Effect of rounding errors on the variable metric method

It has become customary to compare the performance of unconstrained optimization algorithms on families of extended symmetric test functions. In this paper, results are presented which indicate that the performance of the variable metric algorithm on such functions is greatly distorted by rounding errors that destroy the special nature of these functions. A simple method of overcoming this difficulty is demonstrated, and it confirms the theoretical result that the number of iterations required to solve such problems is independent of the dimension.This research was supported by the Science and Engineering Research Council.     

带平行约束的多处理机调度问题的模拟平衡方法

对带平行约束的多处理机调度问题给出一种随机状态转移方式，并通过大量的计算结果表明，这种状态转移方式具有根强的倾向性和遍历性，进而对通常的问题仅需经过很少的状态转移次数就可以获得精度很高的近似解。     

Extended formulations for matroid polytopes through randomized protocols

The hitting number of a polytope P is the smallest size of a subset of vertices of P such that every facet of P has a vertex in the subset. We show that, if P is the base polytope of any matroid, then P admits an extended formulation of linear size on the hitting number of P. Our results generalize those of the spanning tree polytope given by Martin and Wong, and extend to polymatroids.     

Markov Incremental Constructions

A classic result asserts that many geometric structures can be constructed optimally by successively inserting their constituent parts in random order. These randomized incremental constructions (RICs) still work with imperfect randomness: the dynamic operations need only be “locally” random. Much attention has been given recently to inputs generated by Markov sources. These are particularly interesting to study in the framework of RICs, because Markov chains provide highly nonlocal randomness, which incapacitates virtually all known RIC technology. We generalize Mulmuley’s theory of Θ-series and prove that Markov incremental constructions with bounded spectral gap are optimal within polylog factors for trapezoidal maps, segment intersections, and convex hulls in any fixed dimension. The main contribution of this work is threefold: (i) extending the theory of abstract configuration spaces to the Markov setting; (ii) proving Clarkson–Shor-type bounds for this new model; (iii) applying the results to classical geometric problems. We hope that this work will pioneer a new approach to randomized analysis in computational geometry. This work was supported in part by NSF grants CCR-0306283, CCF-0634958.     

On the strength of Gomory mixed-integer cuts as group cuts

Gomory mixed-integer (GMI) cuts generated from optimal simplex tableaus are known to be useful in solving mixed-integer programs. Further, it is well-known that GMI cuts can be derived from facets of Gomory’s master cyclic group polyhedron and its mixed-integer extension studied by Gomory and Johnson. In this paper we examine why cutting planes derived from other facets of master cyclic group polyhedra (group cuts) do not seem to be as useful when used in conjunction with GMI cuts. For many practical problem instances, we numerically show that once GMI cuts from different rows of the optimal simplex tableau are added to the formulation, all other group cuts from the same tableau rows are satisfied.     

一次同余组的关键因子求解算法

本文在文 [1 ]的基础上定义了关键因子的概念 ,提出了一次同余组的关键因子求解算法 ,并给出了解的舍入误差估计 .通过实例说明了该算法的求解过程 ,分析了该算法在计算机求解时相对于传统解法的优点 .     

On minimum k-modal partitions of permutations

Partitioning a permutation into a minimum number of monotone subsequences is -hard. We extend this complexity result to minimum partitioning into k-modal subsequences; here unimodal is the special case k=1. Based on a network flow interpretation we formulate both, the monotone and the k-modal version, as mixed integer programs. This is the first proposal to obtain provably optimal partitions of permutations. LP rounding gives a 2-approximation for minimum monotone partitions and a (k+1)-approximation for minimum (upper) k-modal partitions. For the online problem, in which the permutation becomes known to an algorithm sequentially, we derive a logarithmic lower bound on the competitive ratio for minimum monotone partitions, and we analyze two (bin packing) online algorithms. These immediately apply to online cocoloring of permutation graphs.     

Fault Tolerant Sorting—Theoretical and Empirical Analyses of the Randomized QuickMergesort Algorithm

This paper introduces a new paradigm in the design of sorting algorithms, viz., fault tolerance. Fault tolerance is an important concept in modern day computing and design workflows must accommodate this need. In general, there are a number of avenues for faults to occur and techniques to address the same; this paper focusses on only one source of faulty behavior, viz., process termination. Process termination, as a cause of faulty behavior, is important from the perspective of various applications in real-time scheduling. In order to measure the effectiveness of a fault tolerant protocol, it is necessary to define a suitable metric and analyze the performance of the protocol with respect to that metric. We measure the “unsortedness” of an array, as characterized by the number of inversion pairs that remain when the sorting algorithm (process) terminates. This paper proposes a new algorithm for sorting called the Randomized QuickMergesort (RQMS) algorithm. RQMS has a higher degree of fault tolerance than either Randomized Quicksort (RQS) or Mergesort (MS), in that fewer inversion pairs remain when it terminates. Likewise, RQMS has a lower comparison overhead than RQS and is more space-efficient than MS. Our empirical analysis, which was conducted over a wide variety of distributions, conclusively establishes that RQMS is the algorithm of choice, when fault tolerance is paramount in the application. This research was supported in part by the Air Force Office of Scientific Research under Contract FA9550-06-1-0050.     

An iterative rounding 2-approximation algorithm for the k-partial vertex cover problem

We study a generalization of the vertex cover problem. For a given graph with weights on the vertices and an integer k, we aim to find a subset of the vertices with minimum total weight, so that at least k edges in the graph are covered. The problem is called the k-partial vertex cover problem. There are some 2-approximation algorithms for the problem. In the paper we do not improve on the approximation ratios of the previous algorithms, but we derive an iterative rounding algorithm. We present our technique in two algorithms. The first is an iterative rounding algorithm and gives a （2 ＋ Q/OPT ）-approximation for the k-partial vertex cover problem where Q is the largest finite weight in the problem definition and OPT is the optimal value for the instance. The second algorithm uses the first as a subroutine and achieves an approximation ratio of 2.