A k-chordalisation of a graph G = (V, E) is a graph H = (V, F) obtained by adding edges to G, such that H is a chordal graph with maximum clique size at most k. This note considers the problem: given a graph G = (V, E...
详细信息
A k-chordalisation of a graph G = (V, E) is a graph H = (V, F) obtained by adding edges to G, such that H is a chordal graph with maximum clique size at most k. This note considers the problem: given a graph G = (V, E) which pairs of vertices, non-adjacent in G, will be an edge in every k-chordalisation of G. Such a pair is called necessary for treewidth k. An equivalent formulation is: which edges can one add to a graph G such that every tree decomposition of G of width at most k is also a tree decomposition of the resulting graph G'. Some sufficient, and some necessary and sufficient conditions are given for pairs of vertices to be necessary for treewidth k. For a fixed k, one can find in linear time for a given graph G the set of all necessary pairs for treewidth k. If k is given as part of the input, then this problem is coNP-hard. A few similar results are given when interval graphs (and hence pathwidth) are used instead of chordal graphs and treewidth.
We present an efficient and scalable coarse grained multicomputer (CGM) coloring algorithm that colors a graph G with at most Delta + 1 colors where A is the maximum degree in G. This algorithm is given in two variant...
详细信息
We present an efficient and scalable coarse grained multicomputer (CGM) coloring algorithm that colors a graph G with at most Delta + 1 colors where A is the maximum degree in G. This algorithm is given in two variants: randomized and deterministic. We show that on a p-processor CGM model the proposed algorithms require a parallel time of O(\G\/p) and a total work and overall communication cost of O(\G\). These bounds correspond to the average case for the randomized version and to the worst case for the deterministic variant. (C) 2003 Elsevier B.V. All rights reserved.
The P-4 is the induced path with vertices a, b, c, d and edges ab, bc, cd. The chair (co-P, gem) has a fifth vertex adjacent to b (a and b, a, b, c and d, respectively). We give a complete structure description of pri...
详细信息
The P-4 is the induced path with vertices a, b, c, d and edges ab, bc, cd. The chair (co-P, gem) has a fifth vertex adjacent to b (a and b, a, b, c and d, respectively). We give a complete structure description of prime chair-, co-P- and gem-free graphs which implies bounded clique width for this graph class. It is known that this has some nice consequences;very roughly speaking, every problem expressible in a certain kind of Monadic Second Order Logic (quantifying only over vertex set predicates) can be solved efficiently for graphs of bounded clique width. In particular, we obtain linear time for the problems Vertex Cover, Maximum Weight Stable Set (MWS), Maximum Weight Clique, Steiner Tree, Domination and Maximum Induced Matching on chair-, co-P- and gem-free graphs and a slightly larger class of graphs. This drastically improves a recently published O(n(4)) time bound for Maximum Stable Set on butterfly-, chair-, co-P- and gem-free graphs. (C) 2003 Elsevier Science B.V. All rights reserved.
We consider the problem of patrolling-i.e. ongoing exploration of a network by a decentralized group of simple memoryless robotic agents. The model for the network is an undirected graph, and our goal, beyond complete...
详细信息
We consider the problem of patrolling-i.e. ongoing exploration of a network by a decentralized group of simple memoryless robotic agents. The model for the network is an undirected graph, and our goal, beyond complete exploration, is to achieve close to uniform frequency of traversal of the graph's edges. A simple multi-agent exploration algorithm is presented and analyzed. It is shown that a single agent following this procedure enters, after a transient period, a periodic motion which is an extended Eulerian cycle, during which all edges are traversed an identical number of times. We further prove that if the network is Eulerian, a single agent goes into an Eulerian cycle within 2\E\D steps, \E\ being the number of edges in the graph and D being its diameter. For a team of k agents, we show that after at most 2(l + 1/k)\E\D steps the numbers of edge visits in the network are balanced up to a factor of two. In addition, various aspects of the algorithm are demonstrated by simulations.
Given a 2-edge-connected, real weighted graph G with n vertices and m edges, the 2-edge-connectivity augmentation problem is that of finding a minimum weight set of edges of G to be added to a spanning subgraph H of G...
详细信息
Given a 2-edge-connected, real weighted graph G with n vertices and m edges, the 2-edge-connectivity augmentation problem is that of finding a minimum weight set of edges of G to be added to a spanning subgraph H of G to make it 2-edge-connected. While the general problem is NP-hard and 2-approximable, in this paper we prove that it becomes polynomial time solvable if H is a depth-first search tree of G. More precisely, we provide an efficient algorithm for solving this special case which runs in O(M . alpha(M, n)) time, where alpha is the classic inverse of Ackermann's function and M = m . alpha(m, n). This algorithm has two main consequences: first, it provides a faster 2-approximation algorithm for the general 2-edge-connectivity augmentation problem;second, it solves in 0(m . alpha(m, n)) time the problem of restoring, by means of a minimum weight set of replacement edges, the 2-edge-connectivity of a 2-edge-connected communication network undergoing a link failure.
We consider the problem of finding a long, simple path in an undirected graph. We present a polynomial-time algorithm that finds a path of length Omega((log L/log log L)(2)), where L denotes the length of the longest ...
详细信息
We consider the problem of finding a long, simple path in an undirected graph. We present a polynomial-time algorithm that finds a path of length Omega((log L/log log L)(2)), where L denotes the length of the longest simple path in the graph. This establishes the performance ratio O(n( log log n/log n)(2)) for the longest path problem, where n denotes the number of vertices in the graph.
Planning in nondeterministic domains is typically intractable due to the large number of contingencies. Two techniques for speeding up planning in nondeterministic domains are agent-centered search and assumption-base...
详细信息
Planning in nondeterministic domains is typically intractable due to the large number of contingencies. Two techniques for speeding up planning in nondeterministic domains are agent-centered search and assumption-based planning. Both techniques interleave planning in deterministic domains with plan execution. They differ in how they make planning deterministic. To determine how suboptimal their plans are, we study two planning methods for robot navigation in initially unknown terrain that have successfully been used on mobile robots but not been analyzed before. The planning methods differ both in the technique they use to speed up planning and in the robot-navigation task they solve. Greedy Mapping uses agent-centered search to map unknown terrain. Dynamic A* uses assumption-based planning to navigate to a given goal location in unknown terrain. When we formalize abstractions of these planning methods on undirected graphs G = (V, E), they turn out to be similar enough that we are able to analyze their travel distance in a unified way. We discover that neither method is optimal in a worst-case sense, by a factor of Omega (log \V\ / log log \V\). We also derive factor O(root\V\) upper bounds to show that these methods are not very badly suboptimal in this sense. These results provide a first step towards explaining the good empirical results that have been reported about Greedy Mapping and Dynamic A* in the experimental literature. More generally, they show how to use tools from graph theory to analyze the plan quality of practical planning methods for nondeterministic domains. (C) 2003 Published by Elsevier Science B.V.
In this paper, we give a relatively simple though very efficient way to color the d-dimensional grid G(n(1), n(2),..., n(d)) (with n(i) vertices in each dimension 1 less than or equal to i less than or equal to d), fo...
详细信息
In this paper, we give a relatively simple though very efficient way to color the d-dimensional grid G(n(1), n(2),..., n(d)) (with n(i) vertices in each dimension 1 less than or equal to i less than or equal to d), for two different types of vertex colorings: (1) acyclic coloring of graphs, in which we color the vertices such that (i) no two neighbors are assigned the same color and (ii) for any two colors i and j, the subgraph induced by the vertices colored i or j is acyclic;and (2) k-distance coloring of graphs, in which every vertex must be colored in such a way that two vertices lying at distance less than or equal to k must be assigned different colors. The minimum number of colors needed to acyclically color (respectively k-distance color) a graph G is called acyclic chromatic number of G (respectively k-distance chromatic number), and denoted a(G) (respectively chi(k) (G)). The method we propose for coloring the d-dimensional grid in those two variants relies on the representation of the vertices of G(d)(n(1),...,n(d)) thanks to its coordinates in each dimension;this gives us upper bounds on a(G(d) (n(1),..., n(d))) and chi(k) (G(d) (n(1),...., n(d))). We also give lower bounds on a(G(d)(n(1),...., n(d))) and chi(k)(G(d)(n(1),...., n(d))). In particular, we give a lower bound on a(G) for any graph G;surprisingly, as far as we know this result was never mentioned before. Applied to the d-dimensional grid G(d) (n(1),..., n(d)), the lower and upper bounds for a(G(d) (n(1),...., n(d))) match (and thus give an optimal result) when the lengths in each dimension are "sufficiently large" (more precisely, if Sigma(i=1)(d) 1/n(i) less than or equal to 1). If this is not the case, then these bounds differ by an additive constant at most equal to 1 - [Sigma(i=1)(d) 1/n(i)]. Concerning chi(k) (G(d)(n(1),..., n(d))), we give exact results on its value for (1) k=2 and any d greater than or equal to 1, and (2)d=2 and any k greater than or equal to 1. In the case of acyclic
We study the problem of incrementally maintaining a topological sorting in a large DAG. The Discovery Algorithm (DA) of Alpern et al. [Proc. 1st Annual ACM-SIAM Symp. on Discrete algorithms, 1990, pp. 32-42] computes ...
详细信息
We study the problem of incrementally maintaining a topological sorting in a large DAG. The Discovery Algorithm (DA) of Alpern et al. [Proc. 1st Annual ACM-SIAM Symp. on Discrete algorithms, 1990, pp. 32-42] computes a cover kappa of nodes such that a solution to the modified problem can be found by changing node priorities within kappa only. It achieves a runtime complexity that is polynomially bounded in terms of the minimal cover size k. The temporary space complexity of DA grows quickly with increasing number of added edges and cover size. We introduce the Depth-First Discovery Algorithm (DFDA), which uses depth-first search to reduce the temporary space of DA from O(\A\ x \\kappa\\) to O(\A\ + \\kappa\\), where \A\ is the number of edges to add and \\kappa\\ is the extended size of the cover. DFDA is simpler than DA and performs better in our empirical tests. (C) 2003 Elsevier B.V. All fights reserved.
In the Minimum Label Spanning Tree problem, the input consists of an edge-colored undirected graph, and the goal is to find a spanning tree with the minimum number of different colors. We investigate the special case ...
详细信息
In the Minimum Label Spanning Tree problem, the input consists of an edge-colored undirected graph, and the goal is to find a spanning tree with the minimum number of different colors. We investigate the special case where every color appears at most r times in the input graph. This special case is polynomially solvable for r = 2, and NP- and APX-complete for any fixed r greater than or equal to 3. We analyze local search algorithms that are allowed to switch up to k of the colors used in a feasible solution. We show that for k = 2 any local optimum yields an (r + 1)/2-approximation of the global optimum, and that this bound is tight. For every k greater than or equal to 3, there exist instances for which some local optima are a factor of r/2 away from the global optimum. (C) 2003 Published by Elsevier Science B.V.
暂无评论