An efficient parallel algorithm for the efficient domination problem on distance-hereditary graphs

In the literature, there are quite a few sequential and parallel algorithms for solving problems on distance-hereditary graphs. With an n-vertex and m-edge distance-hereditary graph G, we show that the efficient domination problem on G can be solved in O(log/sup 2/ n) time using O(n + m) processors on a CREW PRAM. Moreover, if a binary tree representation of G is given, the problem can be optimally solved in O(log n) time using O(n/log n) processors on an EREW PRAM.     

A New Approach to Graph Recognition and Applications to Distance-Hereditary Graphs

Algorithms used in data mining and bioinformatics have to deal with huge amount of data efficiently.In many applications,the data are supposed to have explicit or implicit structures.To develop efficient algorithms for such data,we have to propose possible structure models and test if the models are feasible.Hence,it is important to make a compact model for structured data,and enumerate all instances efficiently.There are few graph classes besides trees that can be used for a model.In this paper,we inves...     

A simple and fast parallel coloring for distance-hereditary graphs

In the literature, there are quite a few sequential and parallel algorithms to solve problems on distance-hereditary graphs. Two well-known classes of graphs, which contain trees and cographs, belong to distance-hereditary graphs. We consider the vertex-coloring problem on distance-hereditary graphs. Let T/sub d/(|V|, |E|) and P/sub d/d(|V|, |E|) denote the time and processor complexities, respectively, required to construct a decomposition tree representation of a distance-hereditary graph G=(V,E) on a PRAM model M/sub d/. Our algorithm runs in O(T/sub d/(|V|, |E|)+log|V|) time using O(P/sub d/(|V|, |E|)+|V|/log|V|) processors on M/sub d/. The best known result for constructing a decomposition tree needs O(log/sup 2/ |V|) time using O(|V|+|E|) processors on a CREW PRAM. If a decomposition tree is provided as input, we solve the problem in O(log |V|) time using O(|V|/log |V|) processors on an EREW PRAM. To the best of our knowledge, there is no parallel algorithm for this problem on distance-hereditary graphs.     

A parity domination problem in graphs with bounded treewidth and distance-hereditary graphs

This paper concerns a domination problem in graphs with parity constraints. The task is to find a subset of the vertices with minimum cost such that for every vertex the number of chosen vertices in its neighbourhood has a prespecified parity. This problem is known to be ${\mathcal NP}$ -hard for general graphs. A linear time algorithm was developed for series-parallel graphs and trees with unit cost and restricted to closed neighbourhoods. We present a linear time algorithm for the parity domination problem with open and closed neighbourhoods and arbitrary cost functions on graphs with bounded treewidth and distance-hereditary graphs.     

On the induced matching problem

We study extremal questions on induced matchings in certain natural graph classes. We argue that these questions should be asked for twinless graphs, that is graphs not containing two vertices with the same neighborhood. We show that planar twinless graphs always contain an induced matching of size at least n/40 while there are planar twinless graphs that do not contain an induced matching of size (n+10)/27. We derive similar results for outerplanar graphs and graphs of bounded genus. These extremal results can be applied to the area of parameterized computation. For example, we show that the induced matching problem on planar graphs has a kernel of size at most 40k that is computable in linear time; this significantly improves the results of Moser and Sikdar (2007). We also show that we can decide in time O(k⁹¹+n) whether a planar graph contains an induced matching of size at least k.     

Space-Efficient Counting in Graphs on Surfaces

We consider the problem of counting the number of spanning trees in planar graphs. We prove tight bounds on the complexity of the problem, both in general and especially in the modular setting. We exhibit the problem to be complete for Logspace when the modulus is 2^k, for constant k. On the other hand, we show that for any other modulus and in the non-modular case, our problem is as hard in the planar case as for the case of arbitrary graphs. The techniques used are algebraic topological that may be useful in many other problems involving planar or higher genus graphs – such as higher genus graph recognition in Logspace. In the spirit of counting problems modulo 2^k, we also exhibit a highly parallel ?L\oplus {\bf L} algorithm for finding the value of a permanent modulo 2^k. Previously, the best known result in this direction was Valiant’s result that this problem lies in P. We also show that we can count the number of perfect matchings modulo 2^k in an arbitrary graph in P. This extends Valiant’s result for the permanent, since the Permanent may be modeled as counting the number of perfect matchings in bipartite graphs.     

The 1-Fixed-Endpoint Path Cover Problem is Polynomial on Interval Graphs

We consider a variant of the path cover problem, namely, the k-fixed-endpoint path cover problem, or kPC for short, on interval graphs. Given a graph G and a subset T\mathcal{T} of k vertices of V(G), a k-fixed-endpoint path cover of G with respect to T\mathcal{T} is a set of vertex-disjoint paths ℘ that covers the vertices of G such that the k vertices of T\mathcal{T} are all endpoints of the paths in ℘. The kPC problem is to find a k-fixed-endpoint path cover of G of minimum cardinality; note that, if T\mathcal{T} is empty the stated problem coincides with the classical path cover problem. In this paper, we study the 1-fixed-endpoint path cover problem on interval graphs, or 1PC for short, generalizing the 1HP problem which has been proved to be NP-complete even for small classes of graphs. Motivated by a work of Damaschke (Discrete Math. 112:49–64, 1993), where he left both 1HP and 2HP problems open for the class of interval graphs, we show that the 1PC problem can be solved in polynomial time on the class of interval graphs. We propose a polynomial-time algorithm for the problem, which also enables us to solve the 1HP problem on interval graphs within the same time and space complexity.     

Parallel Algorithms for Maximal Acyclic Sets

Given a graph G=(V,E), the well-known spanning forest problem of G can be viewed as the problem of finding a maximal subset F of edges in G such that the subgraph induced by F is acyclic. Although this problem has well-known efficient NC algorithms, its vertex counterpart, the problem of finding a maximal subset U of vertices in G such that the subgraph induced by U is acyclic, has not been shown to be in NC (or even in RNC) and is not believed to be parallelizable in general. In this paper we present NC algorithms for solving the latter problem for two special cases. First, we show that, for a planar graph with n vertices, the problem can be solved in time with O(n) processors on an EREW PRAM. Second, we show that the problem is solvable in NC if the input graph G has only vertex-induced paths of length polylogarithmic in the number of vertices of G. As a consequence of this result, we show that certain natural extensions of the well-studied maximal independent set problem remain solvable in NC. Moreover, we show that, for a constant-degree graph with n vertices, the problem can be solved in time with O(n ² ) processors on an EREW PRAM. Received July 3, 1995; revised April 1, 1996.     

On approximating the longest path in a graph

We consider the problem of approximating the longest path in undirected graphs. In an attempt to pin down the best achievable performance ratio of an approximation algorithm for this problem, we present both positive and negative results. First, a simple greedy algorithm is shown to find long paths in dense graphs. We then consider the problem of finding paths in graphs that are guaranteed to have extremely long paths. We devise an algorithm that finds paths of a logarithmic length in Hamiltonian graphs. This algorithm works for a much larger class of graphs (weakly Hamiltonian), where the result is the best possible. Since the hard case appears to be that of sparse graphs, we also consider sparse random graphs. Here we show that a relatively long path can be obtained, thereby partially answering an open problem of Broderet al. To explain the difficulty of obtaining better approximations, we also prove hardness results. We show that, for any ε<1, the problem of finding a path of lengthn-n ^ε in ann-vertex Hamiltonian graph isNP-hard. We then show that no polynomial-time algorithm can find a constant factor approximation to the longest-path problem unlessP=NP. We conjecture that the result can be strengthened to say that, for some constant δ>0, finding an approximation of ration ^δ is alsoNP-hard. As evidence toward this conjecture, we show that if any polynomial-time algorithm can approximate the longest path to a ratio of , for any ε>0, thenNP has a quasi-polynomial deterministic time simulation. The hardness results apply even to the special case where the input consists of bounded degree graphs. D. Karger was supported by an NSF Graduate Fellowship, NSF Grant CCR-9010517, and grants from the Mitsubishi Corporation and OTL. R. Motwani was supported by an Alfred P. Sloan Research Fellowship, an IBM Faculty Development Award, grants from Mitsubishi and OTL, NSF Grant CCR-9010517, and NSF Young Investigator Award CCR-9357849, with matching funds from IBM, the Schlumberger Foundation, the Shell Foundation, and the Xerox Corporation, G. D. S. Ramkumar was supported by a grant from the Toshiba Corporation. Communicated by M. X. Goemans.     

Efficient parallel algorithms for path problems in directed graphs

In this paper we describe a technique for finding efficient parallel algorithms for problems on directed graphs that involve checking the existence of certain kinds of paths in the graph. This technique provides efficient algorithms for finding dominators in flow graphs, performing interval and loop analysis on reducible flow graphs, and finding the feedback vertices of a digraph. Each of these algorithms takesO(log² n) time using the same number of processors needed for fast matrix multiplication. All of these bounds are for an EREW PRAM.     

Short Cycles Make W-hard Problems Hard: FPT Algorithms for W-hard Problems in Graphs with no Short Cycles

We show that several problems that are hard for various parameterized complexity classes on general graphs, become fixed parameter tractable on graphs with no small cycles. More specifically, we give fixed parameter tractable algorithms for Dominating Set, t -Vertex Cover (where we need to cover at least t edges) and several of their variants on graphs with girth at least five. These problems are known to be W[i]-hard for some i≥1 in general graphs. We also show that the Dominating Set problem is W[2]-hard for bipartite graphs and hence for triangle free graphs. In the case of Independent Set and several of its variants, we show these problems to be fixed parameter tractable even in triangle free graphs. In contrast, we show that the Dense Subgraph problem where one is interested in finding an induced subgraph on k vertices having at least l edges, parameterized by k, is W[1]-hard even on graphs with girth at least six. Finally, we give an O(log p) ratio approximation algorithm for the Dominating Set problem for graphs with girth at least 5, where p is the size of an optimum dominating set of the graph. This improves the previous O(log n) factor approximation algorithm for the problem, where n is the number of vertices of the input graph. A preliminary version of this paper appeared in the Proceedings of 10th Scandinavian Workshop on Algorithm Theory (SWAT), Lecture Notes in Computer Science, vol. 4059, pp. 304–315, 2006.     

Linear-Time Algorithm for the Paired-Domination Problem in Convex Bipartite Graphs

A bipartite graph G=(U,W,E) with vertex set V=U∪W is convex if there exists an ordering of the vertices of W such that for each u∈U, the neighbors of u are consecutive in W. A compact representation of a convex bipartite graph for specifying such an ordering can be computed in O(|V|+|E|) time. The paired-domination problem on bipartite graphs has been shown to be NP-complete. The complexity of the paired-domination problem on convex bipartite graphs has remained unknown. In this paper, we present an O(|V|) time algorithm to solve the paired-domination problem on convex bipartite graphs given a compact representation. As a byproduct, we show that our algorithm can be directly applied to solve the total domination problem on convex bipartite graphs in the same time bound.     

Optimal Reconstruction of Graphs under the Additive Model

We study the problem of reconstructing unknown graphs under the additive combinatorial search model. The main result concerns the reconstruction of bounded degree graphs, i.e., graphs with the degree of all vertices bounded by a constant d . We show that such graphs can be reconstructed in O(dn) nonadaptive queries, which matches the information-theoretic lower bound. The proof is based on the technique of separating matrices. A central result here is a new upper bound for a general class of separating matrices. As a particular case, we obtain a tight upper bound for the class of d -separating matrices, which settles an open question stated by Lindstr?m in [20]. Finally, we consider several particular classes of graphs. We show how an optimal nonadaptive solution of O(n ² / log n) queries for general graphs can be obtained. We also prove that trees with unbounded vertex degree can be reconstructed in a linear number of queries by a nonadaptive algorithm. Received August 1997; revised January 1999.     

Implementing data structures on a hypercube multiprocessor, and applications in parallel computational geometry

In this paper, we study the problem of implementing standard data structures on a hypercube multiprocessor. We present a technique for efficiently executing multiple independent search processes on a class of graphs called ordered h-level graphs. We show how this technique can be utilized to implement a segment tree on a hypercube, thereby obtaining O(long²n) time algorithms for solving the next element search problem, the trapezoidal composition problem, and the triangulation problem.     

Capacitated Domination Problem

We consider a generalization of the well-known domination problem on graphs. The (soft) capacitated domination problem with demand constraints is to find a dominating set D of minimum cardinality satisfying both the capacity and demand constraints. The capacity constraint specifies that each vertex has a capacity that it can use to meet the demands of dominated vertices in its closed neighborhood, and the number of copies of each vertex allowed in D is unbounded. The demand constraint specifies the demand of each vertex in V to be met by the capacities of vertices in D dominating it. In this paper, we study the capacitated domination problem on trees from an algorithmic point of view. We present a linear time algorithm for the unsplittable demand model, and a pseudo-polynomial time algorithm for the splittable demand model. In addition, we show that the capacitated domination problem on trees with splittable demand constraints is NP-complete (even for its integer version) and provide a polynomial time approximation scheme (PTAS). We also give a primal-dual approximation algorithm for the weighted capacitated domination problem with splittable demand constraints on general graphs.     

The Longest Path Problem has a Polynomial Solution on Interval Graphs

The longest path problem is the problem of finding a path of maximum length in a graph. Polynomial solutions for this problem are known only for small classes of graphs, while it is NP-hard on general graphs, as it is a generalization of the Hamiltonian path problem. Motivated by the work of Uehara and Uno (Proc. of the 15th Annual International Symp. on Algorithms and Computation (ISAAC), LNCS, vol. 3341, pp. 871–883, 2004), where they left the longest path problem open for the class of interval graphs, in this paper we show that the problem can be solved in polynomial time on interval graphs. The proposed algorithm uses a dynamic programming approach and runs in O(n ⁴) time, where n is the number of vertices of the input graph.     

Longest (s,t)-paths in L-shaped grid graphs

The longest path problem, that is, finding a simple path with the maximum number of vertices, is a well-known NP-hard problem with many applications. However, for some classes of graphs, including solid grid graphs and grid graphs with some holes, it is open. An L-shaped grid graph is a special kind of a rectangular grid graph with a rectangular hole. In this paper, we show that a longest path between two given vertices s and t of an L-shaped grid graph can be computed in linear time.     

Greedy partitioned algorithms for the shortest-path problem

A partitioned, priority-queue algorithm for solving the single-source best-path problem is defined and evaluated. Finding single-source paths for sparse graphs is notable because of its definitelack of parallelism-no known algorithms are scalable. Qualitatively, we discuss the close relationships between our algorithm and previous work by Quinn, Chikayama, and others. Performance measurements of variations of the algorithm, implemented both in concurrent and imperative programming languages on a shared-memory multiprocessor, are presented. This quantitative analysis of the algorithms provides insights into the tradeoffs between complexity and overhead in graph-searching executed in high-level parallel languages with automatic task scheduling.Presently at Intel-PCED.     

A New combinatorial approach to optimal embeddings of rectangles

An important problem in graph embeddings and parallel computing is to embed a rectangular grid into other graphs. We present a novel, general, combinatorial approach to (one-to-one) embedding rectangular grids into their ideal rectangular grids and optimal hypercubes. In contrast to earlier approaches of Aleliunas and Rosenberg, and Ellis, our approach is based on a special kind of doubly stochastic matrix. We prove that any rectangular grid can be embedded into its ideal rectangular grid with dilation equal to the ceiling of the compression ratio, which is bothoptimal up to a multiplicative constant and a substantial generalization of previous work. We also show that any rectangular grid can be embedded into its nearly ideal square grid with dilation at most 3. Finally, we show that any rectangular grid can be embedded into itsoptimal hypercube withoptimal dilation 2, a result previously obtained, after much research, through anad hoc approach. Our results also imply optimal simulations of two-dimensional mesh-connected parallel machines by hypercubes and mesh-connected machines, where each processor in the guest machine is simulated by exactly one processor in the host.A preliminary version of this paper appeared at the IEEE International Parallel Processing Symposium, 1994. The research of the third author was supported in part by NSF Grants CCR-9010366 and CCR-9303011.     

An efficient parallel strategy for the two-fixed-endpoint Hamiltonian path problem on distance-hereditary graphs

In this paper, we solve the two-fixed-endpoint Hamiltonian path problem on distance-hereditary graphs efficiently in parallel. Let T_d(|V|,|E|) and P_d(|V|,|E|) denote the parallel time and processor complexities, respectively, required to construct a decomposition tree of a distance-hereditary graph G=(V,E) on a PRAM model M_d. We show that this problem can be solved in O(T_d(|V|,|E|)+log|V|) time using O(P_d(|V|,|E|)+(|V|+|E|)/log|V|) processors on M_d. Moreover, if G is represented by its decomposition tree form, the problem can be solved optimally in O(log|V|) time using O((|V|+|E|)/log|V|) processors on an EREW PRAM. We also obtain a linear-time algorithm which is faster than the previous known O(|V|³) sequential algorithm.