Computing Vertex Connectivity: New Bounds from Old Techniques

The vertex connectivity κ of a graph is the smallest number of vertices whose deletion separates the graph or makes it trivial. We present the fastest known deterministic algorithm for finding the vertex connectivity and a corresponding separator. The time for a digraph having n vertices and m edges is O(min{κ³ + n, κn}m); for an undirected graph the term m can be replaced by κn. A randomized algorithm finds κ with error probability 1/2 in time O(nm). If the vertices have nonnegative weights the weighted vertex connectivity is found in time O(κ₁nmlog(n²/m)) where κ₁ ≤ m/n is the unweighted vertex connectivity or in expected time O(nmlog(n²/m)) with error probability 1/2. The main algorithm combines two previous vertex connectivity algorithms and a generalization of the preflow-push algorithm of Hao and Orlin (1994, J. Algorithms17, 424–446) that computes edge connectivity.     

Finding Scores in Tournaments

A tournamentT_nis an orientation of the complete graph onnvertices. We continue the algorithmic study initiated by10of recognizing various directed trees in tournaments. Hell and Rosenfeld studied the complexity of finding various oriented paths in tournaments by probing edge directions. Here, we investigate the complexity of finding a vertex of prescribed outdegree (or indegree) in the same model. We show that the complexity of finding a vertex of outdegreek( ≤ (n − 1)/2) inT_nis Θ(nk). This bound is in sharp contrast to the Θ(n) bound for selection in the case of transitive tournaments. We also establish tight bounds for finding vertices of prescribed degree from the adjacency matrix of general directed/undirected graphs. These bounds generalize the classical bound of11for finding a sink (a vertex of outdegree 0 and indegreen − 1) in a directed graph.     

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.     

An exact algorithm for subgraph homeomorphism

The subgraph homeomorphism problem is to decide if there is an injective mapping of the vertices of a pattern graph into vertices of a host graph so that the edges of the pattern graph can be mapped into (internally) vertex-disjoint paths in the host graph. The restriction of subgraph homeomorphism where an injective mapping of the vertices of the pattern graph into vertices of the host graph is already given in the input instance is termed fixed-vertex subgraph homeomorphism.We show that fixed-vertex subgraph homeomorphism for a pattern graph on p vertices and a host graph on n vertices can be solved in time 2^n−pn^O(1) or in time 3^n−pn^O(1) and polynomial space. In effect, we obtain new non-trivial upper bounds on the time complexity of the problem of finding k vertex-disjoint paths and general subgraph homeomorphism.     

Maximum hitting time for random walks on graphs

For x and y vertices of a connected graph G, let T_G(x, y) denote the expected time before a random walk starting from x reaches y. We determine, for each n > 0, the n-vertex graph G and vertices x and y for which T_G(x, y) is maximized. the extremal graph consists of a clique on ?(2n + 1)/3?) (or ?)(2_n ? 2)/3?) vertices, including x, to which a path on the remaining vertices, ending in y, has been attached; the expected time T_G(x, y) to reach y from x in this graph is approximately 4_n³/27.     

A linear-time algorithm for edge-disjoint paths in planar graphs

In this paper we discuss the problem of finding edge-disjoint paths in a planar, undirected graph such that each path connects two specified vertices on the boundary of the graph. We will focus on the “classical” case where an instance additionally fulfills the so-calledevenness-condition. The fastest algorithm for this problem known from the literature requiresO (n ^5/3(loglogn)^1/3) time, wheren denotes the number of vertices. In this paper now, we introduce a new approach to this problem, which results in anO(n) algorithm. The proof of correctness immediately yields an alternative proof of the Theorem of Okamura and Seymour, which states a necessary and sufficient condition for solvability.     

Tiling Transitive Tournaments and Their Blow-ups

Let TT _k denote the transitive tournament on k vertices. Let TT(h,k) denote the graph obtained from TT _k by replacing each vertex with an independent set of size h≥1. The following result is proved: Let c ₂=1/2, c ₃=5/6 and c _k =1−2^{−k−log k} for k≥4. For every ∈>0 there exists N=N(∈,h,k) such that for every undirected graph G with n>N vertices and with δ(G)≥c _k n, every orientation of G contains vertex disjoint copies of TT(h,k) that cover all but at most ∈n vertices. In the cases k=2 and k=3 the result is asymptotically tight. For k≥4, c _k cannot be improved to less than 1−2^{−0.5k(1+o(1))}. This revised version was published online in June 2006 with corrections to the Cover Date.     

Augmenting Edge-Connectivity over the Entire Range inÕ(nm) Time

For a given undirected graphG = (V, E, c_G) with edges weighted by nonnegative realsc_G:E → R ⁺ , let Λ_G(k) stand for the minimum amount of weights which needs to be added to makeG k-edge-connected, and letG*(k) be the resulting graph obtained fromG. This paper first shows that function Λ_Gover the entire rangek [0, +∞] can be computed inO(nm + n² log n) time, and then shows that allG*(k) in the entire range can be obtained fromO(n log n) weighted cycles, and such cycles can be computed inO(nm + n² log n) time, wherenandmare the numbers of vertices and edges, respectively.     

A polynomial method for the pos/neg weighted 3-median problem on a tree

Let a connected undirected graph G  =  (V, E) be given. In the classical p-median problem we want to find a set X containing p points in G such that the sum of weighted distances from X to all vertices in V is minimized. We consider the semi-obnoxious case where every vertex has either a positive or negative weight. In this case we have two different objective functions: the sum of the minimum weighted distances from X to all vertices and the sum of the weighted minimum distances. In this paper we show that for the case p = 3 an optimal solution for the second model in a tree can be found in O(n ⁵) time. If the 3-median is restricted to vertices or if the tree is a path then the complexity can be reduced to O(n ³). This research has partially been supported by the Spezialforschungsbereich F 003 “Optimierung und Kontrolle”, Projektbereich Diskrete Optimierung.     

Efficient algorithms for finding minimum spanning trees in undirected and directed graphs

Recently, Fredman and Tarjan invented a new, especially efficient form of heap (priority queue). Their data structure, theFibonacci heap (or F-heap) supports arbitrary deletion inO(logn) amortized time and other heap operations inO(1) amortized time. In this paper we use F-heaps to obtain fast algorithms for finding minimum spanning trees in undirected and directed graphs. For an undirected graph containingn vertices andm edges, our minimum spanning tree algorithm runs inO(m logβ (m, n)) time, improved fromO(mβ(m, n)) time, whereβ(m, n)=min {i|log⁽ⁱ⁾ n ≦m/n}. Our minimum spanning tree algorithm for directed graphs runs inO(n logn + m) time, improved fromO(n log n +m log log log_(m/n+2) n). Both algorithms can be extended to allow a degree constraint at one vertex. Research supported in part by National Science Foundation Grant MCS-8302648. Research supported in part by National Science Foundation Grant MCS-8303139. Research supported in part by National Science Foundation Grant MCS-8300984 and a United States Army Research Office Program Fellowship, DAAG29-83-GO020.     

Induced disjoint paths problem in a planar digraph

As an extension of the disjoint paths problem, we introduce a new problem which we call the induced disjoint paths problem. In this problem we are given a graph G and a collection of vertex pairs {(s₁,t₁),…,(s_k,t_k)}. The objective is to find k paths P₁,…,P_k such that P_i is a path from s_i to t_i and P_i and P_j have neither common vertices nor adjacent vertices for any distinct i,j.The induced disjoint paths problem has several variants depending on whether k is a fixed constant or a part of the input, whether the graph is directed or undirected, and whether the graph is planar or not. We investigate the computational complexity of several variants of the induced disjoint paths problem. We show that the induced disjoint paths problem is (i) solvable in polynomial time when k is fixed and G is a directed (or undirected) planar graph, (ii) NP-hard when k=2 and G is an acyclic directed graph, (iii) NP-hard when k=2 and G is an undirected general graph.As an application of our first result, we show that we can find in polynomial time certain structures called a “hole” and a “theta” in a planar graph.     

Fly Cheaply: On the Minimum Fuel Consumption Problem

In planning a flight, stops at intermediate airports are sometimes necessary to minimize fuel consumption, even if a direct flight is available. We investigate the problem of finding the cheapest path from one airport to another, given a set of n airports in ² and a function l: ² × ² → ⁺ representing the cost of a direct flight between any pair. Given a source airport s, the cheapest-path map is a subdivision of ² where two points lie in the same region iff their cheapest paths from s use the same sequence of intermediate airports. We show a quadratic lower bound on the combinatorial complexity of this map for a class of cost functions. Nevertheless, we are able to obtain subquadratic algorithms to find the cheapest path from s to all other airports for any well-behaved cost function l: our general algorithm runs in O(n^{4/3 +} ) time, and a simpler, more practical variant runs in O(n^{3/2 +} ) time, while a special class of cost functions requires just O(n log n) time.     

A Randomized Parallel Algorithm for Planar Graph Isomorphism

We present a parallel randomized algorithm running on aCRCW PRAM, to determine whether two planar graphs are isomorphic, and if so to find the isomorphism. We assume that we have a tree of separators for each planar graph (which can be computed by known algorithms inO(log² n) time withn^{1 + ε}processors, for any ε > 0). Ifnis the number of vertices, our algorithm takesO(log(n)) time with processors and with a probability of failure of 1/nat most. The algorithm needs 2 · log(m) − log(n) + O(log(n)) random bits. The number of random bits can be decreased toO(log(n)) by increasing the number of processors ton^{3/2 + ε}, for any ε > 0. Our parallel algorithm has significantly improved processor efficiency, compared to the previous logarithmic time parallel algorithm of Miller and Reif (Siam J. Comput.20(1991), 1128–1147), which requiresn⁴randomized processors orn⁵deterministic processors.     

Vertex Set Partitions Preserving Conservativeness

Let G be an undirected graph and ={X₁, …, X_n} be a partition of V(G). Denote by G/ the graph which has vertex set {X₁, …, X_n}, edge set E, and is obtained from G by identifying vertices in each class X_i of the partition . Given a conservative graph (G, w), we study vertex set partitions preserving conservativeness, i.e., those for which (G/ , w) is also a conservative graph. We characterize the conservative graphs (G/ , w), where is a terminal partition of V(G) (a partition preserving conservativeness which is not a refinement of any other partition of this kind). We prove that many conservative graphs admit terminal partitions with some additional properties. The results obtained are then used in new unified short proofs for a co-NP characterization of Seymour graphs by A. A. Ageev, A. V. Kostochka, and Z. Szigeti (1997, J. Graph Theory34, 357–364), a theorem of E. Korach and M. Penn (1992, Math. Programming55, 183–191), a theorem of E. Korach (1994, J. Combin. Theory Ser. B62, 1–10), and a theorem of A. V. Kostochka (1994, in “Discrete Analysis and Operations Research. Mathematics and its Applications (A. D. Korshunov, Ed.), Vol. 355, pp. 109–123, Kluwer Academic, Dordrecht).     

A new graph triconnectivity algorithm and its parallelization

We present a new algorithm for finding the triconnected components of an undirected graph. The algorithm is based on a method of searching graphs called open ear decomposition. A parallel implementation of the algorithm on a CRCW PRAM runs inO(log² n) parallel time usingO(n+m) processors, wheren is the number of vertices andm is the number of edges in the graph.A preliminary version of this paper was presented at the19th Annual ACM Symposium on Theory of Computing, New York, NY, May 1987.Supported by NSF Grant DCR 8514961.Supported by NSF Grant ECS 8404866 and the Semiconductor Research Corporation Grant 86-12-109.     

The List-Chromatic Number of Infinite Graphs Defined on Euclidean Spaces

If D is a countable set of positive reals, 2≤n<ω, let X _n (D) be the graph with the points of R ⁿ as vertices where two vertices are joined iff their distance is in D. We determine the list-chromatic number of X _n (D) as much as possible.     

On the Range of Possible Integrities of Graphs G(n, k)

We discuss the range of values for the integrity of a graphs G(n, k) where G(n, k) denotes a simple graph with n vertices and k edges. Let I _max(n, k) and I _min(n, k) be the maximal and minimal value for the integrity of all possible G(n, k) graphs and let the difference be D(n, k) = I _max(n, k) − I _min(n, k). In this paper we give some exact values and several lower bounds of D(n, k) for various values of n and k. For some special values of n and for s < n ^1/4 we construct examples of graphs G _n  = G _n (n, n + s) with a maximal integrity of I(G _n ) = I(C _n ) + s where C _n is the cycle with n vertices. We show that for k = n ²/6 the value of D(n, n ²/6) is at least \frac?6-13n{\frac{\sqrt{6}-1}{3}n} for large n.     

The Conditional Covering Problem on Unweighted Interval Graphs with Nonuniform Coverage Radius

Let G = (V, E) be an interval graph with n vertices and m edges. A positive integer R(x) is associated with every vertex x ? V{x\in V}. In the conditional covering problem, a vertex x ? V{x \in V} covers a vertex y ? V{y \in V} (x ≠ y) if d(x, y) ≤ R(x) where d(x, y) is the shortest distance between the vertices x and y. The conditional covering problem (CCP) finds a minimum cardinality vertex set C í V{C\subseteq V} so as to cover all the vertices of the graph and every vertex in C is also covered by another vertex of C. This problem is NP-complete for general graphs. In this paper, we propose an efficient algorithm to solve the CCP with nonuniform coverage radius in O(n ²) time, when G is an interval graph containing n vertices.     

Fully Dynamic Algorithms for Maintaining Shortest Paths Trees

We propose fully dynamic algorithms for maintaining the distances and the shortest paths from a single source in either a directed or an undirected graph with positive real edge weights, handling insertions, deletions, and weight updates of edges. The algorithms require linear space and optimal query time. The cost of the update operations depends on the class of the considered graph and on the number of the output updates, i.e., on the number of vertices that, due to an edge modification, either change the distance from the source or change the parent in the shortest paths tree. We first show that, if we deal only with updates on the weights of edges, then the update procedures require O(log n) worst case time per output update for several classes of graphs, as in the case of graphs with bounded genus, bounded arboricity, bounded degree, bounded treewidth, and bounded pagenumber. For general graphs with n vertices and m edges the algorithms require worst case time per output update. We also show that, if insertions and deletions of edges are allowed, then similar amortized bounds hold.