Approximate maximum weight branchings

We consider a special subgraph of a weighted directed graph: one comprising only the k heaviest edges incoming to each vertex. We show that the maximum weight branching in this subgraph closely approximates the maximum weight branching in the original graph. Specifically, it is within a factor of k/(k+1). Our interest in finding branchings in this subgraph is motivated by a data compression application in which calculating edge weights is expensive but estimating which are the heaviest k incoming edges is easy. An additional benefit is that since algorithms for finding branchings run in time linear in the number of edges our results imply faster algorithms although we sacrifice optimality by a small factor. We also extend our results to the case of edge-disjoint branchings of maximum weight and to maximum weight spanning forests.     

Fast leader election in anonymous rings with bounded expected delay

We propose a probabilistic network model, called asynchronous bounded expected delay (ABE), which requires a known bound on the expected message delay. In ABE networks all asynchronous executions are possible, but executions with extremely long delays are less probable. Thus, the ABE model captures asynchrony that occurs in sensor networks and ad-hoc networks.At the example of an election algorithm, we show that the minimal assumptions of ABE networks are sufficient for the development of efficient algorithms. For anonymous, unidirectional ABE rings of known size n we devise a probabilistic election algorithm having average message and time complexity O(n).     

Spanners of bounded degree graphs

A k-spanner of a graph G is a spanning subgraph of G in which the distance between any pair of vertices is at most k times the distance in G. We prove that for fixed k,w, the problem of deciding if a given graph has a k-spanner of treewidth w is fixed-parameter tractable on graphs of bounded degree. In particular, this implies that finding a k-spanner that is a tree (a tree k-spanner) is fixed-parameter tractable on graphs of bounded degree. In contrast, we observe that if the graph has only one vertex of unbounded degree, then Treek-Spanner is NP-complete for k?4.     

Randomized approximation of bounded multicovering problems

The problem of finding approximate solutions for a subclass of multicovering problems denoted byILP(k, b) is considered. The problem involves findingx∈{0,1} ⁿ that minimizes ∑ _j x _j subject to the constraintAx≥b, whereA is a 0–1m×n matrix with at mostk ones per row,b is an integer vector, andb is the smallest entry inb. This subclass includes, for example, the Bounded Set Cover problem whenb=1, and the Vertex Cover problem whenk=2 andb=1. An approximation ratio ofk−b+1 is achievable by known deterministic algorithms. A new randomized approximation algorithm is presented, with an approximation ratio of (k−b+1)(1−(c/m)^1/(k−b+1)) for a small constantc>0. The analysis of this algorithm relies on the use of a new bound on the sum of independent Bernoulli random variables, that is of interest in its own right. The first author was supported in part by a Walter and Elise Haas Career Development Award and by a grant from the Israeli Science Foundation. This work was done white the third author was at the Department of Applied Mathematics and Computer Science, The Weizmann Institute, Rehovot 76100, Israel.     

The complexity of changing colourings with bounded maximum degree

Let G be a graph, x,y∈V(G), and ?:V(G)→[k] a k-colouring of G such that ?(x)=?(y). If then the following question is NP-complete: Does there exist a k-colouring ?^′ of G such that ?^′(x)≠?^′(y)? Conversely, if then the problem is polynomial time.     

Fast consensus in networks of bounded degree

Summary TheDistributed Consensus problem involvesn processors each of which holds an initial binary vlaue. At mostt of the processors may be faulty and ignore any protocol (even behaving maliciously), yet it is required that the non-faulty processors eventually agree on a value that was initially held by one of them. In this paper we focus on consensus in networks whose degree is bounded, following the work of Dwork, Peleg, Pippenger and Upfal [8]. In such a context, complete consensus among all the correct processors is not possible and some exceptions must be allowed. We first show how to achieve consensus in the butterfly network usingO(t+lognloglogn) one-bit parallel transmission steps, while tolerating the asymptotically optimal number of faulty processors (O(n/logn)) and having the asymptotically minimal number of exceptions (O(tlogt)). This result considerably improves on the running time of existing butterfly consensus protocols [2, 8]. In particular, it replaces the running time ofO(nlognloglogn) of [2] with an asymptotically optimal one. As in [8], we can then decrease the number of exceptions toO(t) by using additional links, while maintaining the same running time. The protocol is derived from a consensus protocol for completely connected networks that is interesting in its own right: it achieves Distributed Consensus with optimal number of processors, asymptotically optimal total bit transfer and nearly optimal number of rounds. Piotr Berman was born in 1955 in Warsaw, Poland, where here progressed from day-care to the degree of Master of Mathematics obtained from the University of Warsaw in 1978. He later studied at the Polish Academy of Sciences and MIT. He received a Ph.D. in Mathematics from MIT in 1985. From 1982 he has been teaching at Penn State, where currently he has a permanent position. His research interests are in two areas: fault-tolerant distributed computing and approximation algorithms. His non-professional hobbies include ancient history and mountain hiking. Juan Alberto Garay is originally from Rosario, Argentina, where he received his degree of Electrical Engineering from the Universidad Nacional de Rosario in 1976, at the age of 21. He then received his Master's degree in Electronic Engineering from the Eindhoven International Institute of the Eindhoven University of Technology (Eindhoven, Holland) in 1981, and his Ph.D. in Computer Science at Penn State University (University Park, PA) in 1989. Between his first and second degrees he worked as a digital design engineer for SOMISA (San Nicola's, Argentina), and between his second and Ph.D. degrees as a systems engineer for IBM Argentina. During the 1989/1990 academic year he was a visiting Assistant Professor at Bucknell University (Lewisburg, PA), and since 1990 he is with IBM's T.J. Watson Research Center (Yorktown Heights, NY). In 1992 he spent 6 months at The Weizmann Institute of Science (Rehovot, Israel) as a postdoctoral fellow. His professional interests include algorithms and lower bounds, distributed computation and fault tolerance. Dr. Garay enjoys tennis, music and philosophy.Preliminary version appeared in Proc 4th International Workshop on Distributed Algorithms, LNCS 486 (Springer-Verlag), pp 321–333, 1990     

Approximating maximum satisfiable subsystems of linear equations of bounded width

We consider the problem known as MAX-SATISFY: given a system of m linear equations over the rationals, find a maximum set of equations that can be satisfied. Let r be the width of the system, that is, the maximum number of variables in an equation. We give an Ω(m−1+1/r)-approximation algorithm for any fixed r. Previously the best approximation ratio for this problem was Ω((logm)/m) even for r=2. In addition, we slightly improve the hardness results for MAX-SATISFY.     

Hybrid backtracking bounded by tree-decomposition of constraint networks

We propose a framework for solving CSPs based both on backtracking techniques and on the notion of tree-decomposition of the constraint networks. This mixed approach permits us to define a new framework for the enumeration, which we expect that it will benefit from the advantages of two approaches: a practical efficiency of enumerative algorithms and a warranty of a limited time complexity by an approximation of the tree-width of the constraint networks. Finally, experimental results allow us to show the advantages of this approach.     

Dependency trees in sub-linear time and bounded memory

We focus on the problem of efficient learning of dependency trees. Once grown, they can be used as a special case of a Bayesian network, for PDF approximation, and for many other uses. Given the data, a well-known algorithm can fit an optimal tree in time that is quadratic in the number of attributes and linear in the number of records. We show how to modify it to exploit partial knowledge about edge weights. Experimental results show running time that is near-constant in the number of records, without significant loss in accuracy of the generated trees. Work done at Carnegie-Mellon university. This research was sponsored by the National Science Foundation (NSF) under grant no. ACI-0121671 and no. DMS-9873442.     

An adjustable linear time parallel algorithm for maximum weight bipartite matching

We present a parallel algorithm for finding a maximum weight matching in general bipartite graphs with an adjustable time complexity of using O(nmax(2ω,4+ω)) processing elements for ω?1. Parameter ω is not bounded. This is the fastest known strongly polynomial parallel algorithm to solve this problem. This is also the first adjustable parallel algorithm for the maximum weight bipartite matching problem in which the execution time can be reduced by an unbounded factor. We also present a general approach for finding efficient parallel algorithms for the maximum matching problem.     

A new analysis of a self-stabilizing maximum weight matching algorithm with approximation ratio 2

The maximum weight matching problem is a fundamental problem in graph theory with a variety of important applications. Recently Manne and Mjelde presented the first self-stabilizing algorithm computing a 2-approximation of the optimal solution. They established that their algorithm stabilizes after O(2ⁿ) (resp. O(3ⁿ)) moves under a central (resp. distributed) scheduler. This paper contributes a new analysis, improving these bounds considerably. In particular it is shown that the algorithm stabilizes after O(nm) moves under the central scheduler and that a modified version of the algorithm also stabilizes after O(nm) moves under the distributed scheduler. The paper presents a new proof technique based on graph reduction for analyzing the complexity of self-stabilizing algorithms.     

Consensus seeking in multiagent cooperative control systems with bounded control input

Consensus algorithms in multiagent cooperative control systems with bounded control input are studied in this paper.Consensus algorithms are considered for the single-integrator dynamics and double-integrator dynamics under different communication interaction topologies,and show that consensus is reached asymptotically using the algorithm proposed in this paper for the single-integrator dynamics if the undirected interaction graph is connected,and consensus is reached asymptotically if the directed interaction graph is strongly connected,respectively.In addition,the paper further shows that consensus is reached asymptotically using the algorithm proposed for the double-integrator dynamics if the directed interaction graph is strongly connected.The effectiveness of these algorithms is demonstrated through simulations.     

Game chromatic number of graphs with locally bounded number of cycles

We consider the coloring game and the marking game on graphs with bounded number of cycles passing through any edge. We prove that the game coloring number of a graph G is at most c+4, if every edge of G belongs to at most c different cycles. This result covers two earlier bounds on the game coloring number: for trees (c=0) and for cactuses (c=1).     

Approximate minimum weight matching on points ink-dimensional space

We study the problem of finding a minimum weight complete matching in the complete graph on a set V ofn points ink-dimensional space. The points are the vertices of the graph and the weight of an edge between any two points is the distance between the points under someL _q,-metric. We give anO((2c _q )^{1.5k –1.5k} ((n, n))^0.5 n ^1.5(logn)^2.5) algorithm for finding an almost minimum weight complete matching in such a graph, wherec _q =6k ^1/q for theL _q -metric, is the inverse Ackermann function, and 1. The weight of the complete matching obtained by our algorithm is guaranteed to be at most (1 + ) times the weight of a minimum weight complete matching.This research was supported by a fellowship from the Shell Foundation.     

Improved approximation of the largest common subtree of two unordered trees of bounded height

We present a polynomial time 1.5h-approximation algorithm for the problem of finding the largest common subtree between two rooted, labeled, and unordered trees of height at most h, where a tree S is called a subtree of a tree T if S is obtained from T by deletion of some nodes in T. This result improves the previous 2h-approximation algorithm.     

Maximum weight k-independent set problem on permutation graphs

Based on a Directed Acyclic Graph approach, an O(kn ²) time sequential algorithm is presented to solve the maximum weight k-independent set problem on weighted-permutation graphs. The weights considered here are all non-negative and associated with each of the n vertices of the graph. This problem has many applications in practical problems like k-machines job scheduling problem, k-colourable subgraph problem, VLSI design and routing problem.     

Acyclic edge colourings of graphs with the number of edges linearly bounded by the number of vertices

Let G be any finite graph. A mapping c:E(G)→{1,…,k} is called an acyclic edge k-colouring of G, if any two adjacent edges have different colours and there are no bichromatic cycles in G. In other words, for every pair of distinct colours i and j, the subgraph induced in G by all the edges that have colour i or j is acyclic. The smallest number k of colours such that G has an acyclic edge k-colouring is called the acyclic chromatic index of G and is denoted by .Determining the acyclic chromatic index of a graph is a hard problem, both from theoretical and algorithmical point of view. In 1991, Alon et al. proved that for any graph G of maximum degree Δ(G). This bound was later improved to 16Δ(G) by Molloy and Reed. In general, the problem of computing the acyclic chromatic index of a graph is NP-complete. Only a few algorithms for finding acyclic edge colourings have been known by now. Moreover, these algorithms work only for graphs from particular classes.In our paper, we prove that for every graph G which satisfies the condition that |E(G^′)|?t|V(G^′)|−1 for each subgraph G^′⊆G, where t?2 is a given integer, the constant p=2t³−3t+2. Based on that result, we obtain a polynomial algorithm which computes such a colouring. The class of graphs covered by our theorem is quite rich, for example, it contains all t-degenerate graphs.