An optimal time algorithm for finding a maximum weight independent set in a tree

The maximum weight independent set problem for a general graph is NP-hard. But for some special classes of graphs, polynomial time algorithms do exist for solving it. Based on the divide-and-conquer strategy, Pawagi has presented anO(|V|log|V|) time algorithm for solving this problem on a tree. In this paper, we propose anO(|V|) time algorithm to improve Pawagi's result. The proposed algorithm is based on the dynamic programming strategy and is time optimal within a constant factor.     

Finding thek smallest spanning trees

We give improved solutions for the problem of generating thek smallest spanning trees in a graph and in the plane. Our algorithm for general graphs takes timeO(m log(m, n)=k ²); for planar graphs this bound can be improved toO(n+k ²). We also show that thek best spanning trees for a set of points in the plane can be computed in timeO(min(k ² n+n logn,k ²+kn log(n/k))). Thek best orthogonal spanning trees in the plane can be found in timeO(n logn+kn log log(n/k)+k ²).     

Embeddings on a boolean cube

In this paper, we characterize a class of graphs which can be embedded on a boolean cube. Some of the graphs in this class are identified with the well known graphs such asmulti-dimensional mesh of trees, tree of meshes, etc. We suggest (i) an embedding of anr-dimensional mesh of trees ofn ^r (r+1)–rn ^r–1 nodes on a boolean cube of (2n) ^r nodes, and (ii) an embedding of a tree of meshes with 2n ² logn+n ² nodes on a boolean cube withn ² exp₂ (log (2 logn+1)]) nodes.     

Approximating maximum independent sets by excluding subgraphs

An approximation algorithm for the maximum independent set problem is given, improving the best performance guarantee known toO(n/(logn)²). We also obtain the same performance guarantee for graph coloring. The results can be combined into a surprisingly strongsimultaneous performance guarantee for the clique and coloring problems.The framework ofsubgraph-excluding algorithms is presented. We survey the known approximation algorithms for the independent set (clique), coloring, and vertex cover problems and show how almost all fit into that framework. We show that among subgraph-excluding algorithms, the ones presented achieve the optimal asymptotic performance guarantees.A preliminary version of this paper appeared in [9].Supported in part by National Science Foundation Grant CCR-8902522 and PYI Award CCR-9057488.Research done at Rutgers University. Supported in part by Center for Discrete Mathematics and Theoretical Computer Science (DIMACS) fellowship.     

AnO(n 2) algorithm for finding the compact sets of a graph

A special clustering problem is discussed in this paper, called the compact set problem. The goal of the problem is to find all compact sets in a complete, weighted, undirected graph withn vertices. A subsetC of vertices is called a compact set if 1<|C|<n and the maximum weight among all edges inC is smaller than the minimum weight among all edges connecting one vertex inC and the other vertex not inC. An algorithm with complexityO(n ²) is given for the problem improving the previous results.This research was partially supported by the National Science Council of the Republic of China under Grant NSC81-0408-E-216-502.     

Efficient reduction for path problems on circular-arc graphs

An efficient reduction process for path problems on circular-arc graphs is introduced. For the parity path problem, this reduction gives anO(n+m) algorithm for proper circular-arc graphs, and anO(n+m) algorithm for general circular-arc graphs. This reduction also gives anO(n+m) algorithm for the two path problem on circular-arc graphs.     

On some subclasses of well-covered graphs

A set of points in a graph is independent if no two points in the set are adjacent. A graph is well covered if every maximal independent set is a maximum independent set or, equivalently, if every independent set is contained in a maximum independent set. The well-covered graphs are classified by the W_n property: For a positive integer n, a graph G belongs to class W_n if ≥ n and any n disjoint independent sets are contained in n disjoint maximum independent sets. Constructions are presented that show how to build infinite families of W_n graphs containing arbitrarily large independent sets. A characterization of W_n graphs in terms of well-covered subgraphs is given, as well as bounds for the size of a maximum independent set and the minimum and maximum degrees of points in W_n graphs.     

The most vital edges with respect to the number of spanning trees in two-terminal series-parallel graphs

A setE ofk edges in a multigraphG=(V,E) is said to be ak most vital edge set (k-MVE set) if these edges being removed fromG, the resultant graphG=(V,E –E) has minimum number of spanning trees. The problem of finding ak-MVE set for two-terminal series-parallel graphs is considered in this paper. We present anO (|E|) time algorithm for the casek=1, and anO(|V| ^k +|E|) time algorithm for arbitraryk.     

Maximal independent sets in bipartite graphs

A maximal independent set of a graph G is an independent set that is not contained properly in any other independent set of G. In this paper, we determine the maximum number of maximal independent sets among all bipartite graphs of order n and the extremal graphs as well as the corresponding results for connected bipartite graphs. © 1993 John Wiley & Sons, Inc.     

Optimal parallel quicksort on EREW PRAM

In this paper, we present parallel quicksort algorithms running inO((n/p+logp) logn) expected time andO((n/p+logp+log logn) logn) deterministic time respectively, and both withO(n) space by usingp processors on EREW PRAM. Whenp=O(n/logn), the cost is optimal, in terms of the product of time and number of processors. These algorithms can be used to obtain parallel algorithms for constructing balanced binary search trees without using sorting algorithms. One of our quicksort algorithms leads to a parallel quickhull algorithm on EREW PRAM.The work of this author was partially supported by a fellowship from the College of Science, Old Dominion University, Norfolk, VA 23529, USA.     

A generalized hypergreedy algorithm for weighted perfect matching

We give a generalization of the hypergreedy algorithm for minimum weight perfect matching on a complete edge weighted graph whose weights satisfy the triangle inequality. With a modified version of this algorithm we obtain a logn-approximate perfect matching heuristic for points in the Euclidean plane, inO(n log² n) time.This research was supported in part by the DIMACS Grant No. NSF-STC88-09648.This research was supported in part by the NSF under Grant No. CCR 88-07518.     

Constraints on the number of maximal independent sets in graphs

A maximal independent set of a graph G is an independent set that is not contained properly in any other independent set of G. Let i(G) denote the number of maximal independent sets of G. Here, we prove two conjectures, suggested by P. Erdös, that the maximum number of maximal independent sets among all graphs of order n in a family Φ is o(3^n/3) if Φ is either a family of connected graphs such that the largest value of maximum degrees among all graphs of order n in Φ is o(n) or a family of graphs such that the approaches infinity as n → ∞.     

A divide-and-conquer algorithm for constructing relative neighborhood graph

AnO(n logn) divide-and-conquer algorithm for finding the relative neighborhood graph RNG(V) of a set V ofn points in Euclidean space is presented. If implemented in parallel, its time complexity isO(n) and it requiresO(logn) processors.     

An algorithm for outerplanar graphs with parameter

For n-vertex outerplanar graphs, it is proven that O(n^2.87) is an upper bound on the number of breakpoints of the function which gives the maximum weight of an independent set, where the vertex weights vary as linear functions of a parameter. An O(n^2.87) algorithm for finding the solution is proposed.     

Average-case results on heapsort

A new algorithm for rearranging a heap is presented and analysed in the average case. The average case upper bound for deleting the maximum element of a random heap is improved, and is shown to be less than [logn]+0.299+M(n) comparisons, *) whereM(n) is between 0 and 1. It is also shown that a heap can be constructed using 1.650n+O(logn) comparisons with this algorithm, the best result for any algorithm which does not use any extra space. The expected time to sortn elements is argued to be less thann logn+0.670n+O(logn), while simulation result points at an average case ofn log n+0.4n which will make it the fastest in-place sorting algorithm. The same technique is used to show that the average number of comparisons when deleting the maximum element of a heap using Williams' algorithm for rearrangement is 2([logn]–1.299+L(n)) whereL(n) also is between 0 and 1, and the average cost for Floyd-Williams Heapsort is at least 2nlogn–3.27n, counting only comparisons. An analysis of the number of interchanges when deleting the maximum element of a random heap, which is the same for both algorithms, is also presented.     

A sublogarithmic convex hull algorithm

We present a parallel algorithm for finding the convex hull of a sorted set of points in the plane. Our algorithm runs inO(logn/log logn) time usingO(n log logn/logn) processors in theCommon crcw pram computational model, which is shown to be time and cost optimal. The algorithm is based onn ^1/3 divide-and-conquer and uses a simple pointer-based data structure.Part of this work was done when the last three authors were at the Department of Computer and Information Science, Linköping University. The research of the second author was supported by the Academy of Finland.     

Applications of a semi-dynamic convex hull algorithm

We obtain new results for manipulating and searching semi-dynamic planar convex hulls (subject to deletions only), and apply them to derive improved bounds for two problems in geometry and scheduling. The new convex hull results are logarithmic time bounds for set splitting and for finding a tangent when the two convex hulls are not linearly separated. Using these results, we solve the following two problems optimally inO(n logn) time: (1) [matching] givenn red points andn blue points in the plane, find a matching of red and blue points (by line segments) in which no two edges cross, and (2) [scheduling] givenn jobs with due dates, linear penalties for late completion, and a single machine on which to process them, find a schedule of jobs that minimizes the maximum penalty.     

Fast parallel graph searching with applications

This paper presents fast parallel algorithms for the following graph theoretic problems: breadth-depth search of directed acyclic graphs; minimum-depth search of graphs; finding the minimum-weighted paths between all node-pairs of a weighted graph and the critical activities of an activity-on-edge network. The first algorithm hasO(logdlogn) time complexity withO(n ³) processors and the remaining algorithms achieveO(logd loglogn) time bound withO(n ²[n/loglogn]) processors, whered is the diameter of the graph or the directed acyclic graph (which also represents an activity-on-edge network) withn nodes. These algorithms work on an unbounded shared memory model of the single instruction stream, multiple data stream computer that allows both read and write conflicts.     

The analysis of a nested dissection algorithm

Summary Nested dissection is an algorithm invented by Alan George for preserving sparsity in Gaussian elimination on symmetric positive definite matrices. Nested dissection can be viewed as a recursive divide-and-conquer algorithm on an undirected graph; it usesseparators in the graph, which are small sets of vertices whose removal divides the graph approximately in half. George and Liu gave an implementation of nested dissection that used a heuristic to find separators. Lipton and Tarjan gave an algorithm to findn ^1/2-separators in planar graphs and two-dimensional finite element graphs, and Lipton, Rose, and Tarjan used these separators in a modified version of nested dissection, guaranteeing bounds ofO (n logn) on fill andO(n ^3/2) on operation count. We analyze the combination of the original George-Liu nested dissection algorithm and the Lipton-Tarjan planar separator algorithm. This combination is interesting because it is easier to implement than the Lipton-Rose-Tarjan version, especially in the framework of existïng sparse matrix software. Using some topological graph theory, we proveO(n logn) fill andO(n ^3/2) operation count bounds for planar graphs, twodimensional finite element graphs, graphs of bounded genus, and graphs of bounded degree withn ^1/2-separators. For planar and finite element graphs, the leading constant factor is smaller than that in the Lipton-Rose-Tarjan analysis. We also construct a class of graphs withn ^1/2-separators for which our algorithm does not achieve anO(n logn) bound on fill.The work of this author was supported in part by the Hertz Foundation under a graduate fellowship and by the National Science Foundation under Grant MCS 82-02948The work of this author was supported in part by the National Science Foundation under Grant MCS 78-26858 and by the Office of Naval Research under Contract N00014-76-C-0688     

On the complexity of the independent set problem in triangle graphs

We consider the complexity of the maximum (maximum weight) independent set problem within triangle graphs, i.e., graphs G satisfying the following triangle condition: for every maximal independent set I in G and every edge uv in G−I, there is a vertex w∈I such that {u,v,w} is a triangle in G. We also introduce a new graph parameter (the upper independent neighborhood number) and the corresponding upper independent neighborhood set problem. We show that for triangle graphs the new parameter is equal to the independence number. We prove that the problems under consideration are NP-complete, even for some restricted subclasses of triangle graphs, and provide several polynomially solvable cases for these problems within triangle graphs. Furthermore, we show that, for general triangle graphs, the maximum independent set problem and the upper independent neighborhood set problem cannot be polynomially approximated within any fixed constant factor greater than one unless P=NP.