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.     

Max-Stretch Reduction for Tree Spanners

A tree t-spanner T of a graph G is a spanning tree of G whose max-stretch is t, i.e., the distance between any two vertices in T is at most t times their distance in G. If G has a tree t-spanner but not a tree (t−1)-spanner, then G is said to have max-stretch of t. In this paper, we study the Max-Stretch Reduction Problem: for an unweighted graph G=(V,E), find a set of edges not in E originally whose insertion into G can decrease the max-stretch of G. Our results are as follows: (i) For a ring graph, we give a linear-time algorithm which inserts k edges improving the max-stretch optimally. (ii) For a grid graph, we give a nearly optimal max-stretch reduction algorithm which preserves the structure of the grid. (iii) In the general case, we show that it is -hard to decide, for a given graph G and its spanning tree of max-stretch t, whether or not one-edge insertion can decrease the max-stretch to t−1. (iv) Finally, we show that the max-stretch of an arbitrary graph on n vertices can be reduced to s′≥2 by inserting O(n/s′) edges, which can be determined in linear time, and observe that this number of edges is optimal up to a constant.     

Spanners in sparse graphs

A t-spanner of a graph G is a spanning subgraph S in which the distance between every pair of vertices is at most t times their distance in G. If S is required to be a tree then S is called a tree t-spanner of G. In 1998, Fekete and Kremer showed that on unweighted planar graphs deciding whether G admits a tree t-spanner is polynomial time solvable for t?3 and is NP-complete when t is part of the input. They also left as an open problem if the problem is polynomial time solvable for every fixed t?4. In this work we resolve the open question of Fekete and Kremer by proving much more general results:

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  The problem of finding a t-spanner of treewidth at most k in a given planar graph G is fixed parameter tractable parameterized by k and t. Moreover, for every fixed t and k, the running time of our algorithm is linear.

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  Our technique allows to extend the result from planar graphs to much more general classes of graphs. An apex graph is a graph that can be made planar by the removal of a single vertex. We prove that the problem of finding a t-spanner of treewidth k is fixed parameter tractable on graphs that do not contain some fixed apex graph as a minor, i.e. on apex-minor-free graphs. The class of apex-minor-free graphs contains planar graphs and graphs of bounded genus.

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  Finally, we show that the tractability border of the t-spanner problem cannot be extended beyond the class of apex-minor-free graphs and in this sense our results are tight. In particular, for every t?4, the problem of finding a tree t-spanner is NP-complete on K6-minor-free graphs.

     

Additive tree 2-spanners of permutation graphs

Let G be a connected graph. A spanning tree T of G is a tree t-spanner if the distance between any two vertices in T is at most t times their distance in G. If their distances in T and G differ by at most t, then T is an additive tree t-spanner of G. In this paper, we show that any permutation graph has an additive tree 2-spanner, and it can be found in O(n) time sequentially or in O(log n) time with O(n/log n) processors on the EREW PRAM computational model by using a previously published algorithm for finding a tree 3-spanner of a permutation graph.     

Hybrid backtracking bounded by tree-decomposition of constraint networks

We propose a framework for solving CSPs based both on backtracking techniques and on the notion of tree-decomposition of the constraint networks. This mixed approach permits us to define a new framework for the enumeration, which we expect that it will benefit from the advantages of two approaches: a practical efficiency of enumerative algorithms and a warranty of a limited time complexity by an approximation of the tree-width of the constraint networks. Finally, experimental results allow us to show the advantages of this approach.     

Linearly bounded infinite graphs

Linearly bounded Turing machines have been mainly studied as acceptors for context-sensitive languages. We define a natural class of infinite automata representing their observable computational behavior, called linearly bounded graphs. These automata naturally accept the same languages as the linearly bounded machines defining them. We present some of their structural properties as well as alternative characterizations in terms of rewriting systems and context-sensitive transductions. Finally, we compare these graphs to rational graphs, which are another class of automata accepting the context-sensitive languages, and prove that in the bounded-degree case, rational graphs are a strict sub-class of linearly bounded graphs.A preliminary version of this article appeared in MFCS 2005.     

Packing triangles in bounded degree graphs

We consider the two problems of finding the maximum number of node disjoint triangles and edge disjoint triangles in an undirected graph. We show that the first (respectively second) problem is polynomially solvable if the maximum degree of the input graph is at most 3 (respectively 4), whereas it is APX-hard for general graphs and NP-hard for planar graphs if the maximum degree is 4 (respectively 5) or more.     

Speeding up dynamic transitive closure for bounded degree graphs

In this paper we present an algorithm for solving two problems in dynamically maintaining the transitive closure of a digraph: In the first problem a sequence of edge insertions is performed on an initially empty graph, interspersed withp transitive closure queries of the form: is there a path froma tob in the graph. Our algorithm solves this problem in timeO (dm ^*+p), whered is the maximum outdegree of the resulting graphG andm ^* is the number of edges in the transitive closure ofG. In the second problem, a sequence of edge deletions is performed on anacyclic graph, interspersed withp transitive closure queries. Once again we solve this problem in timeO (dm ^*+p), whered is the maximum outdegree of the initial graphG andm ^* is the number of edges in the transitive closure ofG. For bounded degree graphs, this improves upon previous results. Our algorithms also work when insertions and deletions to the graph are intermixed. Finally, we show how to implement the operation findpath (x, y) which retrieves some path fromx toy in time proportional to the length of the path.     

Geometric Spanners for Weighted Point Sets

Let (S,d) be a finite metric space, where each element p∈S has a non-negative weight w (p). We study spanners for the set S with respect to the following weighted distance function:

Improved Algorithms for Constructing Fault-Tolerant Spanners

Abstract. Let S be a set of n points in a metric space, and let k be a positive integer. Algorithms are given that construct k -fault-tolerant spanners for S . If in such a spanner at most k vertices and/ or edges are removed, then each pair of points in the remaining graph is still connected by a ``short' path. First, an algorithm is given that transforms an arbitrary spanner into a k -fault-tolerant spanner. For the Euclidean metric in R ^d , this leads to an O(n log n + c ^k n) -time algorithm that constructs a k -fault-tolerant spanner of degree O(c ^k ) , whose total edge length is O(c ^k ) times the weight of a minimum spanning tree of S , for some constant c . For constant values of k , this result is optimal. In the second part of the paper, algorithms are presented for the Euclidean metric in R ^d . These algorithms construct (i) in O(n log n + k ² n) time, a k -fault-tolerant spanner with O(k ² n) edges, and (ii) in O(k n log n) time, such a spanner with O(k n log n) edges.     

The monadic second-order logic of graphs,II: Infinite graphs of bounded width

A countable graph can be considered as the value of a certain infinite expression, represented itself by an infinite tree. We establish that the set of finite or infinite (expression) trees constructed with a finite number of operators, the value of which is a graph satisfying a property expressed in monadic second-order logic, is itself definable in monadic second-order logic. From Rabin's theorem, the emptiness of this set of (expression) trees is decidable. It follows that the monadic second-order theory of an equational graph, or of the set of countable graphs of width less than an integerm, is decidable. This work has been supported by the “Programme de Recherches Coordonnées: Mathématiques et Informatique.” Reprints can be requested by electronic mail at mcvax!inria!geocub!courcell (on UUCP network) or courcell@geocub.greco-prog.fr. Unité de Recherche associée au C.N.R.S. no. 726.     

Efficient frequent connected subgraph mining in graphs of bounded tree-width

The frequent connected subgraph mining problem, i.e., the problem of listing all connected graphs that are subgraph isomorphic to at least a certain number of transaction graphs of a database, cannot be solved in output polynomial time in the general case. If, however, the transaction graphs are restricted to forests then the problem becomes tractable. In this paper we generalize the positive result on forests to graphs of bounded tree-width. In particular, we show that for this class of transaction graphs, frequent connected subgraphs can be listed in incremental polynomial time. Since subgraph isomorphism remains NP-complete for bounded tree-width graphs, the positive complexity result of this paper shows that efficient frequent pattern mining is possible even for computationally hard pattern matching operators.     

Finite-time consensus for multi-agent systems with globally bounded convergence time under directed communication graphs

In this paper, we study the finite-time consensus problems with globally bounded convergence time also known as fixed-time consensus problems for multi-agent systems subject to directed communication graphs. Two new distributed control strategies are proposed such that leaderless and leader-follower consensus are achieved with convergence time independent on the initial conditions of the agents. Fixed-time formation generation and formation tracking problems are also solved as the generalizations. Simulation examples are provided to demonstrate the performance of the new controllers.     

Game chromatic number of graphs with locally bounded number of cycles

We consider the coloring game and the marking game on graphs with bounded number of cycles passing through any edge. We prove that the game coloring number of a graph G is at most c+4, if every edge of G belongs to at most c different cycles. This result covers two earlier bounds on the game coloring number: for trees (c=0) and for cactuses (c=1).     

Answering exact distance queries on real-world graphs with bounded performance guarantees

The ability to efficiently obtain exact distance information from both directed and undirected graphs is desired by many real-world applications. In this work, we unified the query indexing efforts on directed and undirected graphs into one by proposing the TreeMap approach. Our approach has very tight bounds on query time, index size, and construction time for answering queries on both directed and undirected graphs. The query time complexity is close to constant for graphs with a small width of tree decomposition, and the index construction can be completed without materializing the distance matrix or other high-cost operations. In the empirical study, we demonstrated that the TreeMap approach in general performs much better than competitive methods in indexing real graphs for answering exact distance queries.     

Topological additive numbering of directed acyclic graphs

We propose to study a problem that arises naturally from both Topological Numbering of Directed Acyclic Graphs, and Additive Coloring (also known as Lucky Labeling). Let D be a digraph and f   a labeling of its vertices with positive integers; denote by S(v)$S(v)$ the sum of labels over all neighbors of each vertex v. The labeling f is called topological additive numbering   if S(u)<S(v)$S(u)<S(v)$ for each arc (u,v)$(u,v)$ of the digraph. The problem asks to find the minimum number k for which D   has a topological additive numbering with labels belonging to {1,…,k}$\{1,\dots ,k\}$, denoted by _ηt(D)${\eta }_t(D)$.     

#BIS-hardness for 2-spin systems on bipartite bounded degree graphs in the tree non-uniqueness region

Counting independent sets on bipartite graphs (#BIS) is considered a canonical counting problem of intermediate approximation complexity. It is conjectured that #BIS neither has an FPRAS nor is as hard as #Sat to approximate. We study #BIS in the general framework of two-state spin systems on bipartite graphs. We define two notions, nearly-independent phase-correlated spins and unary symmetry breaking. We prove that it is #BIS-hard to approximate the partition function of any 2-spin system on bipartite graphs supporting these two notions. Consequently, we classify the complexity of approximating the partition function of antiferromagnetic 2-spin systems on bounded-degree bipartite graphs.