1,720,993 research outputs found
When Can Graph Hyperbolicity Be Computed in Linear Time?
Full text linkHyperbolicity measures, in terms of (distance) metrics, how close a given graph is to being a tree. Due to its relevance in modeling real-world networks, hyperbolicity has seen intensive research over the last years. Unfortunately, the best known practical algorithms for computing the hyperbolicity number of a n-vertex graph have running time O(n4)O(n4) . Exploiting the framework of parameterized complexity analysis, we explore possibilities for “linear-time FPT” algorithms to compute hyperbolicity. For instance, we show that hyperbolicity can be computed in time 2O(k)+O(n+m)2O(k)+O(n+m) (m being the number of graph edges, k being the size of a vertex cover) while at the same time, unless the SETH fails, there is no 2o(k)n22o(k)n2 -time algorithm
Smoothed Analysis of Local Search Algorithms
Full text linkSmoothed analysis is a method for analyzing the performance of algorithms for which classical worst-case analysis fails to explain the performance observed in practice. Smoothed analysis has been applied to explain the performance of a variety of algorithms in the last years. One particular class of algorithms where smoothed analysis has been used successfully are local search algorithms. We give a survey of smoothed analysis, in particular applied to local search algorithms
Parameterized complexity of conflict-free graph coloring
Full text linkGiven a graph G, a q-open neighborhood conflict-free coloring or q-ONCF-coloring is a vertex coloring c: V(G) → { 1, 2, …, q} such that for each vertex v∈ V(G) there is a vertex in N(v) that is uniquely colored from the rest of the vertices in N(v). When we replace N(v) by the closed neighborhood N[v], then we call such a coloring a q-closed neighborhood conflict-free coloring or simply q-CNCF-coloring. In this paper, we study the NP-hard decision questions of whether for a constant q an input graph has a q-ONCF-coloring or a q-CNCF-coloring. We will study these two problems in the parameterized setting. First of all, we study running time bounds on FPT-algorithms for these problems, when parameterized by treewidth. We improve the existing upper bounds, and also provide lower bounds on the running time under ETH and SETH. Secondly, we study the kernelization complexity of both problems, using vertex cover as the parameter. We show that both (q≥ 2) -ONCF-coloring and (q≥ 3) -CNCF-coloring cannot have polynomial kernels when parameterized by the size of a vertex cover unless NP⊆ coNP/poly. On the other hand, we obtain a polynomial kernel for 2-CNCF-coloring parameterized by vertex cover. We conclude the study with some combinatorial results. Denote χON(G) and χCN(G) to be the minimum number of colors required to ONCF-color and CNCF-color G, respectively. Upper bounds on χCN(G) with respect to structural parameters like minimum vertex cover size, minimum feedback vertex set size and treewidth are known. To the best of our knowledge only an upper bound on χON(G) with respect to minimum vertex cover size was known. We provide tight bounds for χON(G) with respect to minimum vertex cover size. Also, we provide the first upper bounds on χON(G) with respect to minimum feedback vertex set size and treewidth
Neighborhood-Preserving Mapping between Trees
No full textWe introduce a variation of the graph isomorphism problem, where, given two graphs G = (V,E) and G = (V,E) and three integers l, d, and k, we seek for a set ⊆ V and a one-to-one mapping f:V → V such that |D| ≤ k and for every vertex v ∈ V \ D and every vertex u ∈ N (v) \ D we have f(u) ∈ N (f(v)). Here, for a graph G and a vertex v, we use N(v) to denote the set of vertices which have distance at most i to v in G. We call this problem Neighborhood-Preserving Mapping (NPM). The main result of this paper is a complete dichotomy of the classical complexity of NPM on trees with respect to different values of l,d,k. Additionally, we present two dynamic programming algorithms for the case that one of the input trees is a path
Online Bin Covering with Advice
Full text linkThe bin covering problem asks for covering a maximum number of bins with an online sequence of n items of different sizes in the range (0, 1]; a bin is said to be covered if it receives items of total size at least 1. We study this problem in the advice setting and provide tight bounds for the size of advice required to achieve optimal solutions. Moreover, we show that any algorithm with advice of size o(log log n) has a competitive ratio of at most 0.5. In other words, advice of size o(log log n) is useless for improving the competitive ratio of 0.5, attainable by an online algorithm without advice. This result highlights a difference between the bin covering and the bin packing problems in the advice model: for the bin packing problem, there are several algorithms with advice of constant size that outperform online algorithms without advice. Furthermore, we show that advice of size O(log log n) is sufficient to achieve a competitive ratio that is arbitrarily close to 0.53 3 ¯ and hence strictly better than the best ratio 0.5 attainable by purely online algorithms. The technicalities involved in introducing and analyzing this algorithm are quite different from the existing results for the bin packing problem and confirm the different nature of these two problems. Finally, we show that a linear number of bits of advice is necessary to achieve any competitive ratio better than 15/16 for the online bin covering problem.</p
Most vital segment barriers
Full text linkWe study continuous analogues of “vitality” for discrete network flows/paths, and consider problems related to placing segment barriers that have highest impact on a flow/path in a polygonal domain. This extends the graph-theoretic notion of “most vital arcs” for flows/paths to geometric environments. We give hardness results and efficient algorithms for various versions of the problem, (almost) completely separating hard and polynomially-solvable cases
Graph isomorphism for (H1,H2)-free graphs: an almost complete dichotomy
Full text linkWe almost completely resolve the computational complexity of Graph Isomorphism for classes of graphs characterized by two forbidden induced subgraphs H1 and H2. Schweitzer settled the complexity of this problem restricted to (H1;H2)-free graphs for all but a nite number of pairs (H1;H2), but without explicitly giving the number of open cases. Grohe and Schweitzer proved that Graph Isomorphism is polynomialtime solvable on graph classes of bounded clique-width. By combining known results with a number of new results, we reduce the number of open cases to seven. By exploiting the strong relationship between Graph Isomorphism and clique-width, we simultaneously reduce the number of open cases for boundedness of clique-width for (H1;H2)-free graphs to ve
Rectilinear computational geometry
No full textIn this thesis it is demonstrated that the structure of rectilinear polygons can be exploited to solve a variety of geometric problems efficiently. These problems include: (1) recognizing polygonal properties, such as star-shapedness, monotonicity, and edge-visibility, (2) removing hidden lines, (3) constructing the rectilinear convex hull, (4) decomposing rectilinear polygons into simpler components, and (5) placing guards in rectilinear polygons.A new tool for computational geometry is introduced which extracts information about the winding properties of rectilinear polygons. Employing this tool as a preprocessing step, efficient and conceptually clear algorithms for the above problems have been designed
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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