Electronic Notes in Theoretical Computer Science最新文献

The Bi-objective Spanning Tree (BiST) is an NP-hard extension of the Minimum Spanning Tree (MST) problem. The BiST models situations in which two conflicting objectives need to be optimized simultaneously. The BiST has been studied in the literature and several heuristic algorithms were proposed for it, most of them evolutionary techniques. The transgenetic algorithms are among these evolutionary techniques which were successfully applied to the BiST. However, a priori defined parameters can limit the search mechanisms used within the algorithm. In this study, we propose a new transgenetic algorithm for the BiST in which the decision about the search mechanisms used along its execution is automatically made. An analysis of the results of computational experiments carried on 165 benchmark instances showed that the algorithm proposed in this study produces good approximation sets concerning two different quality indicators.

Graph searches and the corresponding search trees can exhibit important structural properties and are used in various graph algorithms. The problem of deciding whether a given spanning tree of a graph is a search tree of a particular search on this graph was introduced by Hagerup and Nowak in 1985, and independently by Korach and Ostfeld in 1989 where the authors showed that this problem is efficiently solvable for DFS trees. A linear time algorithm for BFS trees was obtained by Manber in 1990. In this paper we prove that the search tree problem is also in P for LDFS, in contrast to LBFS, MCS, and MNS, where we show NP-completeness. We complement our results by providing linear time algorithms for these searches on split graphs.

In this paper we present approximation preserving reductions from the Budgeted and Generalized Maximum Coverage Problems to the Knapsack Problem with Conflict Graphs. The reductions are used to yield Polynomial Time Approximation Schemes for special classes of instances of these problems. Using these approximation schemes, the existence of pseudo-polynomial algorithms are proven and, in more particular cases, these algorithms are shown to have polynomial time complexity. Moreover, the characteristics of the instances that admit these algorithms are analyzed.

In this paper, we deal with counting and enumerating problems for two types of graph orientations: acyclic and totally cyclic orientations. Counting is known to be #P-hard for both of them. To circumvent this issue, we propose Fixed Parameter Tractable (FPT) algorithms. For the enumeration task, we construct a Binary Decision Diagram (BDD) to represent all orientations of the two kinds, instead of explicitly enumerating them. We prove that the running time of this construction is bounded by O*(2^pw2/4+o(pw2)) with respect to the pathwidth pw. We then develop faster FPT algorithms to count acyclic and totally acyclic orientations, running in O*(2^bw2/2+o(bw2)) time, where bw denotes the branch-width of the given graph. These counting algorithms are obtained by applying the observations in our enumerating algorithm to branch decomposition.

Let $F$ be a family of intervals on the real line. An interval graph is the intersection graph of $F$. An interval order is a partial order $(F,\prec )$ such that for all $I_1,I_2\in F$, I₁ ≺ I₂ if and only if I₁ lies entirely at the left of I₂. Such a family $F$ is called a model of the graph (order). The interval count of a given graph (resp. order) is the smallest number of interval lengths needed in any model of this graph (resp. order). The first problem we consider is related to the classes of graphs and orders which can be represented with two interval lengths, regarding to the inclusion hierarchy among such classes. The second problem is an extremal problem which consists of determining the smallest graph or order which has interval count at least k. In particular, we study a conjecture by Fishburn on this extremal problem, verifying its validity when such a conjecture is constrained to the classes of trivially perfect orders and split orders.

The metric dimension MD(G) of an undirected graph G is the cardinality of a smallest set of vertices that allows, through their distances to all vertices, to distinguish any two vertices of G. Many aspects of this notion have been investigated since its introduction in the 70's, including its generalization to digraphs. In this work, we study, for particular graph families, the maximum metric dimension over all strongly-connected orientations, by exhibiting lower and upper bounds on this value. We first exhibit general bounds for graphs with bounded maximum degree. In particular, we prove that, in the case of subcubic n-node graphs, all strongly-connected orientations asymptotically have metric dimension at most $\frac{n}{2}$, and that there are such orientations having metric dimension $\frac{2n}{5}$. We then consider strongly-connected orientations of grids. For a torus with n rows and m columns, we show that the maximum value of the metric dimension of a strongly-connected Eulerian orientation is asymptotically $\frac{nm}{2}$ (the equality holding when n, m are even, which is best possible). For a grid with n rows and m columns, we prove that all strongly-connected orientations asymptotically have metric dimension at most $\frac{2nm}{3}$, and that there are such orientations having metric dimension $\frac{nm}{2}$.

Originally proved in 1986 by Robertson and Seymour, the Grid Theorem is one of the most important tools in the field of structural graph theory, finding numerous applications in the design of algorithms for undirected graphs. An analogous version of the Grid Theorem isn directed graphs was conjectured by Johnson et al. in 2001, and proved recently by Kawarabayashi and Kreutzer in 2015. Namely, they showed that there is a function f(k) such that every directed graph of directed tree-width at least f(k) contains a cylindrical grid of size k as a butterfly minor. Moreover, they claim that their proof can be turned into an XP algorithm, with parameter k, that either constructs a decomposition of the appropriate width, or finds the claimed large cylindrical grid as a butterfly minor. In this article, we adapt some of the steps of the proof of Kawarabayashi and Kreutzer and we improve the XP algorithm into an FPT algorithm.

The first step of the proof is an XP algorithm by Johnson et al. in 2001 that decides whether a directed graph D has directed tree-width at most 3k − 2 or admits a haven of order k. It is worth mentioning that a skecth of an FPT algorithm for this problem appears in Chapter 9 of the book ”Classes of Directed Graphs”, from 2018, with an approximation factor of 5k + 2. Our first contribution is to adapt the proof from Johnson et al. to find either an arboreal decomposition of width at most 3k − 2 or a haven of order k in a directed graph D in FPT time, by making use of important separators. We then follow the roadmap of the proof by Kawarabayashi and Kreutzer by locally improving the complexity at some steps, in particular concerning the problem of finding hitting sets for brambles of large order.

We consider a multicolored version of a problem that was originally proposed by Erdős and Rothschild. For positive integers n and r, we look for n-vertex graphs that admit the maximum number of r-edge-colorings with no copy of a triangle where exactly two colors appear. It turns out that for 2 ≤ r ≤ 12 colors and n sufficiently large, the complete bipartite graph on n vertices with balanced bipartition (the n-vertex Turán graph for the triangle) yields the largest number of such colorings, and this graph is unique with this property.

This paper is inspired in the well known characterization of chordal graphs as the intersection graphs of subtrees of a tree. We consider families of induced trees of any graph and we prove that their recognition is NP-Complete. A consequence of this fact is that the concept of clique tree of chordal graphs cannot be widely generalized. Finally, we consider the fact that every graph is the intersection graph of induced trees of a bipartite graph and we characterize some classes that arise when we impose restrictions on the host bipartite graph.

A path partition $P$ of a digraph D is a collection of directed paths such that every vertex belongs to precisely one path. Given a positive integer k, the k-norm of a path partition $P$ of D is defined as ${\sum }_{P\in P}\min \left\{\right|P_i|,k\}$. A path partition of a minimum k-norm is called k-optimal and its k-norm is denoted by π_k(D). A stable set of a digraph D is a subset of pairwise non-adjacent vertices of V(D). Given a positive integer k, we denote by α_k(D) the largest set of vertices of D that can be decomposed into k disjoint stable sets of D. In 1981, Linial conjectured that π_k(D) ≤ α_k(D) for every digraph. We say that a digraph D is arc-spine if V(D) can be partitioned into two sets X and Y where X is traceable and Y contains at most one arc in A(D). In this paper we show the validity of Linial's Conjecture for arc-spine digraphs.