT (which is a special case of hypergraph 2-colorability) because the reductions are approximation preserving. Here we study the MAXNAESP problem, the optimal solution to which is a truth assignment of the literals that maximizes the number of clauses satisfied. As a main result of the paper, we show that any locally optimal solution (a solution is locally optimal if its value cannot be increased by complementing assignments to literals and pairs of literals) is guaranteed a performance ratio of . This is an improvement over the ratio of attributed to another local improvement heuristic for MAX-CUT (C. Papadimitriou, Computational Complexity, Addison Wesley, 1994). In fact we provide a bound of for this problem, where k 3 is the minimum number of literals in a clause. Such locally optimal algorithms appear to subsume typical greedy algorithms that have been suggested for problems in the general domain of satisfiability. It should be noted that the NAESP problem where each clause has exactly two literals, is equivalent to MAX-CUT. However, obtaining good approximation ratios using semi-definite programming techniques (M. Goemans and D.P. Williamson, in Proceedings of the 26th Annual ACM Symposium on Theory of Computing, 1994a, pp. 422–431) appears difficult. Also, the randomized rounding algorithm as well as the simple randomized algorithm both (M. Goemans and D.P. Williamson, SIAM J. Disc. Math, vol. 7, pp. 656–666, 1994b) yield a bound of for the MAXNAESP problem. In contrast to this, the algorithm proposed in this paper obtains a bound of for this problem.     

Efficient algorithms for finding a longest common increasing subsequence

We study the problem of finding a longest common increasing subsequence (LCIS) of multiple sequences of numbers. The LCIS problem is a fundamental issue in various application areas, including the whole genome alignment. In this paper we give an efficient algorithm to find the LCIS of two sequences in time where n is the length of each sequence andr is the number of ordered pairs of positions at which the two sequences match, ℓ is the length of the LCIS, and Sort(n) is the time to sort n numbers. For m sequences wherem ≥ 3, we find the LCIS in Sort(n)) time where r is the total number of m-tuples of positions at which the m sequences match. The previous results find the LCIS of two sequences in O(n ²) and Sort(n)) time. Our algorithm is faster when r is relatively small, e.g., for .     

New analysis and computational study for the planar connected dominating set problem

The connected dominating set (CDS) problem is a well studied NP-hard problem with many important applications. Dorn et al. (Algorithmica 58:790–810 2010) introduce a branch-decomposition based algorithm design technique for NP-hard problems in planar graphs and give an algorithm (DPBF algorithm) which solves the planar CDS problem in \(O(2^{9.822\sqrt{n}}n+n^3)\) time and \(O(2^{8.11\sqrt{n}}n+n^3)\) time, with a conventional method and fast matrix multiplication in the dynamic programming step of the algorithm, respectively. We show that DPBF algorithm solves the planar CDS problem in \(O(2^{9.8\sqrt{n}}n+n^3)\) time with a conventional method and in \(O(2^{8.08\sqrt{n}}n+n^3)\) time with a fast matrix multiplication. For a graph \(G\), let \({\hbox {bw}}(G)\) be the branchwidth of \(G\) and \(\gamma _c(G)\) be the connected dominating number of \(G\). We prove \({\hbox {bw}}(G)\le 2\sqrt{10\gamma _c(G)}+32\). From this result, the planar CDS problem admits an \(O(2^{23.54\sqrt{\gamma _c(G)}}\gamma _c(G)+n^3)\) time fixed-parameter algorithm. We report computational study results on the practical performance of DPBF algorithm, which show that the size of instances can be solved by the algorithm mainly depends on the branchwidth of the instances, coinciding with the theoretical analysis. For graphs with small or moderate branchwidth, the CDS problem instances with size up to a few thousands edges can be solved in a practical time and memory space.     

On zero error algorithms having oracle access to one query

It is known that (Cai, 2001). The reverse direction of whether ZPP^NP is contained in remains open. We show that if the zero-error algorithm is allowed to ask only one query to the NP oracle (for any input and random string), then it can be simulated in . That is, we prove that . Next we consider whether the above result can be improved as and point out a difficulty in doing so. Via a simple proof, we observe that BPP ⊆ ZPP^NP[1] (a result implicitly proven in some prior work). Thus, achieving the above improvement would imply BPP ⊆ P^NP, settling a long standing open problem. We then argue that the above mentioned improvement can be obtained for the next level of the polynomial time hierarchy. Namely, we prove that . On the other hand, by adapting our proof of our main result it can be shown that . For the purpose of comparing these two results, we prove that . We conclude by observing that the above claims extend to the higher levels of the hierarchy: for k ≥ 2, and . Research supported in part by NSF grant CCR-0208013. A preliminary version of the paper was presented at COCOON′05 Cai and Chakaravarthy (2005). Part of the research was conducted while the author was at the University of Wisconsin, Madison.     

A Further Improved Approximation Algorithm for Breakpoint Graph Decomposition

Breakpoint graph decomposition is a crucial step in all recent approximation algorithms for SORTING BY REVERSALS, which is one of the best-known algorithmic problems in computational molecular biology. Caprara and Rizzi recently improved the approximation ratio for breakpoint graph decomposition from to + 1.4348 + , for any positive . In this paper, we extend the techniques of Caprara and Rizzi and incorporate a balancing argument to further improve the approximation ratio to + 1.4193 + , for any positive . These improvements imply improved approximation results for SORTING BY REVERSALS for almost all random permutations.     

$$(\alpha , \tau )$$-Monitoring for event detection in wireless sensor networks

Detecting abnormal events is one of the fundamental issues in wireless sensor networks (WSNs). In this paper, we investigate \((\alpha ,\tau )\)-monitoring in WSNs. For a given monitored threshold \(\alpha \), we prove that (i) the tight upper bound of \(\Pr [{S(t)} \ge \alpha ]\) is \(O\left( {\exp \left\{ { - n\ell \left( {\frac{\alpha }{{nsup}},\frac{{\mu (t)}}{{nsup}}} \right) } \right\} } \right) \), if \(\mu (t) < \alpha \); and (ii) the tight upper bound of \(\Pr [{S(t)} \le \alpha ]\) is \(O\left( {\exp \left\{ { - n\ell \left( {\frac{\alpha }{{nsup}},\frac{{\mu (t)}}{{nsup}}} \right) } \right\} } \right) \), if \(\mu (t) > \alpha \), where \(\Pr [X]\) is the probability of random event \(X,\, S(t)\) is the sum of the monitored area at time \(t,\, n\) is the number of the sensor nodes, \(sup\) is the upper bound of sensed data, \( \mu (t)\) is the expectation of \(S(t)\), and \(\ell ({x_1},{x_2}) = {x_1}\ln \left( {\frac{{{x_1}}}{{{x_2}}}} \right) + (1 - {x_1})\ln \left( {\frac{{1 - {x_1}}}{{1 - {x_2}}}} \right) \). An instant \((\alpha ,\tau )\)-monitoring scheme is then developed based on the upper bound. Moreover, approximate continuous \((\alpha , \tau )\)-monitoring is investigated. We prove that the probability of false negative alarm is \(\delta \), if the sample size is Open image in new window for a given precision requirement, where Open image in new window is the Open image in new window fractile of a standard normal distribution. Finally, the performance of the proposed algorithms is validated through experiments.