Electronic Notes in Theoretical Computer Science最新文献

{"title":"The Geodesic Classification Problem on Graphs","authors":"Paulo Henrique Macêdo de Araújo ,&nbsp;Manoel Campêlo ,&nbsp;Ricardo C. Corrêa ,&nbsp;Martine Labbé","doi":"10.1016/j.entcs.2019.08.007","DOIUrl":"10.1016/j.entcs.2019.08.007","url":null,"abstract":"<div><p>Motivated by the significant advances in integer optimization in the past decade, Bertsimas and Shioda developed an integer optimization method to the classical statistical problem of classification in a multidimensional space, delivering a software package called <em>CRIO</em> (<em>Classification and Regression via Integer Optimization</em>). Following those ideas, we define a new classification problem, exploring its combinatorial aspects. That problem is defined on graphs using the geodesic convexity as an analogy of the Euclidean convexity in the multidimensional space. We denote such a problem by <em>Geodesic Classification</em> (<em>GC</em>) problem. We propose an integer programming formulation for the <em>GC</em> problem along with a branch-and-cut algorithm to solve it. Finally, we show computational experiments in order to evaluate the combinatorial optimization efficiency and classification accuracy of the proposed approach.</p></div>","PeriodicalId":38770,"journal":{"name":"Electronic Notes in Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.entcs.2019.08.007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133987059","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Hull and Geodetic Numbers for Some Classes of Oriented Graphs","authors":"Julio Araujo,&nbsp;Pedro Arraes","doi":"10.1016/j.entcs.2019.08.008","DOIUrl":"10.1016/j.entcs.2019.08.008","url":null,"abstract":"<div><p>An oriented graph <em>D</em> is an orientation of a simple graph, i.e. a directed graph whose underlying graph is simple. A directed path from <em>u</em> to <em>v</em> with minimum number of arcs in <em>D</em> is an (<em>u</em>, <em>v</em>)-geodesic, for every <em>u</em>, <em>v</em> ∈ <em>V</em>(<em>D</em>). A set <em>S</em> ⊆ <em>V</em>(<em>D</em>) is (geodesically) convex if, for every <em>u</em>, <em>v</em> ∈ <em>S</em>, all the vertices in each (<em>u</em>, <em>v</em>)-geodesic and in each (<em>v</em>, <em>u</em>)-geodesic are in <em>S</em>. For every <em>S</em> ⊆ <em>V</em>(<em>D</em>) the (convex) hull of <em>S</em> is the smallest convex set containing <em>S</em> and it is denoted by [<em>S</em>]. A hull set of <em>D</em> is a set <em>S</em> ⊆ <em>V</em>(<em>D</em>) whose hull is <em>V</em>(<em>D</em>). The cardinality of a minimum hull set is the hull number of <em>D</em> and it is denoted by <span><math><mover><mrow><mi>h</mi><mi>n</mi></mrow><mrow><mo>→</mo></mrow></mover><mo>(</mo><mi>D</mi><mo>)</mo></math></span>. A geodetic set of <em>D</em> is a set <em>S</em> ⊆ <em>V</em>(<em>D</em>) such that each vertex of <em>D</em> lies in an (<em>u</em>, <em>v</em>)-geodesic, for some <em>u</em>, <em>v</em> ∈ <em>S</em>. The cardinality of a minimum geodetic set is the geodetic number of <em>D</em> and it is denoted by <span><math><mover><mrow><mi>g</mi><mi>n</mi></mrow><mrow><mo>→</mo></mrow></mover><mo>(</mo><mi>D</mi><mo>)</mo></math></span>.</p><p>In this work, we first present an upper bound for the hull number of oriented split graphs. Then, we turn our attention to the computational complexity of determining such parameters. We first show that computing <span><math><mover><mrow><mi>h</mi><mi>n</mi></mrow><mrow><mo>→</mo></mrow></mover><mo>(</mo><mi>D</mi><mo>)</mo></math></span> is NP-hard for partial cubes, a subclass of bipartite graphs, and that computing <span><math><mover><mrow><mi>g</mi><mi>n</mi></mrow><mrow><mo>→</mo></mrow></mover><mo>(</mo><mi>D</mi><mo>)</mo></math></span> is also NP-hard for directed acyclic graphs (DAG). Finally, we present a positive result by showing how to compute such parameters in polynomial time when the input graph is an oriented cactus.</p></div>","PeriodicalId":38770,"journal":{"name":"Electronic Notes in Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.entcs.2019.08.008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131398275","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Identifying Codes in the Complementary Prism of Cycles","authors":"Márcia R. Cappelle ,&nbsp;Erika M.M. Coelho ,&nbsp;Hebert Coelho ,&nbsp;Lucia D. Penso,&nbsp;Dieter Rautenbach","doi":"10.1016/j.entcs.2019.08.022","DOIUrl":"10.1016/j.entcs.2019.08.022","url":null,"abstract":"<div><p>We show that an identifying code of minimum order in the complementary prism of a cycle of order <em>n</em> has order 7<em>n/</em>9 + Θ(1). Furthermore, we observe that the clique-width of the complementary prism of a graph of clique-width <em>k</em> is at most 4<em>k</em>, and discuss some algorithmic consequences.</p></div>","PeriodicalId":38770,"journal":{"name":"Electronic Notes in Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.entcs.2019.08.022","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131735532","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Aα-Spectrum of a Firefly Graph","authors":"André Ebling Brondani,&nbsp;Carla Silva Oliveira,&nbsp;Francisca Andrea Macedo França,&nbsp;Leonardo de Lima","doi":"10.1016/j.entcs.2019.08.019","DOIUrl":"10.1016/j.entcs.2019.08.019","url":null,"abstract":"<div><p>Let <em>G</em> be a connected graph of order <em>n</em>, <em>A</em>(<em>G</em>) is the adjacency matrix of <em>G</em> and <em>D</em>(<em>G</em>) is the diagonal matrix of the row-sums of <em>A</em>(<em>G</em>). In 2017, Nikiforov [Nikiforov, V., <em>Merging the A- and Q-Spectral Theories</em>, Applicable Analysis and Discrete Mathematics <strong>11</strong> (2017), pp. 81–107.] defined the convex linear combinations <em>A</em>_<em>α</em>(<em>G</em>) of <em>A</em>(<em>G</em>) and <em>D</em>(<em>G</em>) by<span><span><span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>α</mi><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mo>(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo>)</mo><mi>A</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>,</mo><mspace></mspace><mn>0</mn><mo>≤</mo><mi>α</mi><mo>≤</mo><mn>1</mn><mo>.</mo></math></span></span></span> In this paper, we obtain a partial factorization of the <em>A</em>_<em>α</em>-characteristic polynomial of the firefly graph which explicitly gives some eigenvalues of the graph.</p></div>","PeriodicalId":38770,"journal":{"name":"Electronic Notes in Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.entcs.2019.08.019","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131694451","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Branch-and-Price Algorithm for the Ring-Tree Facility Location Problem","authors":"Fabio H.N. Abe ,&nbsp;Edna A. Hoshino ,&nbsp;Alessandro Hill ,&nbsp;Roberto Baldacci","doi":"10.1016/j.entcs.2019.08.002","DOIUrl":"10.1016/j.entcs.2019.08.002","url":null,"abstract":"<div><p>The ring-tree facility location problem is a generalization of the capacitated ring-tree problem in which additional cost and capacity related to facilities are considered. Applications of this problem arise in the strategic design of bi-level telecommunication networks. We investigate an extended integer programming formulation for the problem and different approaches to deal with the NP-hardness of the pricing problem that appears in a branch-and-price algorithm to solve it. Computational experiments show how heuristics and relaxations improved the performance of a branch-and-price algorithm.</p></div>","PeriodicalId":38770,"journal":{"name":"Electronic Notes in Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.entcs.2019.08.002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126800255","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Equitable Total Chromatic Number of Kr×p for p Even","authors":"Anderson G. da Silva,&nbsp;Simone Dantas,&nbsp;Diana Sasaki","doi":"10.1016/j.entcs.2019.08.060","DOIUrl":"10.1016/j.entcs.2019.08.060","url":null,"abstract":"<div><p>A total coloring is equitable if the number of elements colored by any two distinct colors differs by at most one. The equitable total chromatic number of a graph <span><math><mo>(</mo><msubsup><mrow><mi>χ</mi></mrow><mrow><mi>e</mi></mrow><mrow><mo>″</mo></mrow></msubsup><mo>)</mo></math></span> is the smallest integer for which the graph has an equitable total coloring. Wang (2002) conjectured that <span><math><mi>Δ</mi><mo>+</mo><mn>1</mn><mo>≤</mo><msubsup><mrow><mi>χ</mi></mrow><mrow><mi>e</mi></mrow><mrow><mo>″</mo></mrow></msubsup><mo>≤</mo><mi>Δ</mi><mo>+</mo><mn>2</mn></math></span>. In 1994, Fu proved that there exist equitable (Δ + 2)-total colorings for all complete <em>r</em>-partite <em>p</em>-balanced graphs of odd order. For the even case, he determined that <span><math><msubsup><mrow><mi>χ</mi></mrow><mrow><mi>e</mi></mrow><mrow><mo>″</mo></mrow></msubsup><mo>≤</mo><mi>Δ</mi><mo>+</mo><mn>3</mn></math></span>. Silva, Dantas and Sasaki (2018) verified Wang's conjecture when <em>G</em> is a complete <em>r</em>-partite <em>p</em>-balanced graph, showing that <span><math><msubsup><mrow><mi>χ</mi></mrow><mrow><mi>e</mi></mrow><mrow><mo>″</mo></mrow></msubsup><mo>=</mo><mi>Δ</mi><mo>+</mo><mn>1</mn></math></span> if <em>G</em> has odd order, and <span><math><msubsup><mrow><mi>χ</mi></mrow><mrow><mi>e</mi></mrow><mrow><mo>″</mo></mrow></msubsup><mo>≤</mo><mi>Δ</mi><mo>+</mo><mn>2</mn></math></span> if <em>G</em> has even order. In this work we improve this bound by showing that <span><math><msubsup><mrow><mi>χ</mi></mrow><mrow><mi>e</mi></mrow><mrow><mo>″</mo></mrow></msubsup><mo>=</mo><mi>Δ</mi><mo>+</mo><mn>1</mn></math></span> when <em>G</em> is a complete <em>r</em>-partite <em>p</em>-balanced graph with <em>r</em> ≥ 4 even and <em>p</em> even, and for <em>r</em> odd and <em>p</em> even.</p></div>","PeriodicalId":38770,"journal":{"name":"Electronic Notes in Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.entcs.2019.08.060","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124380384","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Graphs with Girth at Least 8 are b-continuous","authors":"Allen Ibiapina,&nbsp;Ana Silva","doi":"10.1016/j.entcs.2019.08.059","DOIUrl":"10.1016/j.entcs.2019.08.059","url":null,"abstract":"<div><p>A b-coloring of a graph is a proper coloring such that each color class has at least one vertex which is adjacent to each other color class. The b-spectrum of <em>G</em> is the set <em>S</em>_<em>b</em>(<em>G</em>) of integers <em>k</em> such that <em>G</em> has a b-coloring with <em>k</em> colors and <em>b</em>(<em>G</em>) = max<em>S</em>_<em>b</em>(<em>G</em>) is the b-chromatic number of <em>G</em>. A graph is b-continous if <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mo>[</mo><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>,</mo><mi>b</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>]</mo><mo>∩</mo><mi>Z</mi></math></span>. An infinite number of graphs that are not b-continuous is known. It is also known that graphs with girth at least 10 are b-continuous. In this work, we prove that graphs with girth at least 8 are b-continuous, and that the b-spectrum of a graph <em>G</em> with girth at least 7 contains the integers between 2<em>χ</em>(<em>G</em>) and <em>b</em>(<em>G</em>). This generalizes a previous result by Linhares-Sales and Silva (2017), and tells that graphs with girth at least 7 are, in a way, almost b-continuous.</p></div>","PeriodicalId":38770,"journal":{"name":"Electronic Notes in Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.entcs.2019.08.059","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123747113","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the Complexity of Gap-[2]-vertex-labellings of Subcubic Bipartite Graphs","authors":"C.A. Weffort-Santos,&nbsp;C.N. Campos,&nbsp;R.C.S. Schouery","doi":"10.1016/j.entcs.2019.08.063","DOIUrl":"10.1016/j.entcs.2019.08.063","url":null,"abstract":"<div><p>A gap-[<em>k</em>]-vertex-labelling of a simple graph <em>G</em> = (<em>V</em>, <em>E</em>) is a pair (<em>π</em>, <em>c</em>_<em>π</em>) in which <em>π</em> : <em>V</em> (<em>G</em>) → {1, 2, ..., <em>k</em>} is an assignment of labels to the vertices of <em>G</em> and <em>c</em>_<em>π</em> : <em>V</em> (<em>G</em>) → {0, 1, ..., <em>k</em>} is a proper vertex-colouring of <em>G</em> such that, for every <em>v</em> ∈ <em>V</em> (<em>G</em>) of degree at least two, <em>c</em>_<em>π</em>(<em>v</em>) is induced by the largest difference, i.e. the largest gap, between the labels of its neighbours (cases where <em>d</em>(<em>v</em>) = 1 and <em>d</em>(<em>v</em>) = 0 are treated separately). Introduced in 2013 by A. Dehghan et al. [Dehghan, A., M. Sadeghi and A. Ahadi, <em>Algorithmic complexity of proper labeling problems</em>, Theoretical Computer Science <strong>495</strong> (2013), pp. 25–36.], they show that deciding whether a bipartite graph admits a gap-[2]-vertex-labelling is NP-complete and question the computational complexity of deciding whether cubic bipartite graphs admit such a labelling. In this work, we advance the study of the computational complexity for this class, proving that this problem remains NP-complete even when restricted to subcubic bipartite graphs.</p></div>","PeriodicalId":38770,"journal":{"name":"Electronic Notes in Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.entcs.2019.08.063","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121777892","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Approximation Algorithms for Sorting Permutations by Length-Weighted Short Rearrangements","authors":"Alexsandro Oliveira Alexandrino ,&nbsp;Guilherme Henrique Santos Miranda ,&nbsp;Carla Negri Lintzmayer ,&nbsp;Zanoni Dias","doi":"10.1016/j.entcs.2019.08.004","DOIUrl":"10.1016/j.entcs.2019.08.004","url":null,"abstract":"<div><p>Genome rearrangements are events that affect large portions of a genome. When using the rearrangement distance to compare two genomes, one wants to find a minimum cost sequence of rearrangements that transforms one into another. Since we represent genomes as permutations, we can reduce this problem to the problem of sorting a permutation with a minimum cost sequence of rearrangements. In the traditional approach, we consider that all rearrangements are equally likely to occur and we set a unitary cost for all rearrangements. However, there are two variations of the problem motivated by the observation that rearrangements involving large segments of a genome rarely occur. The first variation adds a restriction to the rearrangement's length. The second variation uses a cost function based on the rearrangement's length. In this work, we present approximation algorithms for five problems combining both variations, that is, problems with a length-limit restriction and a cost function based on the rearrangement's length.</p></div>","PeriodicalId":38770,"journal":{"name":"Electronic Notes in Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.entcs.2019.08.004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115154188","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A GRASP for the Convex Recoloring Problem in Graphs","authors":"Ana Paula dos Santos Dantas ,&nbsp;Cid Carvalho de Souza ,&nbsp;Zanoni Dias","doi":"10.1016/j.entcs.2019.08.034","DOIUrl":"10.1016/j.entcs.2019.08.034","url":null,"abstract":"<div><p>In this paper, we consider a coloring as a function that assigns a color to a vertex, regardless of the color of its neighbors. The Convex Recoloring Problem finds the minimum number of recolored vertices needed to turn a coloring convex, that is, every set formed by all the vertices with the same color induces a connected subgraph. The problem is most commonly studied considering trees due to its origins in the study of phylogenetic trees, but in this paper, we focus on general graphs and propose a GRASP heuristic to solve the problem. We present computational experiments for our heuristic and compare it to an Integer Linear Programming model from the literature. In these experiments, the GRASP algorithm recolored a similar number of vertices than the model from the literature, and used considerably less time. We also introduce a set of benchmark instances for the problem.</p></div>","PeriodicalId":38770,"journal":{"name":"Electronic Notes in Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.entcs.2019.08.034","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133238800","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}