On the k-path cover problem for cacti

In this paper we investigate the k-path cover problem for graphs, which is to find the minimum number of vertex disjoint k-paths that cover all the vertices of a graph. The k-path cover problem for general graphs is NP-complete. Though notable applications of this problem to database design, network, VLSI design, ring protocols, and code optimization, efficient algorithms are known for only few special classes of graphs. In order to solve this problem for cacti, i.e., graphs where no edge lies on more than one cycle, we introduce the so-called Steiner version of the k-path cover problem, and develop an efficient algorithm for the Steiner k-path cover problem for cacti, which finds an optimal k-path cover for a given cactus in polynomial time.     

A graph coloring heuristic using partial solutions and a reactive tabu scheme

Most of the recent heuristics for the graph coloring problem start from an infeasible k-coloring (adjacent vertices may have the same color) and try to make the solution feasible through a sequence of color exchanges. In contrast, our approach (called FOO-PARTIALCOL), which is based on tabu search, considers feasible but partial solutions and tries to increase the size of the current partial solution. A solution consists of k disjoint stable sets (and, therefore, is a feasible, partial k-coloring) and a set of uncolored vertices. We introduce a reactive tabu tenure which substantially enhances the performance of both our heuristic as well as the classical tabu algorithm (called TABUCOL) proposed by Hertz and de Werra [Using tabu search techniques for graph coloring, Computing 1987;39:345–51]. We will report numerical results on different benchmark graphs and we will observe that FOO-PARTIALCOL, though very simple, outperforms TABUCOL on some instances, provides very competitive results on a set of benchmark graphs which are known to be difficult, and outperforms the best-known methods on the graph flat300_28_0. For this graph, FOO-PARTIALCOL finds an optimal coloring with 28 colors. The best coloring achieved to date uses 31 colors. Algorithms very close to TABUCOL are still used as intensification procedures in the best coloring methods, which are evolutionary heuristics. FOO-PARTIALCOL could then be a powerful alternative. In conclusion FOO-PARTIALCOL is one of the most efficient simple local search coloring methods yet available.     

Stochastic Local Search Algorithms for Graph Set <Emphasis Type="Italic">T</Emphasis>-colouring and Frequency Assignment

The graph set T-colouring problem (GSTCP) generalises the classical graph colouring problem; it asks for the assignment of sets of integers to the vertices of a graph such that constraints on the separation of any two numbers assigned to a single vertex or to adjacent vertices are satisfied and some objective function is optimised. Among the objective functions of interest is the minimisation of the difference between the largest and the smallest integers used (the span). In this article, we present an experimental study of local search algorithms for solving general and large size instances of the GSTCP. We compare the performance of previously known as well as new algorithms covering both simple construction heuristics and elaborated stochastic local search algorithms. We investigate systematically different models and search strategies in the algorithms and determine the best choices for different types of instance. The study is an example of design of effective local search for constraint optimisation problems.     

Local Search Heuristics for Capacitated p-Median Problems

The search for p-median vertices on a network (graph) is a classical location problem. The p facilities (medians) must be located so as to minimize the sum of the distances from each demand vertex to its nearest facility. The Capacitated p-Median Problem (CPMP) considers capacities for the service to be given by each median. The total service demanded by vertices identified by p-median clusters cannot exceed their service capacity. Primal-dual based heuristics are very competitive and provide simultaneously upper and lower bounds to optimal solutions. The Lagrangean/surrogate relaxation has been used recently to accelerate subgradient like methods. The dual lower bound have the same quality of the usual Lagrangean relaxation dual but is obtained using modest computational times. This paper explores improvements on upper bounds applying local search heuristics to solutions made feasible by the Lagrangean/surrogate optimization process. These heuristics are based on location-allocation procedures that swap medians and vertices inside the clusters, reallocate vertices, and iterate until no improvements occur. Computational results consider instances from the literature and real data obtained using a geographical information system.     

Why Almost All k-Colorable Graphs Are Easy to Color

Coloring a k-colorable graph using k colors (k≥3) is a notoriously hard problem. Considering average case analysis allows for better results. In this work we consider the uniform distribution over k-colorable graphs with n vertices and exactly cn edges, c greater than some sufficiently large constant. We rigorously show that all proper k-colorings of most such graphs lie in a single “cluster”, and agree on all but a small, though constant, portion of the vertices. We also describe a polynomial time algorithm that whp finds a proper k-coloring of such a random k-colorable graph, thus asserting that most such graphs are easy to color. This should be contrasted with the setting of very sparse random graphs (which are k-colorable whp), where experimental results show some regime of edge density to be difficult for many coloring heuristics.     

Selection of programme slots of television channels for giving advertisement: A graph theoretic approach

The present paper aims at developing a linear time algorithm to find a solution to the ‘maximum weight 1 colouring’ problem for an interval graph with interval weight. This algorithm has been applied to solve the problem that involves selecting different programme slots telecast on different television channels in a day so as to reach the maximum number of viewers. It is shown that all programmes of all television channels can be modelled as a weighted interval graph with interval numbers as weights. The programme slots are taken as the vertices of the graph and if the time durations of two programme slots have non-empty intersection, the corresponding vertices are considered to be connected by an edge. The number of viewers of a programme is taken as the weight of the vertex. In reality, the number of viewers of a programme is not a fixed one; generally, it lies in an interval. Thus, the weights of the vertices are taken as interval numbers. We assume that a company sets the objective of selecting the popular programme in different channels so as to make its commercial advertisement reach the maximum number of viewers. However, the constraint imposed is that all selected programmes are mutually exclusive in respect of time scheduling. The objective of the paper is, therefore, to helps the companies to select the programme slots, which are mutually exclusive with respect to the time schedule of telecasting time, in such a way that the total number of viewers of the selected programme slots rises to the optimum level. It is shown that the solution of this problem is obtained by solving the maximum weight colouring problem on an interval graph. An algorithm is designed to solve this optimization problem using O(n) time, where n represents the total number of programmes of all channels.     

Improved heuristics for the bounded-diameter minimum spanning tree problem

Given an undirected, connected, weighted graph and a positive integer k, the bounded-diameter minimum spanning tree (BDMST) problem seeks a spanning tree of the graph with smallest weight, among all spanning trees of the graph, which contain no path with more than k edges. In general, this problem is NP-Hard for 4 ≤ k < n − 1, where n is the number of vertices in the graph. This work is an improvement over two existing greedy heuristics, called randomized greedy heuristic (RGH) and centre-based tree construction heuristic (CBTC), and a permutation-coded evolutionary algorithm for the BDMST problem. We have proposed two improvements in RGH/CBTC. The first improvement iteratively tries to modify the bounded-diameter spanning tree obtained by RGH/CBTC so as to reduce its cost, whereas the second improves the speed. We have modified the crossover and mutation operators and the decoder used in permutation-coded evolutionary algorithm so as to improve its performance. Computational results show the effectiveness of our approaches. Our approaches obtained better quality solutions in a much shorter time on all test problem instances considered.     

Linear-Time Algorithms for Partial \boldmathk -Tree Complements

It is known that a graph decision problem can be solved in linear time over partial k -trees if the problem can be defined in Monadic Second-order (or MS) logic. MS logic allows quantification of vertex and edge subsets, with respect to which logical sentences can encode many different conditions that an input graph must satisfy. It is not always clear, however, which graph problems can be expressed in such a way. In this paper we consider problems stated as logical conditions on subsets of the vertices and nonedges of the input graph. If such a problem can be defined in MS logic (i.e., in terms of the vertices and edges of the input graph), then there is a linear-time algorithm to solve the problem over partial k -trees. This algorithm also provides a solution to some problem over the graph-theoretic complements of partial k -trees. We study several examples of these ``complement-problems.' We introduce a variation of MS logic in which, if a graph-problem can be defined over the class of partial k -tree complements, then there is a linear-time algorithm to solve that problem over partial k -tree complements, and (equivalently) a linear-time algorithm to solve its complement-problem over partial k -trees.     

A refined search tree technique for Dominating Set on planar graphs

We establish a refined search tree technique for the parameterized DOMINATING SET problem on planar graphs. Here, we are given an undirected graph and we ask for a set of at most k vertices such that every other vertex has at least one neighbor in this set. We describe algorithms with running times O(8^kn) and O(8^kk+n³), where n is the number of vertices in the graph, based on bounded search trees. We describe a set of polynomial time data-reduction rules for a more general “annotated” problem on black/white graphs that asks for a set of k vertices (black or white) that dominate all the black vertices. An intricate argument based on the Euler formula then establishes an efficient branching strategy for reduced inputs to this problem. In addition, we give a family examples showing that the bound of the branching theorem is optimal with respect to our reduction rules. Our final search tree algorithm is easy to implement; its analysis, however, is involved.     

Applying Modular Decomposition to Parameterized Cluster Editing Problems

A graph G is said to be a bicluster graph if G is a disjoint union of bicliques (complete bipartite subgraphs), and a cluster graph if G is a disjoint union of cliques (complete subgraphs). In this work, we study the parameterized versions of the NP-hard Bicluster Graph Editing and Cluster Graph Editing problems. The former consists of obtaining a bicluster graph by making the minimum number of modifications in the edge set of an input bipartite graph. When at most k modifications are allowed (Bicluster(k) Graph Editing problem), this problem is FPT, and can be solved in O(4 ^k nm) time by a standard search tree algorithm. We develop an algorithm of time complexity O(4 ^k +n+m), which uses a strategy based on modular decomposition techniques; we slightly generalize the original problem as the input graph is not necessarily bipartite. The algorithm first builds a problem kernel with O(k ²) vertices in O(n+m) time, and then applies a bounded search tree. We also show how this strategy based on modular decomposition leads to a new way of obtaining a problem kernel with O(k ²) vertices for the Cluster(k) Graph Editing problem, in O(n+m) time. This problem consists of obtaining a cluster graph by modifying at most k edges in an input graph. A previous FPT algorithm of time O(1.92 ^k +n ³) for this problem was presented by Gramm et al. (Theory Comput. Syst. 38(4), 373–392, 2005, Algorithmica 39(4), 321–347, 2004). In their solution, a problem kernel with O(k ²) vertices is built in O(n ³) time.     

Deciding k-Colorability of P 5-Free Graphs in Polynomial Time

The problem of computing the chromatic number of a P ₅-free graph (a graph which contains no path on 5 vertices as an induced subgraph) is known to be NP-hard. However, we show that for every fixed integer k, there exists a polynomial-time algorithm determining whether or not a P ₅-free graph admits a k-coloring, and finding one, if it does.     

Maximumk-covering of weighted transitive graphs with applications

Consider a weighted transitive graph, where each vertex is assigned a positive weight. Given a positive integerk, the maximumk-covering problem is to findk disjoint cliques covering a set of vertices with maximum total weight. An 0(kn ²)-time algorithm to solve the problem in a transitive graph is proposed, wheren is the number of vertices. Based on the proposed algorithm the weighted version of a number of problems in VLSI layout (e.g.,k-layer topological via minimization), computational geometry (e.g., maximum multidimensionalk-chain), graph theory (e.g., maximumk-independent set in interval graphs), and sequence manipulation (e.g., maximum increasingk-subsequence) can be solved inO(kn ²), wheren is the input size.This Work was supported in part by the National Science Foundation under Grant MIP-8709074 and MIP-8921540.     

Polynomial Approximation and Graph-Coloring

Consider a graph G=(V,E) of order n. In the minimum graph-coloring problem we try to color V with as few colors as possible so that no two adjacent vertices receive the same color. This problem is among the first ones proved to be intractable, and hence, it is very unlikely that an optimal polynomial-time algorithm could ever be devised for it. In this paper, we survey the main polynomial time approximation algorithms (the ones for which theoretical approximability bounds have been studied) for the minimum graph-coloring and we discuss their approximation performance and their complexity. Finally, we further improve the approximation ratio for graph-coloring. Received October 5, 2001; revised November 15, 2002 Published online: February 20, 2003     

Fixed-Parameter Enumerability of Cluster Editing and Related Problems

Cluster Editing is transforming a graph by at most k edge insertions or deletions into a disjoint union of cliques. This problem is fixed-parameter tractable (FPT). Here we compute concise enumerations of all minimal solutions in O(2.27 ^k +k ² n+m) time. Such enumerations support efficient inference procedures, but also the optimization of further objectives such as minimizing the number of clusters. In an extended problem version, target graphs may have a limited number of overlaps of cliques, measured by the number t of edges that remain when the twin vertices are merged. This problem is still in FPT, with respect to the combined parameter k and t. The result is based on a property of twin-free graphs. We also give FPT results for problem versions avoiding certain artificial clusterings. Furthermore, we prove that all solutions with minimal edit sequences differ on a so-called full kernel with at most k ²/4+O(k) vertices, that can be found in polynomial time. The size bound is tight. We also get a bound for the number of edges in the full kernel, which is optimal up to a (large) constant factor. Numerous open problems are mentioned.     

A survey of local search methods for graph coloring

Tabucol   is a tabu search algorithm that tries to determine whether the vertices of a given graph can be colored with a fixed number k$k$ of colors such that no edge has both endpoints with the same color. This algorithm was proposed in 1987, one year after Fred Glover's article that launched tabu search. While more performing local search algorithms have now been proposed, Tabucol remains very popular and is often chosen as a subroutine in hybrid algorithms that combine a local search with a population based method. In order to explain this unfailing success, we make a thorough survey of local search techniques for graph coloring problems, and we point out the main differences between all these techniques.     

基于随机kNN图的批量边删除聚类算法

建立邻接图上的批量边删除聚类算法通用框架,提出基于高斯平滑模型的批量边删除判定准则,定义了适于聚类的邻接图的一般性质,提出并证明在kNN图基础上引入随机因子构造的随机kNN图,可以增强顶点之间的局部连通性,使聚类结果不再强烈依赖于某条边或某些边的保留或删除.RkNNClus算法简洁高效,依赖参数少,无需指定类簇数目,模拟和真实数据上的实验均有证明.     

Using the Redundant Chords of the k-Fault-Tolerant Graph to Eliminate the Effect of Failed Components

Studies of the structural graphs of the fault-tolerant systems designed by the methods of M.F. Karavai demonstrated that if the number of the redundant graph vertices coincides with the degree k of fault-tolerance, then the number of redundant chords grows rapidly with k. Described was a method of reconfiguration of the redundant graph which is a variant of the sliding redundantization allowing one to use the redundant chords in order to eliminate the effect of some m (> k) failed chords of the graph, its value being dependent on the structure of the objective graph, the magnitude of k, and the topology of failures. Presented were examples of realizing this method which suggest that greater values of m are more frequently obtained for the longest chords of the k-fault-tolerant graph.     

Fixed-Parameter Complexity of Minimum Profile Problems

The profile of a graph is an integer-valued parameter defined via vertex orderings; it is known that the profile of a graph equals the smallest number of edges of an interval supergraph. Since computing the profile of a graph is an NP-hard problem, we consider parameterized versions of the problem. Namely, we study the problem of deciding whether the profile of a connected graph of order n is at most n−1+k, considering k as the parameter; this is a parameterization above guaranteed value, since n−1 is a tight lower bound for the profile. We present two fixed-parameter algorithms for this problem. The first algorithm is based on a forbidden subgraph characterization of interval graphs. The second algorithm is based on two simple kernelization rules which allow us to produce a kernel with linear number of vertices and edges. For showing the correctness of the second algorithm we need to establish structural properties of graphs with small profile which are of independent interest. A preliminary version of the paper is published in Proc. IWPEC 2006, LNCS vol. 4169, 60–71.     

Finding optimal solutions to the graph partitioning problem with heuristic search

As search spaces become larger and as problems scale up, an efficient way to speed up the search is to use a more accurate heuristic function. A better heuristic function might be obtained by the following general idea. Many problems can be divided into a set of subproblems and subgoals that should be achieved. Interactions and conflicts between unsolved subgoals of the problem might provide useful knowledge which could be used to construct an informed heuristic function. In this paper we demonstrate this idea on the graph partitioning problem (GPP). We first show how to format GPP as a search problem and then introduce a sequence of admissible heuristic functions estimating the size of the optimal partition by looking into different interactions between vertices of the graph. We then optimally solve GPP with these heuristics. Experimental results show that our advanced heuristics achieve a speedup of up to a number of orders of magnitude. Finally, we experimentally compare our approach to other states of the art graph partitioning optimal solvers on a number of classes of graphs. The results obtained show that our algorithm outperforms them in many cases.     

Hamiltonicity in connected regular graphs

In 1980, Jackson proved that every 2-connected k-regular graph with at most 3k vertices is Hamiltonian. This result has been extended in several papers. In this note, we determine the minimum number of vertices in a connected k-regular graph that is not Hamiltonian, and we also solve the analogous problem for Hamiltonian paths. Further, we characterize the smallest connected k-regular graphs without a Hamiltonian cycle.