软容量约束带随机需求的设施选址问题的近似算法

文考虑了软容量约束带随机需求的设施选址问题,根据此问题构造出一个无容量约束带随机需求的设施选址问题,通过求解无容量约束情形给出软容量情形的一个可行解,分析出近似比为6。     

平方度量的k层设施选址问题的近似算法

本文中,我们研究平方度量的k层设施选址问题,该问题中设施分为k层,每个顾客都要连接到位于不同层上的k个设施,顾客与设施以及设施与设施之间的距离是平方度量的.目标是使得开设费用与连接费用之和最小.基于线性规划舍入技巧,我们给出了9-近似算法.进一步,我们研究了平方度量的k层软容量设施选址问题,并给出了线性规划舍入12.2216-近似算法.     

软容量约束的动态设施选址问题的近似算法

考虑软容量约束的动态设施选址问题.假设设施的开放费用及连接费用都与时间有关,而且每一个设施均有容量约束.对此问题给出了第一个近似比为6的原始对偶(组合)算法.运行贪婪增加程序后,近似比进一步改进到3.7052.     

鲁棒动态设施选址问题的近似算法

设施选址问题是组合优化中重要问题之一。动态设施选址问题是传统设施选址问题的推广,其中度量空间中设施的开设费用和顾客的需求均随着时间的变化而变化。更多地,经典设施选址问题假设所有的顾客都需要被服务。在这个模型假设下,所有的顾客都需要服务。但事实上,有时为服务距离较远的顾客,需要单独开设设施,导致了资源的浪费。因此,在模型设置中,可以允许一些固定数目的顾客不被服务 (带异常点的设施选址问题),此外也可以通过支付一些顾客的惩罚费用以达到不服务的目的 (带惩罚的设施选址问题)。本文将综合以上两种鲁棒设置考虑同时带有异常点和惩罚的动态设施选址问题,通过原始-对偶框架得到近似比为3的近似算法。     

带次模惩罚的优先设施选址问题的近似算法

研究带次模惩罚的优先设施选址问题, 每个顾客都有一定的服务水平要求, 开设的设施只有满足了顾客的服务水平要求, 才能为顾客提供服务, 没被服务的顾客对应一定的次模惩罚费用. 目标是使得开设费用、连接费用与次模惩罚费用之和最小. 给出该问题的整数规划、 线性规划松弛及其对偶规划. 基于原始对偶和贪婪增广技巧, 给出该问题的两个近似算法, 得到的近似比分别为3和2.375.     

平方度量动态设施选址问题的近似算法

研究了单阶段度量设施选址问题的推广问题平方度量动态设施选址问题. 研究中首先利用原始对偶技巧得到 9-近似算法, 然后利用贪婪增广技巧将近似比改进到2.606, 最后讨论了该问题的相应变形问题.     

带有异常点的平方度量设施选址问题

传统的设施选址问题一般假设所有顾客都被服务,考虑到异常点的存在不仅会增加总费用（设施的开设费用与连接费用之和）,也会影响到对其他顾客的服务质量。研究异常点在最终方案中允许不被服务的情况,称之为带有异常点的平方度量设施选址问题。该问题是无容量设施选址问题的推广。问题可描述如下：给定设施集合、顾客集,以及设施开设费用和顾客连接费用,目标是选择设施的子集开设以满足顾客的需求,使得设施开设费用与连接费用之和最小。利用原始对偶技巧设计了近似算法,证明了该算法的近似比是9。     

带惩罚的动态设施选址问题的近似算法

本文研究带惩罚的动态设施选址问题,在该问题中假设不同时段内设施的开放费用、用户的需求及连接费用可以不相同,而且允许用户的需求不被满足,但是要有惩罚.对此问题我们给出了第-个近似比为1.8526的原始对偶(组合)算法.     

带次模惩罚和随机需求的设施选址问题

考虑带次模惩罚和随机需求的设施选址问题,目的是开设设施集合的一个子集,把客户连接到开设的设施上并对没有连接的客户进行惩罚,使得开设费用、连接费用、库存费用、管理费用和惩罚费用之和达到最小. 根据该问题的特殊结构,给出原始对偶3-近似算法. 在算法的第一步,构造了一组对偶可行解;在第二步中构造了对应的一组原始整数可行解,这组原始整数可行解给出了最后开设的设施集合和被惩罚的客户集合. 最后,证明了算法在多项式时间内可以完成,并且算法所给的整数解不会超过最优解的3倍.     

考虑中断风险与库存成本的分销网络设计模型

分销网络可能面临各种中断风险,而分销网络设计属于战略决策问题,短期内难以改变,因而有必要在选址设计阶段就考虑中断风险。考虑中断风险,对传统的分销网络设计问题进行扩展,基于非线性0－1整数规划方法建立了一个有容量约束的设施定位-库存模型。采用遗传算法予以求解。算例分析证明了遗传算法的有效性。结果表明：在网络设计阶段就考虑中断风险可以显著降低将来可能发生的应急成本;系统对中断风险、惩罚成本因子等因素的反应敏感。     

Semi-preemptive routing on trees

We study a variant of the pickup-and-delivery problem (PDP) in which the objects that have to be transported can be reloaded at most d times, for a given d∈N. This problem is known to be polynomially solvable on paths or cycles and NP-complete on trees. We present a (4/3+ε)-approximation algorithm if the underlying graph is a tree. By using a result of Charikar et al. [M. Charikar, C. Chekuri, A. Goel, S. Guha, S. Plotkin, Approximating a finite metric by a small number of tree metrics, in: FOCS ’98: Proceedings of the 39th Annual Symposium on Foundations of Computer Science, IEEE Computer Society, Washington, DC, USA, 1998, pp. 379-388], this can be extended to a O(lognloglogn)-approximation for general graphs.     

List decoding of number field codes

This paper presents a list decoding algorithm for the number field codes of Guruswami (IEEE Trans Inf Theory 49:594–603, 2003). The algorithm is an implementation of the unified framework for list decoding of algebraic codes of Guruswami, Sahai and Sudan (Proceedings of the 41st Annual Symposium on Foundations of Computer Science, 2000), specialised for number field codes. The computational complexity of the algorithm is evaluated in terms of the size of the inputs and field invariants.     

Sums of squares based approximation algorithms for MAX-SAT

We investigate the Semidefinite Programming based sums of squares (SOS) decomposition method, designed for global optimization of polynomials, in the context of the (Maximum) Satisfiability problem. To be specific, we examine the potential of this theory for providing tests for unsatisfiability and providing MAX-SAT upper bounds. We compare the SOS approach with existing upper bound and rounding techniques for the MAX-2-SAT case of Goemans and Williamson [Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, J. Assoc. Comput. Mach. 42(6) (1995) 1115-1145] and Feige and Goemans [Approximating the value of two prover proof systems, with applications to MAX2SAT and MAXDICUT, in: Proceedings of the Third Israel Symposium on Theory of Computing and Systems, 1995, pp. 182-189] and the MAX-3-SAT case of Karloff and Zwick [A 7/8-approximation algorithm for MAX 3SAT? in: Proceedings of the 38th Annual IEEE Symposium on Foundations of Computer Science, Miami Beach, FL, USA, IEEE Press, New York, 1997], which are based on Semidefinite Programming as well. We prove that for each of these algorithms there is an SOS-based counterpart which provides upper bounds at least as tight, but observably tighter in particular cases. Also, we propose a new randomized rounding technique based on the optimal solution of the SOS Semidefinite Program (SDP) which we experimentally compare with the appropriate existing rounding techniques. Further we investigate the implications to the decision variant SAT and compare experimental results with those yielded from the higher lifting approach of Anjos [On semidefinite programming relaxations for the satisfiability problem, Math. Methods Oper. Res. 60(3) (2004) 349-367; An improved semidefinite programming relaxation for the satisfiability problem, Math. Programming 102(3) (2005) 589-608; Semidefinite optimization approaches for satisfiability and maximum-satisfiability problems, J. Satisfiability Boolean Modeling Comput. 1 (2005) 1-47].We give some impression of the fraction of the so-called unit constraints in the various SDP relaxations. From a mathematical viewpoint these constraints should be easily dealt within an algorithmic setting, but seem hard to be avoided as extra constraints in an SDP setting. Finally, we briefly indicate whether this work could have implications in finding counterexamples to uncovered cases in Hilbert's Positivstellensatz.     

Metric structures in : dimension, snowflakes, and average distortion

We study the metric properties of finite subsets of L₁. The analysis of such metrics is central to a number of important algorithmic problems involving the cut structure of weighted graphs, including the Sparsest Cut Problem, one of the most compelling open problems in the field of approximation algorithms. Additionally, many open questions in geometric non-linear functional analysis involve the properties of finite subsets of L₁.We present some new observations concerning the relation of L₁ to dimension, topology, and Euclidean distortion. We show that every n-point subset of L₁ embeds into L₂ with average distortion , yielding the first evidence that the conjectured worst-case bound of is valid. We also address the issue of dimension reduction in L_p for p(1,2). We resolve a question left open by M. Charikar and A. Sahai [Dimension reduction in the ℓ₁ norm, in: Proceedings of the 43rd Annual IEEE Conference on Foundations of Computer Science, ACM, 2002, pp. 251–260] concerning the impossibility of dimension reduction with a linear map in the above cases, and we show that a natural variant of the recent example of Brinkman and Charikar [On the impossibility of dimension reduction in ℓ₁, in: Proceedings of the 44th Annual IEEE Conference on Foundations of Computer Science, ACM, 2003, pp. 514–523], cannot be used to prove a lower bound for the non-linear case. This is accomplished by exhibiting constant-distortion embeddings of snowflaked planar metrics into Euclidean space.     

On the “piano movers” problem. II. General techniques for computing topological properties of real algebraic manifolds

This paper continues the discussion, begun in J. Schwartz and M. Sharir [Comm. Pure Appl. Math., in press], of the following problem, which arises in robotics: Given a collection of bodies B, which may be hinged, i.e., may allow internal motion around various joints, and given a region bounded by a collection of polyhedral or other simple walls, decide whether or not there exists a continuous motion connecting two given positions and orientations of the whole collection of bodies. We show that this problem can be handled by appropriate refinements of methods introduced by A. Tarski [“A Decision Method for Elementary Algebra and Geometry,” 2nd ed., Univ. of Calif. Press, Berkeley, 1951] and G. Collins [in “Second GI Conference on Automata Theory and Formal Languages,” Lecture Notes in Computer Science, Vol. 33, pp. 134–183, Springer-Verlag, Berlin, 1975], which lead to algorithms for this problem which are polynomial in the geometric complexity of the problem for each fixed number of degrees of freedom (but exponential in the number of degrees of freedom). Our method, which is also related to a technique outlined by J. Reif [in “Proceedings, 20th Symposium on the Foundations of Computer Science,” pp. 421–427, 1979], also gives a general (but not polynomial time) procedure for calculating all of the homology groups of an arbitrary real algebraic variety. Various algorithmic issues concerning computations with algebraic numbers, which are required in the algorithms presented in this paper, are also reviewed.     

An Approximate Max-Steiner-Tree-Packing Min-Steiner-Cut Theorem*

Given an undirected multigraph G and a subset of vertices S ⊆ V (G), the STEINER TREE PACKING problem is to find a largest collection of edge-disjoint trees that each connects S. This problem and its generalizations have attracted considerable attention from researchers in different areas because of their wide applicability. This problem was shown to be APX-hard (no polynomial time approximation scheme unless P=NP). In fact, prior to this paper, not even an approximation algorithm with asymptotic ratio o(n) was known despite several attempts. In this work, we present the first polynomial time constant factor approximation algorithm for the STEINER TREE PACKING problem. The main theorem is an approximate min-max relation between the maximum number of edge-disjoint trees that each connects S (S-trees) and the minimum size of an edge-cut that disconnects some pair of vertices in S (S-cut). Specifically, we prove that if every S-cut in G has at least 26k edges, then G has at least k edge-disjoint S-trees; this answers Kriesells conjecture affirmatively up to a constant multiple. * A preliminary version appeared in the Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science (FOCS) 2004. † The author was supported by an Ontario Graduate Scholarship and a University of Toronto Fellowship.     

Approximability of flow shop scheduling

Shop scheduling problems are notorious for their intractability, both in theory and practice. In this paper, we construct a polynomial approximation scheme for the flow shop scheduling problem with an arbitrary fixed number of machines. For the three common shop models (open, flow, and job), this result is the only known approximation scheme. Since none of the three models can be approximated arbitrarily closely in the general case (unless P = NP), the result demonstrates the approximability gap between the models in which the number of machines is fixed, and those in which it is part of the input of the instance. The result can be extended to flow shops with job release dates and delivery times and to flow shops with a fixed number of stages, where the number of machines at any stage is fixed. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.A preliminary version of this paper appeared in theProceedings of the 36th Annual IEEE Symposium on the Foundations of Computer Science, 1995.Research supported by NSF grant DMI-9496153.     

A greedy approximation algorithm for the group Steiner problem

In the group Steiner problem we are given an edge-weighted graph G=(V,E,w) and m subsets of vertices . Each subset g_i is called a group and the vertices in ?_ig_i are called terminals. It is required to find a minimum weight tree that contains at least one terminal from every group.We present a poly-logarithmic ratio approximation for this problem when the input graph is a tree. Our algorithm is a recursive greedy algorithm adapted from the greedy algorithm for the directed Steiner tree problem [Approximating the weight of shallow Steiner trees, Discrete Appl. Math. 93 (1999) 265-285, Approximation algorithms for directed Steiner problems, J. Algorithms 33 (1999) 73-91]. This is in contrast to earlier algorithms that are based on rounding a linear programming based relaxation for the problem [A polylogarithmic approximation algorithm for the Group Steiner tree problem, J. Algorithms 37 (2000) 66-84, preliminary version in Proceedings of SODA, 1998 pp. 253-259, On directed Steiner trees, Proceedings of SODA, 2002, pp. 59-63]. We answer in positive a question posed in [A polylogarithmic approximation algorithm for the Group Steiner tree problem, J. Algorithms 37 (2000) 66-84, preliminary version in Proceedings of SODA, 1998 pp. 253-259] on whether there exist good approximation algorithms for the group Steiner problem that are not based on rounding linear programs. For every fixed constant ε>0, our algorithm gives an approximation in polynomial time. Approximation algorithms for trees can be extended to arbitrary undirected graphs by probabilistically approximating the graph by a tree. This results in an additional multiplicative factor of in the approximation ratio, where |V| is the number of vertices in the graph. The approximation ratio of our algorithm on trees is slightly worse than the ratio of O(log(max_i|g_i|)·logm) provided by the LP based approaches.     

带惩罚的容错设施布局问题的近似算法

在带惩罚的容错设施布局问题中, 给定顾客集合、地址集合、以及每个顾客和各个地址之间的连接费用, 这里假设连接费用是可度量的. 每位顾客有各自的服务需求, 每个地址可以开设任意多个设施, 顾客可以被安排连接到某些地址的一些开设的设施上以满足其需求, 也可以被拒绝, 但这时要支付拒绝该顾客所带来的惩罚费用. 目标是确定哪些顾客的服务需求被拒绝并开设一些设施, 将未被拒绝的顾客连接到不同的开设设施上, 使得开设费用、连接费用和惩罚费用总和最小. 给出了带惩罚的容错设施布局问题的线性整数规划及其对偶规划, 进一步, 给出了基于其线性规划和对偶规划舍入的4-近似算法.     

On routing in VLSI design and communication networks

In this paper, we study the global routing problem in VLSI design and the multicast routing problem in communication networks. First we propose new and realistic models for both problems. In the global routing problem in VLSI design, we are given a lattice graph and subsets of the vertex set. The goal is to generate trees spanning these vertices in the subsets to minimize a linear combination of overall wirelength (edge length) and the number of bends of trees with respect to edge capacity constraints. In the multicast routing problem in communication networks, a graph is given to represent the network, together with subsets of the vertex set. We are required to find trees to span the given subsets and the overall edge length is minimized with respect to capacity constraints. Both problems are APX-hard. We present the integer linear programming (LP) formulation of both problems and solve the LP relaxations by the fast approximation algorithms for min-max resource-sharing problems in [K. Jansen, H. Zhang, Approximation algorithms for general packing problems and their application to the multicast congestion problem, Math. Programming, to appear, doi:10.1007/s10107-007-0106-8] (which is a generalization of the approximation algorithm proposed by Grigoriadis and Khachiyan [Coordination complexity of parallel price-directive decomposition, Math. Oper. Res. 2 (1996) 321-340]). For the global routing problem, we investigate the particular property of lattice graphs and propose a combinatorial technique to overcome the hardness due to the bend-dependent vertex cost. Finally, we develop asymptotic approximation algorithms for both problems with ratios depending on the best known approximation ratio for the minimum Steiner tree problem. They are the first known theoretical approximation bound results for the problems of minimizing the total costs (including both the edge and the bend costs) while spanning all given subsets of vertices.