Definability Equals Recognizability of Partial 3-Trees and \sl k -Connected Partial \sl k -Trees

We consider graph decision problems on partial 3-trees that can be solved by a finite-state, leaf-to-root tree automaton processing a width-3 tree decomposition of the given graph. The class of yes-instances of such a problem is said to be recognizable by the tree automaton. In this paper we show that any such class of recognizable partial 3-trees is definable by a sentence in CMS logic. Here, CMS logic is the extension of Monadic Second-order logic with predicates to count the cardinality of sets modulo fixed integers. We also generalize this result to show that recognizability (by a tree automaton that processes width-k tree decompositions) implies CMS-definability for k -connected partial k -trees. The converse--definability implies recognizability--is known to hold for any class of partial k -trees, and even for any graph class with an appropriate definition of recognizability. It has been conjectured that recognizability implies definability over partial k -trees; but a proof was previously known only for k h 2 . This paper proves the conjecture, and hence the equivalence of definability and recognizability, over partial 3-trees and k -connected partial k -trees. We use techniques that may lead to a proof of this equivalence over all partial k -trees.     

An Efficient Parallel Strategy for ComputingK-Terminal Reliability and Finding Most Vital Edges in 2-Trees and Partial 2-Trees

In this paper, we first develop a parallel algorithm for computingK-terminal reliability, denoted byR(G_K), in 2-trees. Based on this result, we can also computeR(G_K) in partial 2-trees using a method that transforms, in parallel, a given partial 2-tree into a 2-tree. Finally, we solve the problem of finding most vital edges with respect toK-terminal reliability in partial 2-trees. Our algorithms takeO(log n) time withC(m, n) processors on a CRCW PRAM, whereC(m, n) is the number of processors required to find the connected components of a graph withmedges andnvertices in logarithmic time.     

A Linear Algorithm for Finding \boldmath[{g,f }]-Colorings of Partial \boldmath{k }-Trees

In an ordinary edge-coloring of a graph each color appears at each vertex v at most once. A [g,f] -coloring is a generalized edge-coloring in which each color appears at each vertex v at least g(v) and at most f(v) times, where g(v) and f(v) are respectively nonnegative and positive integers assigned to v . This paper gives a linear-time algorithm to find a [g,f] -coloring of a given partial k -tree using the minimum number of colors if there exists a [g,f] -coloring.     

Finding Edge-Disjoint Paths in Partial k -Trees

For a given graph G and p pairs (s _i ,t _i ) , , of vertices in G , the edge-disjoint paths problem is to find p pairwise edge-disjoint paths P _i , , connecting s _i and t _i . Many combinatorial problems can be efficiently solved for partial k -trees (graphs of treewidth bounded by a fixed integer k ), but the edge-disjoint paths problem is NP-complete even for partial 3 -trees. This paper gives two algorithms for the edge-disjoint paths problem on partial k -trees. The first one solves the problem for any partial k -tree G and runs in polynomial time if p=O( log n) and in linear time if p=O(1) , where n is the number of vertices in G . The second one solves the problem under some restriction on the location of terminal pairs even if . Received January 21, 1977; revised September 19, 1997.     

Easy Cases of Probabilistic Satisfiability

The Probabilistic Satisfiability problem (PSAT) can be considered as a probabilistic counterpart of the classical SAT problem. In a PSAT instance, each clause in a CNF formula is assigned a probability of being true; the problem consists in checking the consistency of the assigned probabilities. Actually, PSAT turns out to be computationally much harder than SAT, e.g., it remains difficult for some classes of formulas where SAT can be solved in polynomial time. A column generation approach has been proposed in the literature, where the pricing sub-problem reduces to a Weighted Max-SAT problem on the original formula. Here we consider some easy cases of PSAT, where it is possible to give a compact representation of the set of consistent probability assignments. We follow two different approaches, based on two different representations of CNF formulas. First we consider a representation based on directed hypergraphs. By extending a well-known integer programming formulation of SAT and Max-SAT, we solve the case in which the hypergraph does not contain cycles; a linear time algorithm is provided for this case. Then we consider the co-occurrence graph associated with a formula. We provide a solution method for the case in which the co-occurrence graph is a partial 2-tree, and we show how to extend this result to partial k-trees with k>2.     

Bijective Linear Time Coding and Decoding for k-Trees

The problem of coding labeled trees has been widely studied in the literature and several bijective codes that realize associations between labeled trees and sequences of labels have been presented. k-trees are one of the most natural and interesting generalizations of trees and there is considerable interest in developing efficient tools to manipulate this class of graphs, since many NP-Complete problems have been shown to be polynomially solvable on k-trees and partial k-trees. In 1970 Rényi and Rényi generalized the Prüfer code, the first bijective code for trees, to a subset of labeled k-trees. Subsequently, non redundant codes that realize bijection between k-trees (or Rényi k-trees) and a well defined set of strings were produced. In this paper we introduce a new bijective code for labeled k-trees which, to the best of our knowledge, produces the first coding and decoding algorithms running in linear time with respect to the size of the k-tree.     

Linear-time algorithms for problems on planar graphs with fixed disk dimension

The disk dimension of a planar graph G is the least number k for which G embeds in the plane minus k open disks, with every vertex on the boundary of some disk. Useful properties of graphs with a given disk dimension are derived, leading to an algorithm to obtain an outerplanar subgraph of a graph with disk dimension k by removing at most 2k−2 vertices. This reduction is used to obtain linear-time exact and approximation algorithms on graphs with fixed disk dimension. In particular, a linear-time approximation algorithm is presented for the pathwidth problem.     

A Fixed-Parameter Approach to 2-Layer Planarization

A bipartite graph is biplanar if the vertices can be placed on two parallel lines (layers) in the plane such that there are no edge crossings when edges are drawn as line segments between the layers. In this paper we study the 2-Layer Planarization problem: Can k edges be deleted from a given graph G so that the remaining graph is biplanar? This problem is NP-complete, and remains so if the permutation of the vertices in one layer is fixed (the 1-Layer Planarization problem). We prove that these problems are fixed-parameter tractable by giving linear-time algorithms for their solution (for fixed k). In particular, we solve the 2-Layer Planarization problem in O(k · 6^k + |G|) time and the 1-Layer Planarization problem in O(3^k · |G|) time. We also show that there are polynomial-time constant-approximation algorithms for both problems.     

k-tuple domination in graphs

In a graph G, a vertex is said to dominate itself and all of its neighbors. For a fixed positive integer k, the k-tuple domination problem is to find a minimum sized vertex subset in a graph such that every vertex in the graph is dominated by at least k vertices in this set. The current paper studies k-tuple domination in graphs from an algorithmic point of view. In particular, we give a linear-time algorithm for the k-tuple domination problem in strongly chordal graphs, which is a subclass of chordal graphs and includes trees, block graphs, interval graphs and directed path graphs. We also prove that the k-tuple domination problem is NP-complete for split graphs (a subclass of chordal graphs) and for bipartite graphs.     

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.     

Mining collective knowledge: inferring functional labels from online review for business

Uncertain graph has been widely used to represent graph data with inherent uncertainty in structures. Reliability search is a fundamental problem in uncertain graph analytics. This paper investigates on a new problem with broad real-world applications, the top-k reliability search problem on uncertain graphs, that is, finding the k vertices v with the highest reliabilities of connections from a source vertex s to v. Note that the existing algorithm for the threshold-based reliability search problem is inefficient for the top-k reliability search problem. We propose a new algorithm to efficiently solve the top-k reliability search problem. The algorithm adopts two important techniques, namely the BFS sharing technique and the offline sampling technique. The BFS sharing technique exploits overlaps among different sampled possible worlds of the input uncertain graph and performs a single BFS on all possible worlds simultaneously. The offline sampling technique samples possible worlds offline and stores them using a compact structure. The algorithm also takes advantages of bit vectors and bitwise operations to improve efficiency. In addition, we generalize the top-k reliability search problem from single-source case to the multi-source case and show that the multi-source case of the problem can be equivalently converted to the single-source case of the problem. Moreover, we define two types of the reverse top-k reliability search problems with different semantics on uncertain graphs. We propose appropriate solutions for both of them. Extensive experiments carried out on both real and synthetic datasets verify that the optimized algorithm outperforms the baselines by 1–2 orders of magnitude in execution time while achieving comparable accuracy. Meanwhile, the optimized algorithm exhibits linear scalability with respect to the size of the input uncertain graph.     

Finding the closed partition of a planar graph

In this paper we consider the problem of finding aclosed partition in a directed graph. This problem has applications in concurrent probabilistic program verification. The best sequential algorithm known for this problem runs inO(mn) time wherem is the number of directed edges andn is the number of vertices in the given digraph. In this paper we present a linear-time sequential algorithm to solve the closed partition problem for planar digraphs that arecompact. We then build on this algorithm to obtain an O(n^1.5)-time sequential algorithm to solve the closed partition problem for a general planar digraph.This work was supported in part by NSF Grant CCR 89-10707.     

A decomposition approach for the p-median problem on disconnected graphs

The p-median problem seeks for the location of p facilities on the vertices (customers) of a graph to minimize the sum of transportation costs for satisfying the demands of the customers from the facilities. In many real applications of the p-median problem the underlying graph is disconnected. That is the case of p-median problem defined over split administrative regions or regions geographically apart (e.g. archipelagos), and the case of problems coming from industry such as the optimal diversity management problem. In such cases the problem can be decomposed into smaller p-median problems which are solved in each component k for different feasible values of p_k, and the global solution is obtained by finding the best combination of p_k medians. This approach has the advantage that it permits to solve larger instances since only the sizes of the connected components are important and not the size of the whole graph. However, since the optimal number of facilities to select from each component is not known, it is necessary to solve p-median problems for every feasible number of facilities on each component. In this paper we give a decomposition algorithm that uses a procedure to reduce the number of subproblems to solve. Computational tests on real instances of the optimal diversity management problem and on simulated instances are reported showing that the reduction of subproblems is significant, and that optimal solutions were found within reasonable time.     

On the Parameterized Complexity of Layered Graph Drawing

We consider graph drawings in which vertices are assigned to layers and edges are drawn as straight line-segments between vertices on adjacent layers. We prove that graphs admitting crossing-free h-layer drawings (for fixed h) have bounded pathwidth. We then use a path decomposition as the basis for a linear-time algorithm to decide if a graph has a crossing-free h-layer drawing (for fixed h). This algorithm is extended to solve related problems, including allowing at most k crossings, or removing at most r edges to leave a crossing-free drawing (for fixed k or r). If the number of crossings or deleted edges is a non-fixed parameter then these problems are NP-complete. For each setting, we can also permit downward drawings of directed graphs and drawings in which edges may span multiple layers, in which case either the total span or the maximum span of edges can be minimized. In contrast to the Sugiyama method for layered graph drawing, our algorithms do not assume a preassignment of the vertices to layers. Research initiated at the International Workshop on Fixed Parameter Tractability in Graph Drawing, Bellairs Research Institute of McGill University, Holetown, Barbados, Feb. 9–16, 2001, organized by S. Whitesides. Research of Canada-based authors is supported by NSERC; research of Quebec-based authors is also supported by a grant from FCAR. Research of D.R. Wood completed while visiting McGill University. Research of G. Liotta supported by CNR and MURST.     

Acyclic colorings of subcubic graphs

It is known that the acyclic chromatic number of a subcubic graph is at most four, and its acyclic edge chromatic number is at most five. We present algorithms that prove these two facts. Let n be the number of vertices of a graph. Our first algorithm takes O(n) time and uses four colors to properly color the vertices of any subcubic graph so that there is no 2-colored cycle. Our second algorithm takes O(n) time and uses five colors to properly color the edges of any subcubic graph so that there is no 2-colored cycle. Both are the first linear-time algorithms for the problems they solve.     

A Monadic Second-Order Definition of the Structure of Convex Hypergraphs

We consider finite hypergraphs with hyperedges defined as sets of vertices of unbounded cardinality. Each such hypergraph has a unique modular decomposition, which is a tree, the nodes of which correspond to certain subhypergraphs (induced by certain sets of vertices called strong modules) of the considered hypergraph. One can define this decomposition by monadic second-order (MS) logical formulas. Such a hypergraph is convex if the vertices are linearly ordered in such a way that the hyperedges form intervals. Our main result says that the unique linear order witnessing the convexity of a prime hypergraph (i.e., of one, the modular decomposition of which is trivial) can be defined in MS logic. As a consequence, we obtain that if a set of bipartite graphs that correspond (in the usual way) to convex hypergraphs has a decidable monadic second-order theory (which means that one can decide whether a given MS formula is satisfied in some graph of the set) then it has bounded clique-width. This yields a new case of validity of a conjecture which is still open.     

Maximum Dispersion and Geometric Maximum Weight Cliques

We consider a facility location problem, where the objective is to “disperse” a number of facilities, i.e., select a given number k of locations from a discrete set of n candidates, such that the average distance between selected locations is maximized. In particular, we present algorithmic results for the case where vertices are represented by points in d-dimensional space, and edge weights correspond to rectilinear distances. Problems of this type have been considered before, with the best result being an approximation algorithm with performance ratio 2. For the case where k is fixed, we establish a linear-time algorithm that finds an optimal solution. For the case where k is part of the input, we present a polynomial-time approximation scheme.     

Smallest Bipartite Bridge-Connectivity Augmentation

This paper addresses two augmentation problems related to bipartite graphs. The first, a fundamental graph-theoretical problem, is how to add a set of edges with the smallest possible cardinality so that the resulting graph is 2-edge-connected, i.e., bridge-connected, and still bipartite. The second problem, which arises naturally from research on the security of statistical data, is how to add edges so that the resulting graph is simple and does not contain any bridges. In both cases, after adding edges, the graph can be either a simple graph or, if necessary, a multi-graph. Our approach then determines whether or not such an augmentation is possible. We propose a number of simple linear-time algorithms to solve both problems. Given the well-known bridge-block data structure for an input graph, the algorithms run in O(log n) parallel time on an EREW PRAM using a linear number of processors, where n is the number of vertices in the input graph. We note that there is already a polynomial time algorithm that solves the first augmentation problem related to graphs with a given general partition constraint in O(n(m+nlog n)log n) time, where m is the number of distinct edges in the input graph. We are unaware of any results for the second problem. H.-W. Wei, W.-C. Lu and T.-s. Hsu research supported in part by NSC of Taiwan Grants 94-2213-E-001-014, 95-2221-E-001-004 and 96-2221-E-001-004.     

New Graph Classes of Bounded Clique-Width

The clique-width of graphs is a major new concept with respect to the efficiency of graph algorithms; it is known that every problem expressible in a certain kind of Monadic Second Order Logic called LinEMSOL(τ_1,L) by Courcelle et al. is linear-time solvable on any graph class with bounded clique-width for which a k-expression for the input graph can be constructed in linear time. The notion of clique-width extends the one of treewidth since bounded treewidth implies bounded clique-width. We give a complete classification of all graph classes defined by forbidden one-vertex extensions of the P₄ (i.e., the path with four vertices a,b,c,d and three edges ab,bc,cd) with respect to bounded clique-width. Our results extend and improve recent structural and complexity results in a systematic way.     

Parameterized algorithm for eternal vertex cover

In this paper we initiate the study of a “dynamic” variant of the classical Vertex Cover problem, the Eternal Vertex Cover problem introduced by Klostermeyer and Mynhardt, from the perspective of parameterized algorithms. This problem consists in placing a minimum number of guards on the vertices of a graph such that these guards can protect the graph from any sequence of attacks on its edges. In response to an attack, each guard is allowed either to stay in his vertex, or to move to a neighboring vertex. However, at least one guard has to fix the attacked edge by moving along it. The other guards may move to reconfigure and prepare for the next attack. Thus at every step the vertices occupied by guards form a vertex cover. We show that the problem admits a kernel of size k⁴(k+1)+2k, which shows that the problem is fixed parameter tractable when parameterized by the number of available guards k. Finally, we also provide an algorithm with running time O(²O(k²)+nm) for Eternal Vertex Cover, where n is the number of vertices and m the number of edges of the input graph. In passing we also observe that Eternal Vertex Cover is NP-hard, yet it has a polynomial time 2-approximation algorithm.