Solving shortest paths efficiently on nearly acyclic directed graphs

Shortest path problems can be solved very efficiently when a directed graph is nearly acyclic. Earlier results defined a graph decomposition, now called the 1-dominator set, which consists of a unique collection of acyclic structures with each single acyclic structure dominated by a single associated trigger vertex. In this framework, a specialised shortest path algorithm only spends delete-min operations on trigger vertices, thereby making the computation of shortest paths through non-trigger vertices easier. A previously presented algorithm computed the 1-dominator set in O(mn) worst-case time, which allowed it to be integrated as part of an O(mn+nrlogr) time all-pairs algorithm. Here m and n respectively denote the number of edges and vertices in the graph, while r denotes the number of trigger vertices. A new algorithm presented in this paper computes the 1-dominator set in just O(m) time. This can be integrated as part of the O(m+rlogr) time spent solving single-source, improving on the value of r obtained by the earlier tree-decomposition single-source algorithm. In addition, a new bidirectional form of 1-dominator set is presented, which further improves the value of r by defining acyclic structures in both directions over edges in the graph. The bidirectional 1-dominator set can similarly be computed in O(m) time and included as part of the O(m+rlogr) time spent computing single-source. This paper also presents a new all-pairs algorithm under the more general framework where r is defined as the size of any predetermined feedback vertex set of the graph, improving the previous all-pairs time complexity from O(mn+nr²) to O(mn+r³).     

Degree-constrained decompositions of graphs: Bounded treewidth and planarity

We study the problem of decomposing the vertex set V$V$ of a graph into two nonempty parts _V1,_V2$V_1,V_2$ which induce subgraphs where each vertex v∈_V1$v\in V_1$ has degree at least a(v)$a(v)$ inside _V1$V_1$ and each v∈_V2$v\in V_2$ has degree at least b(v)$b(v)$ inside _V2$V_2$. We give a polynomial-time algorithm for graphs with bounded treewidth which decides if a graph admits a decomposition, and gives such a decomposition if it exists. This result and its variants are then applied to designing polynomial-time approximation schemes for planar graphs where a decomposition does not necessarily exist but the local degree conditions should be met for as many vertices as possible.     

A distance measure for large graphs based on prime graphs

Graphs are universal modeling tools. They are used to represent objects and their relationships in almost all domains: they are used to represent DNA, images, videos, social networks, XML documents, etc. When objects are represented by graphs, the problem of their comparison is a problem of comparing graphs. Comparing objects is a key task in our daily life. It is the core of a search engine, the backbone of a mining tool, etc. Nowadays, comparing objects faces the challenge of the large amount of data that this task must deal with. Moreover, when graphs are used to model these objects, it is known that graph comparison is very complex and computationally hard especially for large graphs. So, research on simplifying graph comparison gainedan interest and several solutions are proposed. In this paper, we explore and evaluate a new solution for the comparison of large graphs. Our approach relies on a compact encoding of graphs called prime graphs. Prime graphs are smaller and simpler than the original ones but they retain the structure and properties of the encoded graphs. We propose to approximate the similarity between two graphs by comparing the corresponding prime graphs. Simulations results show that this approach is effective for large graphs.     

Fuzzy graphs and networks repairs

In Ref. [3] McAllister L. N. (1992) A simulation technique under uncertainty Proceedings of the 1992 NAFIPS Conference Puerto Vallarta Mexico pp. 555–563  [Google Scholar], this author provided a new definition of fuzzy graph that differs from the original definition.

In this paper, the author shows how the new definition provides with efficient numerical techniques to repair a failed link or a failed node of a network.     

Stable sets in k-colorable -free graphs

We show that, for fixed k, there is a polynomial-time algorithm that finds a maximum (or maximum-weight) stable set in any graph that belongs to the class of k-colorable P₅-free graphs, or, more generally, to the class of P₅-free graphs that contain no clique of size k+1. This is based on the following structural result: every connected k-colorable P₅-free graph has a vertex whose non-neighbors induce a (k−1)-colorable subgraph.     

Editing to Eulerian graphs

The Eulerian Editing problem asks, given a graph G and an integer k, whether G can be modified into an Eulerian graph using at most k edge additions and edge deletions. We show that this problem is polynomial-time solvable for both undirected and directed graphs. We generalize these results for problems with degree parity constraints and degree balance constraints, respectively. We also consider the variants where vertex deletions are permitted. Combined with known results, this leads to full complexity classifications for both undirected and directed graphs and for every subset of the three graph operations.     

Linear structure of bipartite permutation graphs and the longest path problem

The class of bipartite permutation graphs is the intersection of two well known graph classes: bipartite graphs and permutation graphs. A complete bipartite decomposition of a bipartite permutation graph is proposed in this note. The decomposition gives a linear structure of bipartite permutation graphs, and it can be obtained in O(n) time, where n is the number of vertices. As an application of the decomposition, we show an O(n) time and space algorithm for finding a longest path in a bipartite permutation graph.     

Self-assembling graphs

A self-assembly algorithm for synchronising agents and have them arrange according to a particular graph is given. This algorithm, expressed using an ad hoc rule-based process algebra, extends Klavins’ original proposal (Klavin, 2002: Automatic synthesis of controllers for assembly and formation forming. In: Proceedings of the International Conference on Robotics and Automation), in that it relies only on point-to-point communication, and can deal with any assembly graph whereas Klavins’ method dealt only with trees.     

AnO(n 2) incremental algorithm for modular decomposition of graphs and 2-structures

This paper gives anO(n ²) incremental algorithm for computing the modular decomposition of 2-structures [1], [2]. A 2-structure is a type of edge-colored graph, and its modular decomposition is also known as the prime tree family. Modular decomposition of 2-structures arises in the study of relational systems. The modular decomposition of undirected graphs and digraphs is a special case, and has applications in a number of combinatorial optimization problems. This algorithm generalizes elements of a previousO(n ²) algorithm of Muller and Spinrad [3] for the decomposition of undirected graphs. However, Muller and Spinrad's algorithm employs a sophisticated data structure that impedes its generalization to digraphs and 2-structures, and limits its practical use. We replace this data structure with a scheme that labels each edge with at most one node, thereby obtaining an algorithm that is both practical and general to 2-structures.     

Boolean Algebra as a Fragment of the Theory of Boolean Toposes

Basic identities of Boolean algebra are considered, and their categorical analogs are proved.     

Finding multiple induced disjoint paths in general graphs

In an undirected graph, paths P₁,P₂,…,Pk are induced disjoint if each one of them is chordless (i.e., is an induced path) and any two of them have neither common nodes nor adjacent nodes. This paper investigates the Maximum Induced Disjoint Paths (MIDP) problem: in an undirected graph G=(V,E), given k node pairs {s₁,t₁},…,{sk,tk}, connect maximum number of these node pairs via induced disjoint paths. Till now, the only things known about MIDP are: i) it is NP-hard; ii) it is NP-hard even when k=2; iii) it can be solved in polynomial time when k is a fixed constant and the given graph is a directed planar graph (Kobayashi, 2009 [9]). This paper proves that for general k and any ?>0, it is NP-hard to approximate MIDP within m1/2−?, where m=|E|. Two algorithms for MIDP are given by this paper: a greedy algorithm whose approximation ratio is and an on-line algorithm which has a good lower bound.     

On acyclic edge coloring of toroidal graphs

Let c be a proper edge coloring of a graph G. If there exists no bicolored cycle in G with respect to c, then c is called an acyclic edge coloring of G. Let G be a planar graph with maximum degree Δ and girth g. In Dong and Xu (2010) [8], Dong and Xu proved that G admits an acyclic edge coloring with Δ(G) colors if Δ?8 and g?7, or Δ?6 and g?8, or Δ?5 and g?9, or Δ?4 and g?10, or Δ?3 and g?14. In this note, we fix a small gap in the proof of Dong and Xu (2010) [8], and generalize the above results to toroidal graphs.     

Efficient approximation algorithms for shortest cycles in undirected graphs

We describe a simple combinatorial approximation algorithm for finding a shortest (simple) cycle in an undirected graph. Given an adjacency-list representation of an undirected graph G with n vertices and unknown girth k, our algorithm returns with high probability a cycle of length at most 2k for even k and 2k+2 for odd k, in time . Thus, in general, it yields a approximation. For a weighted, undirected graph, with non-negative edge weights in the range {1,2,…,M}, we present a simple combinatorial 2-approximation algorithm for a minimum weight (simple) cycle that runs in time O(n²logn(logn+logM)).     

Shortest paths in euclidean graphs

We analyze a simple method for finding shortest paths inEuclidean graphs (where vertices are points in a Euclidean space and edge weights are Euclidean distances between points). For many graph models, the average running time of the algorithm to find the shortest path between a specified pair of vertices in a graph withV vertices andE edges is shown to beO(V) as compared withO(E +V logV) required by the classical algorithm due to Dijkstra.Support for the first author was provided in part by NSF Grant MCS-83-08806. Support for the second author was provided in part by NSF Grants MCS-81-05324 and DCR-84-03613, an NSF Presidential Young Investigator Award, an IBM research contract, and an IBM Faculty Development Award. Support for this research was also provided in part by an ONR and DARPA under Contract N00014-83-K-0146 and ARPA Order No. 4786. Equipment support was provided by NSF Grant MCS-81-218106.     

Spanners of bounded degree graphs

A k-spanner of a graph G is a spanning subgraph of G in which the distance between any pair of vertices is at most k times the distance in G. We prove that for fixed k,w, the problem of deciding if a given graph has a k-spanner of treewidth w is fixed-parameter tractable on graphs of bounded degree. In particular, this implies that finding a k-spanner that is a tree (a tree k-spanner) is fixed-parameter tractable on graphs of bounded degree. In contrast, we observe that if the graph has only one vertex of unbounded degree, then Treek-Spanner is NP-complete for k?4.     

Computing the pathwidth of directed graphs with small vertex cover

We give an algorithm that computes the pathwidth of a given directed graph D   in 3^τ(D)n^O(1)$3^{\tau (D)}{n}^{O(1)}$ time where n is the number of vertices of D   and τ(D)$\tau (D)$ is the number of vertices of a minimum vertex cover of the underlying undirected graph of D. This result extends that of [5] for undirected graphs to directed graphs. Moreover, our algorithm is based on a standard dynamic programming with a simple tree-pruning trick, which is extremely simple and easy to implement. An additional advantage of our algorithm is that a minimum vertex cover appears in the analysis but not in the algorithm itself, in contrast to the algorithm of [5] which needs to compute a minimum vertex cover.     

Node overlap removal in clustered directed acyclic graphs

Graph drawing and visualization represent structural information as diagrams of abstract graphs and networks. An important subset of graphs is directed acyclic graphs (DAGs). This paper presents a new E-Spring algorithm, extended from the popular spring embedder model, which eliminates node overlaps in clustered DAGs. In this framework, nodes are modeled as non-uniform charged particles with weights, and a final drawing is derived by adjusting the positions of the nodes according to a combination of spring forces and repulsive forces derived from electrostatic forces between the nodes. The drawing process needs to reach a stable state when the average distances of separation between nodes are near optimal. We introduce a stopping condition for such a stable state, which reduces equilibrium distances between nodes and therefore results in a significantly reduced area for DAG visualization. It imposes an upper bound on the repulsive forces between nodes based on graph geometry. The algorithm employs node interleaving to eliminate any residual node overlaps. These new techniques have been validated by visualizing eBay buyer–seller relationships and has resulted in overall area reductions in the range of 45–79%.     

A Representation Theorem for MV-algebras

An MV-pair is a pair (B,G) where B is a Boolean algebra and G is a subgroup of the automorphism group of B satisfying certain conditions. Let ~ _G be the equivalence relation on B naturally associated with G. We prove that for every MV-pair (B,G), the effect algebra B/ ~ _G is an MV-effect algebra. Moreover, for every MV-effect algebra M there is an MV-pair (B,G) such that M is isomorphic to B/ ~ _G . This research is supported by grant VEGA G-1/3025/06 of MŠ SR, Slovakia and by the Science and Technology Assistance Agency under the contract No. APVT-51-032002.