A trust region method for solving the decentralized static output feedback design problem

The shortest-paths problem is a fundamental problem in graph theory and finds diverse applications in various fields. This is why shortest path algorithms have been designed more thoroughly than any other algorithm in graph theory. A large number of optimization problems are mathematically equivalent to the problem of finding shortest paths in a graph. The shortest-path between a pair of vertices is defined as the path with shortest length between the pair of vertices. The shortest path from one vertex to another often gives the best way to route a message between the vertices. This paper presents anO(n ²) time sequential algorithm and anO(n ²/p+logn) time parallel algorithm on EREW PRAM model for solving all pairs shortest paths problem on circular-arc graphs, wherep andn represent respectively the number of processors and the number of vertices of the circular-arc graph.     

Graph algorithms on a tree-structured parallel computer

Parallel algorithms for some graph-theoretic problems on a tree-structured computer are presented. In particular, ifp denotes the number of processing elements, algorithms that run inO(n ²/p) time for finding connected components, transitive closure and the minimum spanning tree of an undirected graph withn vertices are obtained.     

The greedy coloring is a bad probabilistic algorithm

Given a graph G and an ordering p of its vertices, denote by A(G, p) the number of colors used by the greedy coloring algorithm when applied to G with vertices ordered by p. Let , , Δ be positive constants. It is proved that for each n there is a graph G_n such that the chromatic number of G_n is at most n, but the probability that A(G_n, p) < (1 − )n/log₂ n for a randomly chosen ordering p is O(n^−Δ).     

Average‐case complexity of shortest‐paths problems in the vertex‐potential model

We study the average‐case complexity of shortest‐paths problems in the vertex‐potential model. The vertex‐potential model is a family of probability distributions on complete directed graphs with arbitrary real edge lengths, but without negative cycles. We show that on a graph with n vertices and with respect to this model, the single‐source shortest‐paths problem can be solved in O(n²) expected time, and the all‐pairs shortest‐paths problem can be solved in O(n² log n) expected time. ©2000 John Wiley & Sons, Inc. Random Struct. Alg., 16, 33–46, 2000     

On brittle graphs

Chvátal defined a graph G to be brittle if each induced subgraph F of G contains a vertex that is not a midpoint of any P₄ or not an endpoint of any P₄. Every brittle graph is perfectly orderable. In this paper, we prove that a graph is brittle whenever it is HHD-free (containing no chordless cycle with at least five vertices, no cycle on six vertices with a long chord, and no complement of the chordless path on five vertices). We also design an O(n₄) algorithm to recognize HHD-free graphs, and also an O(n₄) algorithm to construct a perfect order of an HHD-free graph. It follows from this result that an optimal coloring and a largest clique of an HHD-free graph can be found in O(n₄) time.     

Algorithms for graphs of bounded treewidth via orthogonal range searching

We show that, for any fixed constant k3, the sum of the distances between all pairs of vertices of an abstract graph with n vertices and treewidth at most k can be computed in O(nlog^k−1n) time.We also show that, for any fixed constant k2, the dilation of a geometric graph (i.e., a graph drawn in the plane with straight-line segments) with n vertices and treewidth at most k can be computed in O(nlog^k+1n) expected time. The dilation (or stretch-factor) of a geometric graph is defined as the largest ratio, taken over all pairs of vertices, between the distance measured along the graph and the Euclidean distance.The algorithms for both problems are based on the same principle: data structures for orthogonal range searching in bounded dimension provide a compact representation of distances in abstract graphs of bounded treewidth.     

An optimal algorithm for finding depth-first spanning tree on permutation graphs

LetG be a connected graph ofn vertices. The problem of finding a depth-first spanning tree ofG is to find a connected subgraph ofG with then vertices andn − 1 edges by depth-first-search. In this paper, we propose anO(n) time algorithm to solve this problem on permutation graphs.     

Preemptive Scheduling of Equal-Length Jobs in Polynomial Time

In this paper we consider the k-fixed-endpoint path cover problem on proper interval graphs, which is a generalization of the path cover problem. Given a graph G and a set T of k vertices, a k-fixed-endpoint path cover of G with respect to T is a set of vertex-disjoint simple paths that covers the vertices of G, such that the vertices of T are all endpoints of these paths. The goal is to compute a k-fixed-endpoint path cover of G with minimum cardinality. We propose an optimal algorithm for this problem with runtime O(n), where n is the number of intervals in G. This algorithm is based on the Stair Normal Interval Representation (SNIR) matrix that characterizes proper interval graphs. In this characterization, every maximal clique of the graph is represented by one matrix element; the proposed algorithm uses this structural property, in order to determine directly the paths in an optimal solution.     

Extremal subgraphs of random graphs

We shall prove that if L is a 3-chromatic (so called “forbidden”) graph, and —Rⁿ is a random graph on n vertices, whose edges are chosen independently, with probability p, and —Bⁿ is a bipartite subgraph of Rⁿ of maximum size, —Fⁿ is an L-free subgraph of Rⁿ of maximum size, then (in some sense) Fⁿ and Bⁿ are very near to each other: almost surely they have almost the same number of edges, and one can delete O_p(1) edges from Fⁿ to obtain a bipartite graph. Moreover, with p = 1/2 and L any odd cycle, Fⁿ is almost surely bipartite.     

Embedding Planar Graphs at Fixed Vertex Locations

 Let G be a planar graph of n vertices, v ₁,…,v _n , and let {p ₁,…,p _n } be a set of n points in the plane. We present an algorithm for constructing in O(n ²) time a planar embedding of G, where vertex v _i is represented by point p _i and each edge is represented by a polygonal curve with O(n) bends (internal vertices). This bound is asymptotically optimal in the worst case. In fact, if G is a planar graph containing at least m pairwise independent edges and the vertices of G are randomly assigned to points in convex position, then, almost surely, every planar embedding of G mapping vertices to their assigned points and edges to polygonal curves has at least m/20 edges represented by curves with at least m/40³ bends. Received: May 24, 1999 Final version received: April 10, 2000     

Incremental algorithms for minimal length paths

We consider the problem of maintaining on-line a solution to the All Pairs Shortest Paths Problem in a directed graph G = (V,E) where edges may be dynamically inserted or have their cost decreased. For the case of integer edge costs in a given range [1…C], we introduce a new data structure which is able to answer queries concerning the length of the shortest path between any two vertices in constant time and to trace out the shortest path between any two vertices in time linear in the number of edges reported. The total time required to maintain the data structure under a sequence of at most O(n²) edge insertions and at most O(Cn²) edge cost decreases is O(Cn³ log(nC)) in the worst case, where n is the total number of vertices in G. For the case of unit edge costs, the total time required to maintain the data structure under a sequence of at most O(n²) insertions of edges becomes O(n³ logn) in the worst case. The same bounds can be achieved for the problem of maintaining on-line longest paths in directed acyclic graphs. All our algorithms improve previously known algorithms and are only a logarithmic factor away from the best possible bounds.     

Finding a minimum path cover of a distance-hereditary graph in polynomial time

A path cover of a graph G=(V,E) is a set of pairwise vertex-disjoint paths such that the disjoint union of the vertices of these paths equals the vertex set V of G. The path cover problem is, given a graph, to find a path cover having the minimum number of paths. The path cover problem contains the Hamiltonian path problem as a special case since finding a path cover, consisting of a single path, corresponds directly to the Hamiltonian path problem. A graph is a distance-hereditary graph if each pair of vertices is equidistant in every connected induced subgraph containing them. The complexity of the path cover problem on distance-hereditary graphs has remained unknown. In this paper, we propose the first polynomial-time algorithm, which runs in O(|V|⁹) time, to solve the path cover problem on distance-hereditary graphs.     

All-Pairs Small-Stretch Paths

Let G = (V, E) be a weighted undirected graph. A path between u, v  V is said to be of stretch t if its length is at most t times the distance between u and v in the graph. We consider the problem of finding small-stretch paths between all pairs of vertices in the graph G.It is easy to see that finding paths of stretch less than 2 between all pairs of vertices in an undirected graph with n vertices is at least as hard as the Boolean multiplication of two n × n matrices. We describe three algorithms for finding small-stretch paths between all pairs of vertices in a weighted graph with n vertices and m edges. The first algorithm, STRETCH₂, runs in Õ(n^3/2m^1/2) time and finds stretch 2 paths. The second algorithm, STRETCH_7/3, runs in Õ(n^7/3) time and finds stretch 7/3 paths. Finally, the third algorithm, STRETCH₃, runs in Õ(n²) and finds stretch 3 paths.Our algorithms are simpler, more efficient and more accurate than the previously best algorithms for finding small-stretch paths. Unlike all previous algorithms, our algorithms are not based on the construction of sparse spanners or sparse neighborhood covers.     

On Conditional Covering Problem

The conditional covering problem (CCP) aims to locate facilities on a graph, where the vertex set represents both the demand points and the potential facility locations. The problem has a constraint that each vertex can cover only those vertices that lie within its covering radius and no vertex can cover itself. The objective of the problem is to find a set that minimizes the sum of the facility costs required to cover all the demand points. An algorithm for CCP on paths was presented by Horne and Smith (Networks 46(4):177–185, 2005). We show that their algorithm is wrong and further present a correct O(n ³) algorithm for the same. We also propose an O(n ²) algorithm for the CCP on paths when all vertices are assigned unit costs and further extend this algorithm to interval graphs without an increase in time complexity.     

Unavoidable Configurations in Complete Topological Graphs

A topological graph is a graph drawn in the plane so that its vertices are represented by points, and its edges are represented by Jordan curves connecting the corresponding points, with the property that any two curves have at most one point in common. We define two canonical classes of topological complete graphs, and prove that every topological complete graph with n vertices has a canonical subgraph of size at least clog^1/8 n, which belongs to one of these classes. We also show that every complete topological graph with n vertices has a non-crossing subgraph isomorphic to any fixed tree with at most clog^1/6 n vertices.     

Embedding arbitrary finite simple graphs into small strongly regular graphs

It is well known that any finite simple graph Γ is an induced subgraph of some exponentially larger strongly regular graph Γ (e.g., [2, 8]). No general polynomial‐size construction has been known. For a given finite simple graph Γ on υ vertices, we present a construction of a strongly regular graph Γ on O(υ⁴) vertices that contains Γ as its induced subgraph. A discussion is included of the size of the smallest possible strongly regular graph with this property. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 1–8, 2000     

Induced embeddings in Steinhaus graphs

Fix any positive integer n. Let S be the set of all Steinhaus graphs of order n(n − 1)/2 + 1. The vertices for each graph in S are the first n(n − 1)/2 + 1 positive integers. Let I be the set of all labeled graphs of order n with vertices of the form i(i − 1)/2 + 1 for the first n positive integers i. This article shows that the function ϕ : S → I that maps a Steinhaus graph to its induced subgraph is a bijection. Therefore, any graph of order n is isomorphic to an induced subgraph of a Steinhaus graph of order n(n − 1)/2 + 1. This considerably tightens a result of Brigham, Carrington, and Dutton in [Brigham, Carrington, & Dutton, Combin. Inform. System Sci. 17 (1992)], which showed that this could be done with a Steinhaus graph of order 2ⁿ⁻¹. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 1–9, 1998     

An exact algorithm for connected red–blue dominating set

In the Connected Red–Blue Dominating Set problem we are given a graph G whose vertex set is partitioned into two parts R and B (red and blue vertices), and we are asked to find a connected subgraph induced by a subset S of B such that each red vertex of G is adjacent to some vertex in S. The problem can be solved in O^?(²n−|B|) time by reduction to the Weighted Steiner Tree problem. Combining exhaustive enumeration when |B| is small with the Weighted Steiner Tree approach when |B| is large, solves the problem in O^?(n^1.4143). In this paper we present a first non-trivial exact algorithm whose running time is in O^?(n^1.3645). We use our algorithm to solve the Connected Dominating Set problem in O^?(n^1.8619). This improves the current best known algorithm, which used sophisticated run-time analysis via the measure and conquer technique to solve the problem in O^?(n^1.8966).     

O(n2.5) time algorithms for the subgraph homeomorphism problem on trees

The complexity of the subgraph homeomorphism problems have been open. We show O(n^2.5) time algorithms when the problems are restricted to trees, directed or undirected. The algorithm can be applied to the subtree isomorphism problem for unrooted trees with the same complexity, and improves over Reyner's O(n^3.5) algorithm for the subtree isomorphism problem.     

Induced Circuits in Planar Graphs

In McDiarmid, B. Reed, A. Schrijver, and B. Shepherd (Univ. of Waterloo Tech. Rep., 1990) a polynomial-time algorithm is given for the problem of finding a minimum cost circuit without chords (induced circuit) traversing two given vertices of a planar graph. The algorithm is based on the ellipsoid method. Here we give an O(n²) combinatorial algorithm to determine whether two nodes in a planar graph lie on an induced circuit. We also give a min-max relation for the problem of finding a maximum number of paths connecting two given vertices in a planar graph so that each pair of these paths forms an induced circuit.