The structure of obstructions to treewidth and pathwidth

It is known that the class of graphs with treewidth (resp. pathwidth) bounded by a constant w can be characterized by a finite obstruction set obs(TW(w)) (resp. obs(PW(w))). These obstruction sets are known for w3 so far. In this paper we give a structural characterization of graphs from obs(TW(w)) (resp. obs(PW(w))) with a fixed number of vertices in terms of subgraphs of the complement. Our approach also essentially simplifies known characterization of graphs from obs(TW(w)) (resp. obs(PW(w))) with (w+3) vertices.

Also for any w3 a graph from obs(TW(w))obs(PW(w)) is constructed, that solves an open problem.     

Algorithms for finding a Kth best valued assignment

We consider a new problem, the Kth best valued assignment problem. Given a bipartite graph G and a cost vector w on its edge set, this is the problem of finding a perfect matching M_k in G such that there exist perfect matchings M₁,…,M_K−1 satisfying w(M₁) < < w(M_K−1) < w(M_K), and w(M_K) < w(M) for all perfect matchings M with w(M) ≠ w(M₁),…,w(M_K). Here w(M) denotes the sum of costs of edges in M. In this paper, we propose two algorithms for solving this problem and verify the efficiency of our algorithms by our preliminary computational experiments.     

Power optimization for connectivity problems

Given a graph with costs on the edges, the power of a node is the maximum cost of an edge leaving it, and the power of the graph is the sum of the powers of the nodes of this graph. Motivated by applications in wireless multi-hop networks, we consider four fundamental problems under the power minimization criteria: the Min-Power b-Edge-Cover problem (MPb-EC) where the goal is to find a min-power subgraph so that the degree of every node v is at least some given integer b(v), the Min-Power k-node Connected Spanning Subgraph problem (MPk-CSS), Min-Power k-edge Connected Spanning Subgraph problem (MPk-CSS), and finally the Min-Power k-Edge-Disjoint Paths problem in directed graphs (MPk-EDP). We give an O(log⁴ n)-approximation algorithm for MPb-EC. This gives an O(log⁴ n)-approximation algorithm for MPk-CSS for most values of k, improving the best previously known O(k)-approximation guarantee. In contrast, we obtain an approximation algorithm for ECSS, and for its variant in directed graphs (i.e., MPk-EDP), we establish the following inapproximability threshold: MPk-EDP cannot be approximated within O(2^log1-ε n) for any fixed ε > 0, unless NP-hard problems can be solved in quasi-polynomial time. This paper was done when V. S. Mirrokni was at Computer Science and Artificial Intelligence Laboratory, MIT.     

Bounds on a class of partial partitions of a vector space over gf(2):a graph theoretical approach

A collection Q of linearly independent w-suhicfs of the n-dimensional vector space V(n) over GF(2) is a w-quilt if whenever X and Y are distinct elements of Q, then X is disjoint from the linear span of Y. The main problem is to determine the maximum possibility cardinality of a w-quilt in V(n) for fixed w and n. Here a graph T(Q) is associated with each quilt Q. The connected components of T(Q) are shown to be complete graphs and the structure of the subquilts corresponding to these components is completely determined. By use of Ramsey type arguments these results are shown to lead to new upper bounds on the cardinality of a w-quilt in V(n) where n = w + 2, a case of particular interest.     

图的区间边着色的收缩图方法

图G的一个用了颜色1,2,…,t的边着色称为区间t-着色,如果所有t种颜色都被用到,并且关联于G的同一个顶点的边上的颜色是各不相同的,且这些颜色构成了一个连续的整数区间.G称作是可区间着色的,如果对某个正整数t,G有一个区间t-着色.所有可区间着色的图构成的集合记作■.对图G∈■,使得G有一个区间t-着色的t的最小值和最大值分别记作ω(G)和W(G).现给出了图的区间着色的收缩图方法.利用此方法,我们对双圈图G∈■,证明了ω(G)=△(G)或△(G)+1,并且完全确定了ω(G)=△(G)及ω(G)=△(G)+1的双圈图类.     

Extremal graphs for weights

Given a graph G = (V,E) and R, we write w(G)=∑_xyεEdG(x)dG(y), and study the function w(m) = max {w(G): e(G) = m}. Answering a question from Bollobás and Erdös (Graphs of external weights, to appear), we determine w_i(m) for every m, and we also give bounds for the case ≠ 1.     

On the multisearching problem for hypercubes

In this paper we give improved bounds for the multisearch problem on a hypercube. This is a parallel search problem where the elements in the structure S to be searched are totally ordered, but where it is not possible to compare in constant time any two given queries q and q′. More precisely, we are given on a n-processor hypercube a sorted n-element sequence S, and a set Q of n queries, and we need to find for each query q Q its location in the sorted S. We present an improved algorithm for the multisearch problem, one that takes O(log n(log log n)³) time on a n-processor hypercube. This problem is fundamental in computational geometry, for example it models planar point location in a slab. We give as application a trapezoidal decomposition algorithm with the same time complexity on a n log n-processor hypercube. The hypercube model for which we claim our bounds is the standard one, SIMD, with O(1) memory registers per processor, and with one-port communication. Each register can store O(log n) bits, so that a processor knows its ID.     

Decomposably non-negative matrices

Let A be an mn- by - mn symmetric matrix. Partition A into m²n - by - n blocks and suppose that each of these blocks is also symmetric. Suppose that for every decomposable (rank one) tensor ν ⊗ w, we have (ν ⊗ w)^t A(ν otimes; w) ≥ 0. Here, ν is a column m-tuple and w is a column n-tuple. We study the maximum number of negative eigenvalues such a matrix can have, as well as obtaining alternative characterizations of such matrices.     

Weighted domination in triangle-free graphs

A weighted graph (G,w) is a graph G together with a positive weight-function on its vertex set w : V(G)→R^>0. The weighted domination number γ_w(G) of (G,w) is the minimum weight w(D)=∑_vDw(v) of a set DV(G) such that every vertex xV(D)−D has a neighbor in D. If ∑_vV(G)w(v)=|V(G)|, then we speak of a normed weighted graph. Recently, we proved that

for normed weighted bipartite graphs (G,w) of order n such that neither G nor the complement has isolated vertices. In this paper we will extend these Nordhaus–Gaddum-type results to triangle-free graphs.     

Approximate range searching

The range searching problem is a fundamental problem in computational geometry, with numerous important applications. Most research has focused on solving this problem exactly, but lower bounds show that if linear space is assumed, the problem cannot be solved in polylogarithmic time, except for the case of orthogonal ranges. In this paper we show that if one is willing to allow approximate ranges, then it is possible to do much better. In particular, given a bounded range Q of diameter w and >0, an approximate range query treats the range as a fuzzy object, meaning that points lying within distance w of the boundary of Q either may or may not be counted. We show that in any fixed dimension d, a set of n points in can be preprocessed in O(n+logn) time and O(n) space, such that approximate queries can be answered in O(logn(1/)^d) time. The only assumption we make about ranges is that the intersection of a range and a d-dimensional cube can be answered in constant time (depending on dimension). For convex ranges, we tighten this to O(logn+(1/)^d−1) time. We also present a lower bound for approximate range searching based on partition trees of Ω(logn+(1/)^d−1), which implies optimality for convex ranges (assuming fixed dimensions). Finally, we give empirical evidence showing that allowing small relative errors can significantly improve query execution times.     

Bounded-hops power assignment in ad hoc wireless networks

Motivated by topology control in ad hoc wireless networks, Power Assignment is a family of problems, each defined by a certain connectivity constraint (such as strong connectivity). The input consists of a directed complete weighted digraph G=(V,c) (that is, ). The power of a vertex u in a directed spanning subgraph H is given by , and corresponds to the energy consumption required for node u to transmit to all nodes v with uv∈E(H). The power of H is given by . Power Assignment seeks to minimize p(H) while H satisfies the given connectivity constraint.Min-Power Bounded-Hops Broadcast is a power assignment problem which has as additional input a positive integer d and a r∈V. The output H must be a r-rooted outgoing arborescence of depth at most d. We give an (O(logn),O(logn)) bicriteria approximation algorithm for Min-Power Bounded-Hops Broadcast: that is, our output has depth at most O(dlogn) and power at most O(logn) times the optimum solution.For the Euclidean case, when c(u,v)=c(v,u)=∥u,v∥^κ (here ∥u,v∥ is the Euclidean distance and κ is a constant between 2 and 5), the output of our algorithm can be modified to give a O((logn)^κ) approximation ratio. Previous results for Min-Power Bounded-Hops Broadcast are only exact algorithms based on dynamic programming for the case when the nodes lie on the line and c(u,v)=c(v,u)=∥u,v∥^κ.Our bicriteria results extend to Min-Power Bounded-Hops Strong Connectivity, where H must have a path of at most d edges in between any two nodes. Previous work for Min-Power Bounded-Hops Strong Connectivity consists only of constant or better approximation for special cases of the Euclidean case.     

Weight distribution of the bases of a binary matroid

Let M be a weighted binary matroid and w₁ < … < w_m be the increasing sequence of all possible distinct weights of bases of M. We give a sufficient condition for the property that w₁, …, w_m is an arithmetical progression of common difference d. We also give conditions which guarantee that w_i+1 − w_i ≤ d, 1 ≤ i ≤ m −1. Dual forms for these results are given also.     

Valuations defined by Ostrowski nets

Ostrowski proved that if v is a valuation on an algebraically closed field F and if w is a valuation extending v to the field F (x) of rational functions over F, then w is defined by a transcendental pseudo-convergent net (that is, an Ostrowski net) or w is a Rella extension of v centered about x − c for some c in F. In this paper, valuations on the field of rational functions over an arbitrary field determined by Ostrowski nets, along with the topologies they generate, are investigated.     

Global optimization and simulated annealing

In this paper we are concerned with global optimization, which can be defined as the problem of finding points on a bounded subset of ⁿ in which some real valued functionf assumes its optimal (maximal or minimal) value.We present a stochastic approach which is based on the simulated annealing algorithm. The approach closely follows the formulation of the simulated annealing algorithm as originally given for discrete optimization problems. The mathematical formulation is extended to continuous optimization problems, and we prove asymptotic convergence to the set of global optima. Furthermore, we discuss an implementation of the algorithm and compare its performance with other well-known algorithms. The performance evaluation is carried out for a standard set of test functions from the literature.     

Homogeneous renewal equation

Let F be a non-arithmetic distribution on the line , and W be the class of bounded functions w without discontinuity of the second kind such that

.In this paper, we show that the solution of the homogeneous renewal equation w = w F in the class W is a constant-function.     

Bit-parallel approximate string matching algorithms with transposition

Using bit-parallelism has resulted in fast and practical algorithms for approximate string matching under Levenshtein edit distance, which permits a single edit operation to insert, delete or substitute a character. Depending on the parameters of the search, currently the fastest non-filtering algorithms in practice are the O(km/wn) algorithm of Wu and Manber, the O((k+2)(m−k)/wn) algorithm of Baeza-Yates and Navarro, and the O(m/wn) algorithm of Myers, where m is the pattern length, n is the text length, k is the error threshold and w is the computer word size. In this paper we discuss a uniform way of modifying each of these algorithms to permit also a fourth type of edit operation: transposing two adjacent characters in the pattern. This type of edit distance is also known as Damerau edit distance. In the end we also present an experimental comparison of the resulting algorithms.     

Quantum annealing of the graph coloring problem

Quantum annealing extends simulated annealing by introducing artificial quantum fluctuations. The path-integral Monte Carlo version chosen is population-based and designed to be implemented on a classical computer. Its first application to the graph coloring problem is presented in this paper. It is shown by experiments that quantum annealing can outperform classical thermal simulated annealing for this particular problem. Moreover, quantum annealing proved competitive when compared with the best algorithms on most of the difficult instances from the DIMACS benchmarks. The quantum annealing algorithm has even found that the well-known benchmark graph dsjc1000.9 has a chromatic number of at most 222. This is an improvement on its best upper-bound from a large body of literature.     

Molecular conformation on the CM-5 by parallel two-level simulated annealing

In this paper, we propose a new kind of simulated annealing algorithm calledtwo-level simulated annealing for solving certain class of hard combinatorial optimization problems. This two-level simulated annealing algorithm is less likely to get stuck at a non-global minimizer than conventional simulated annealing algorithms. We also propose a parallel version of our two-level simulated annealing algorithm and discuss its efficiency. This new technique is then applied to the Molecular Conformation problem in 3 dimensional Euclidean space. Extensive computational results on Thinking Machines CM-5 are presented. With the full Lennard-Jones potential function, we were able to get satisfactory results for problems for cluster sizes as large as 100,000. A peak rate of over 0.8 giga flop per second in 64-bit operations was sustained on a partition with 512 processing elements. To the best of our knowledge, ground states of Lennard-Jones clusters of size as large as these have never been reported before.Also a researcher at the Army High Performance Computing Research Center, University of Minnesota, Minneapolis, MN 55415     

A simulated annealing technique for multi-objective simulation optimization

In this paper, we present a simulated annealing algorithm for solving multi-objective simulation optimization problems. The algorithm is based on the idea of simulated annealing with constant temperature, and uses a rule for accepting a candidate solution that depends on the individual estimated objective function values. The algorithm is shown to converge almost surely to an optimal solution. It is applied to a multi-objective inventory problem; the numerical results show that the algorithm converges rapidly.     

Simulated N-Body: New Particle Physics-Based Heuristics for a Euclidean Location-Allocation Problem

The general facility location problem and its variants, including most location-allocation and P-median problems, are known to be NP-hard combinatorial optimization problems. Consequently, there is now a substantial body of literature on heuristic algorithms for a variety of location problems, among which can be found several versions of the well-known simulated annealing algorithm. This paper presents an optimization paradigm that, like simulated annealing, is based on a particle physics analogy but is markedly different from simulated annealing. Two heuristics based on this paradigm are presented and compared to simulated annealing for a capacitated facility location problem on Euclidean graphs. Experimental results based on randomly generated graphs suggest that one of the heuristics outperforms simulated annealing both in cost minimization as well as execution time. The particular version of location problem considered here, a location-allocation problem, involves determining locations and associated regions for a fixed number of facilities when the region sizes are given. Intended applications of this work include location problems with congestion costs as well as graph and network partitioning problems.