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1.
Local and variable neighborhood search for the <Emphasis Type="Italic">k</Emphasis>-cardinality subgraph problem 总被引:1,自引:0,他引:1
The minimum weighted k-cardinality subgraph problem consists of finding a connected subgraph of a given graph with exactly k edges whose sum of weights is minimum. For this NP-hard combinatorial problem, only constructive types of heuristics have been suggested in the literature. In this paper we
propose a new heuristic based on variable neighborhood search metaheuristic rules. This procedure uses a new local search
developed by us. Extensive numerical results that include graphs with up to 5,000 vertices are reported. It appears that VNS
outperforms all previous methods. 相似文献
2.
For a bounded integer ℓ, we wish to color all edges of a graph G so that any two edges within distance ℓ have different colors. Such a coloring is called a distance-edge-coloring or an ℓ-edge-coloring of G. The distance-edge-coloring problem is to compute the minimum number of colors required for a distance-edge-coloring of a given graph G. A partial k-tree is a graph with tree-width bounded by a fixed constant k. We first present a polynomial-time exact algorithm to solve the problem for partial k-trees, and then give a polynomial-time 2-approximation algorithm for planar graphs. 相似文献
3.
Hans L. Bodlaender Corinne Feremans Alexander Grigoriev Eelko Penninkx Ren Sitters Thomas Wolle 《Computational Geometry》2009,42(9):939-951
In this paper we discuss the complexity and approximability of the minimum corridor connection problem where, given a rectilinear decomposition of a rectilinear polygon into “rooms”, one has to find the minimum length tree along the edges of the decomposition such that every room is incident to a vertex of the tree. We show that the problem is strongly NP-hard and give a subexponential time exact algorithm. For the special case when the room connectivity graph is k-outerplanar the algorithm running time becomes cubic. We develop a polynomial time approximation scheme for the case when all rooms are fat and have nearly the same size. When rooms are fat but are of varying size we give a polynomial time constant factor approximation algorithm. 相似文献
4.
The cop number c(G) of a graph G is an invariant connected with the genus and the girth. We prove that for a fixed k there is a polynomial-time algorithm which decides whether c(G) ≤ k. This settles a question of T. Andreae. Moreover, we show that every graph is topologically equivalent to a graph with c ≤ 2. Finally we consider a pursuit-evasion problem in Littlewood′s miscellany. We prove that two lions are not always sufficient to catch a man on a plane graph, provided the lions and the man have equal maximum speed. We deal both with a discrete motion (from vertex to vertex) and with a continuous motion. The discrete case is solved by showing that there are plane graphs of cop number 3 such that all the edges can be represented by straight segments of the same length. 相似文献
5.
We consider the problem of finding a smallest set of edges whose addition four-connects a triconnected graph. This is a fundamental graph-theoretic problem that has applications in designing reliable networks and improving statistical database security. We present an O(n · α(m, n) + m)-time algorithm for four-connecting an undirected graph G that is triconnected by adding the smallest number of edges, where n and m are the number of vertices and edges in G, respectively, and α(m, n) is the inverse Ackermann function. This is the first polynomial time algorithm to solve this problem exactly.In deriving our algorithm, we present a new lower bound for the number of edges needed to four-connect a triconnected graph. The form of this lower bound is different from the form of the lower bound known for biconnectivity augmentation and triconnectivity augmentation. Our new lower bound applies for arbitrary k and gives a tighter lower bound than the one known earlier for the number of edges needed to k-connect a (k − 1)-connected graph. For k = 4, we show that this lower bound is tight by giving an efficient algorithm to find a set of edges whose size equals the new lower bound and whose addition four-connects the input triconnected graph. 相似文献
6.
A connected matching in a graph is a collection of edges that are pairwise disjoint but joined by another edge of the graph. Motivated by applications to Hadwiger’s conjecture, Plummer, Stiebitz, and Toft (2003) introduced connected matchings and proved that, given a positive integer , determining whether a graph has a connected matching of size at least is NP-complete. Cameron (2003) proved that this problem remains NP-complete on bipartite graphs, but can be solved in polynomial-time on chordal graphs. We present a polynomial-time algorithm that finds a maximum connected matching in a chordal bipartite graph. This includes a novel edge-without-vertex-elimination ordering of independent interest. We give several applications of the algorithm, including computing the Hadwiger number of a chordal bipartite graph, solving the unit-time bipartite margin-shop scheduling problem in the case in which the bipartite complement of the precedence graph is chordal bipartite, and determining–in a totally balanced binary matrix–the largest size of a square sub-matrix that is permutation equivalent to a matrix with all zero entries above the main diagonal. 相似文献
7.
Francisco Barahona 《Mathematical Programming》2006,105(2-3):181-200
We give an algorithm for the following problem: given a graph G=(V,E) with edge-weights and a nonnegative integer k, find a minimum cost set of edges that contains k disjoint spanning trees. This also solves the following reinforcement problem: given a network, a number k and a set of candidate edges, each of them with an associated cost, find a minimum cost set of candidate edges to be added
to the network so it contains k disjoint spanning trees. The number k is seen as a measure of the invulnerability of a network. Our algorithm has the same asymptotic complexity as |V| applications of the minimum cut algorithm of Goldberg & Tarjan.
Received: April, 2004 相似文献
8.
This paper studies the problem of proper‐walk connection number: given an undirected connected graph, our aim is to colour its edges with as few colours as possible so that there exists a properly coloured walk between every pair of vertices of the graph, that is, a walk that does not use consecutively two edges of the same colour. The problem was already solved on several classes of graphs but still open in the general case. We establish that the problem can always be solved in polynomial time in the size of the graph and we provide a characterization of the graphs that can be properly connected with colours for every possible value of . 相似文献
9.
We address the problem of assigning probabilities at discrete time instants for routing toll-free calls to a given set of call centers to minimize a weighted sum of transmission costs and load variability at the call centers during the next time interval.We model the problem as a tripartite graph and decompose the finding of an optimal probability assignment in the graph into the following problems: (i) estimating the true arrival rates at the nodes for the last time period; (ii) computing routing probabilities assuming that the estimates are correct. We use a simple approach for arrival rate estimation and solve the routing probability assignment by formulating it as a convex quadratic program and using the affine scaling algorithm to obtain an optimal solution.We further address a practical variant of the problem that involves changing routing probabilities associated with k nodes in the graph, where k is a prespecified number, to minimize the objective function. This involves deciding which k nodes to select for changing probabilities and determining the optimal value of the probabilities. We solve this problem using a heuristic that ranks all subsets of k nodes using gradient information around a given probability assignment.The routing model and the heuristic are evaluated for speed of computation of optimal probabilities and load balancing performance using a Monte Carlo simulation. Empirical results for load balancing are presented for a tripartite graph with 99 nodes and 17 call center gates. 相似文献
10.
Harold N. Gabow Haim Kaplan Robert E. Tarjan 《Journal of Algorithms in Cognition, Informatics and Logic》2001,40(2):159
We consider the problem of testing the uniqueness of maximum matchings, both in the unweighted and in the weighted case. For the unweighted case, we have two results. First, given a graph with n vertices and m edges, we can test whether the graph has a unique perfect matching, and find it if it exists, in O(m log4 n) time. This algorithm uses a recent dynamic connectivity algorithm and an old result of Kotzig characterizing unique perfect matchings in terms of bridges. For the special case of planar graphs, we improve the algorithm to run in O(n log n) time. Second, given one perfect matching, we can test for the existence of another in linear time. This algorithm is a modification of Edmonds' blossom-shrinking algorithm implemented using depth-first search. A generalization of Kotzig's theorem proved by Jackson and Whitty allows us to give a modification of the first algorithm that tests whether a given graph has a unique f-factor, and find it if it exists. We also show how to modify the second algorithm to check whether a given f-factor is unique. Both extensions have the same time bounds as their perfect matching counterparts. For the weighted case, we can test in linear time whether a maximum-weight matching is unique, given the output from Edmonds' algorithm for computing such a matching. The method is an extension of our algorithm for the unweighted case. 相似文献
11.
Philip N Klein Serge A Plotkin Satish Rao Éva Tardos 《Journal of Algorithms in Cognition, Informatics and Logic》1997,22(2):241-269
In this paper we consider theSteiner multicutproblem. This is a generalization of the minimum multicut problem where instead of separating nodepairs, the goal is to find a minimum weight set of edges that separates all givensetsof nodes. A set is considered separated if it is not contained in a single connected component. We show anO(log3(kt)) approximation algorithm for the Steiner multicut problem, wherekis the number of sets andtis the maximum cardinality of a set. This improves theO(t log k) bound that easily follows from the previously known multicut results. We also consider an extension of multicuts to directed case, namely the problem of finding a minimum-weight set of edges whose removal ensures that none of the strongly connected components includes one of the prespecifiedknode pairs. In this paper we describe anO(log2 k) approximation algorithm for this directed multicut problem. Ifk ? n, this represents an improvement over theO(log n log log n) approximation algorithm that is implied by the technique of Seymour. 相似文献
12.
《European Journal of Operational Research》1999,118(1):127-138
Given a graph and costs of assigning to each vertex one of k different colors, we want to find a minimum cost assignment such that no color q induces a subgraph with more than a given number (γq) of connected components. This problem arose in the context of contiguity-constrained clustering, but also has a number of other possible applications. We show the problem to be NP-hard. Nevertheless, we derive a dynamic programming algorithm that proves the case where the underlying graph is a tree to be solvable in polynomial time. Next, we propose mixed-integer programming formulations for this problem that lead to branch-and-cut and branch-and-price algorithms. Finally, we introduce a new class of valid inequalities to obtain an enhanced branch-and-cut. Extensive computational experiments are reported. 相似文献
13.
In this paper, we propose a fast heuristic algorithm for the maximum concurrent k-splittable flow problem. In such an optimization problem, one is concerned with maximizing the routable demand fraction across a capacitated network,
given a set of commodities and a constant k expressing the number of paths that can be used at most to route flows for each commodity. Starting from known results on
the k-splittable flow problem, we design an algorithm based on a multistart randomized scheme which exploits an adapted extension
of the augmenting path algorithm to produce starting solutions for our problem, which are then enhanced by means of an iterative
improvement routine. The proposed algorithm has been tested on several sets of instances, and the results of an extensive
experimental analysis are provided in association with a comparison to the results obtained by a different heuristic approach
and an exact algorithm based on branch and bound rules. 相似文献
14.
Greg N Frederickson Roberto Solis-Oba 《Journal of Algorithms in Cognition, Informatics and Logic》1999,33(2):244
The problems of computing the maximum increase in the weight of the minimum spanning trees of a graph caused by the removal of a given number of edges, or by finite increases in the weights of the edges, are investigated. For the case of edge removals, the problem is shown to be NP-hard and an Ω(1/log k)-approximation algorithm is presented for it, where (input parameter) k > 1 is the number of edges to be removed. The second problem is studied, assuming that the increase in the weight of an edge has an associated cost proportional to the magnitude of the change. An O(n3m2 log(n2/m)) time algorithm is presented to solve it. 相似文献
15.
16.
Yukio Shibata 《Journal of Graph Theory》1988,12(3):421-428
We introduce the notion of the boundary clique and the k-overlap clique graph and prove the following: Every incomplete chordal graph has two nonadjacent simplicial vertices lying in boundary cliques. An incomplete chordal graph G is k-connected if and only if the k-overlap clique graph gk(G) is connected. We give an algorithm to construct a clique tree of a connected chordal graph and characterize clique trees of connected chordal graphs using the algorithm. 相似文献
17.
The minimum k-partition (MkP) problem is the problem of partitioning the set of vertices of a graph into k disjoint subsets so as to minimize the total weight of the edges joining vertices in the same partition. The main contribution
of this paper is the design and implementation of a branch-and-cut algorithm based on semidefinite programming (SBC) for the
MkP problem. The two key ingredients for this algorithm are: the combination of semidefinite programming with polyhedral results;
and a novel iterative clustering heuristic (ICH) that finds feasible solutions for the MkP problem. We compare ICH to the hyperplane rounding techniques of Goemans and Williamson and of Frieze and Jerrum, and the
computational results support the conclusion that ICH consistently provides better feasible solutions for the MkP problem. ICH is used in our SBC algorithm to provide feasible solutions at each node of the branch-and-bound tree. The SBC
algorithm computes globally optimal solutions for dense graphs with up to 60 vertices, for grid graphs with up to 100 vertices,
and for different values of k, providing a fast exact approach for k≥3. 相似文献
18.
Chao-Lin Chen 《Journal of Global Optimization》2011,50(3):473-483
Given a set X, we consider the problem of finding a graph G with vertex set X and the minimum number of edges such that for i = 1, . . . , m, the subgraph G i induced from pattern i is a label connected graph with minimum edges. In the paper, we show that the problem is NP hard and develop a heuristic algorithm to get a fewer number of edges to store patterns. 相似文献
19.
Given a hypergraph, a partition of its vertex set, and a nonnegative integer k, find a minimum number of graph edges to be added between different members of the partition in order to make the hypergraph k‐edge‐connected. This problem is a common generalization of the following two problems: edge‐connectivity augmentation of graphs with partition constraints (J. Bang‐Jensen, H. Gabow, T. Jordán, Z. Szigeti, SIAM J Discrete Math 12(2) (1999), 160–207) and edge‐connectivity augmentation of hypergraphs by adding graph edges (J. Bang‐Jensen, B. Jackson, Math Program 84(3) (1999), 467–481). We give a min–max theorem for this problem, which implies the corresponding results on the above‐mentioned problems, and our proof yields a polynomial algorithm to find the desired set of edges. 相似文献
20.
The antibandwidth maximization problem (AMP) consists of labeling the vertices of a n-vertex graph G with distinct integers from 1 to n such that the minimum difference of labels of adjacent vertices is maximized. This problem can be formulated as a dual problem
to the well known bandwidth problem. Exact results have been proved for some standard graphs like paths, cycles, 2 and 3-dimensional
meshes, tori, some special trees etc., however, no algorithm has been proposed for the general graphs. In this paper, we propose
a memetic algorithm for the antibandwidth maximization problem, wherein we explore various breadth first search generated
level structures of a graph—an imperative feature of our algorithm. We design a new heuristic which exploits these level structures
to label the vertices of the graph. The algorithm is able to achieve the exact antibandwidth for the standard graphs as mentioned.
Moreover, we conjecture the antibandwidth of some 3-dimensional meshes and complement of power graphs, supported by our experimental
results. 相似文献