Pub Date : 2019-08-30DOI: 10.1016/j.entcs.2019.08.007
Paulo Henrique Macêdo de Araújo , Manoel Campêlo , Ricardo C. Corrêa , Martine Labbé
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 CRIO (Classification and Regression via Integer Optimization). 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 Geodesic Classification (GC) problem. We propose an integer programming formulation for the GC 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.
在近十年来整数优化研究取得重大进展的推动下,Bertsimas和Shioda针对多维空间分类的经典统计问题,开发了一种整数优化方法,并提供了一个名为CRIO (classification and Regression via integer optimization)的软件包。根据这些思想,我们定义了一个新的分类问题,探索其组合方面。这个问题是在图形上定义的,使用测地线凸性来类比多维空间中的欧几里得凸性。我们用测地线分类(GC)问题来表示这类问题。我们提出了GC问题的整数规划公式,并提出了分支切断算法来解决该问题。最后,通过计算实验验证了该方法的组合优化效率和分类精度。
{"title":"The Geodesic Classification Problem on Graphs","authors":"Paulo Henrique Macêdo de Araújo , Manoel Campêlo , Ricardo C. Corrêa , 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":"346 ","pages":"Pages 65-76"},"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}
Pub Date : 2019-08-30DOI: 10.1016/j.entcs.2019.08.008
Julio Araujo, Pedro Arraes
An oriented graph D is an orientation of a simple graph, i.e. a directed graph whose underlying graph is simple. A directed path from u to v with minimum number of arcs in D is an (u, v)-geodesic, for every u, v ∈ V(D). A set S ⊆ V(D) is (geodesically) convex if, for every u, v ∈ S, all the vertices in each (u, v)-geodesic and in each (v, u)-geodesic are in S. For every S ⊆ V(D) the (convex) hull of S is the smallest convex set containing S and it is denoted by [S]. A hull set of D is a set S ⊆ V(D) whose hull is V(D). The cardinality of a minimum hull set is the hull number of D and it is denoted by . A geodetic set of D is a set S ⊆ V(D) such that each vertex of D lies in an (u, v)-geodesic, for some u, v ∈ S. The cardinality of a minimum geodetic set is the geodetic number of D and it is denoted by .
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 is NP-hard for partial cubes, a subclass of bipartite graphs, and that computing 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.
{"title":"Hull and Geodetic Numbers for Some Classes of Oriented Graphs","authors":"Julio Araujo, 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":"346 ","pages":"Pages 77-88"},"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}
Pub Date : 2019-08-30DOI: 10.1016/j.entcs.2019.08.022
Márcia R. Cappelle , Erika M.M. Coelho , Hebert Coelho , Lucia D. Penso, Dieter Rautenbach
We show that an identifying code of minimum order in the complementary prism of a cycle of order n has order 7n/9 + Θ(1). Furthermore, we observe that the clique-width of the complementary prism of a graph of clique-width k is at most 4k, and discuss some algorithmic consequences.
{"title":"Identifying Codes in the Complementary Prism of Cycles","authors":"Márcia R. Cappelle , Erika M.M. Coelho , Hebert Coelho , Lucia D. Penso, 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":"346 ","pages":"Pages 241-251"},"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}
Pub Date : 2019-08-30DOI: 10.1016/j.entcs.2019.08.019
André Ebling Brondani, Carla Silva Oliveira, Francisca Andrea Macedo França, Leonardo de Lima
Let G be a connected graph of order n, A(G) is the adjacency matrix of G and D(G) is the diagonal matrix of the row-sums of A(G). In 2017, Nikiforov [Nikiforov, V., Merging the A- and Q-Spectral Theories, Applicable Analysis and Discrete Mathematics 11 (2017), pp. 81–107.] defined the convex linear combinations Aα(G) of A(G) and D(G) by In this paper, we obtain a partial factorization of the Aα-characteristic polynomial of the firefly graph which explicitly gives some eigenvalues of the graph.
{"title":"Aα-Spectrum of a Firefly Graph","authors":"André Ebling Brondani, Carla Silva Oliveira, Francisca Andrea Macedo França, 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><sub><em>α</em></sub>(<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><sub><em>α</em></sub>-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":"346 ","pages":"Pages 209-219"},"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}
Pub Date : 2019-08-30DOI: 10.1016/j.entcs.2019.08.002
Fabio H.N. Abe , Edna A. Hoshino , Alessandro Hill , Roberto Baldacci
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.
{"title":"A Branch-and-Price Algorithm for the Ring-Tree Facility Location Problem","authors":"Fabio H.N. Abe , Edna A. Hoshino , Alessandro Hill , 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":"346 ","pages":"Pages 3-14"},"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}
Pub Date : 2019-08-30DOI: 10.1016/j.entcs.2019.08.060
Anderson G. da Silva, Simone Dantas, Diana Sasaki
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 is the smallest integer for which the graph has an equitable total coloring. Wang (2002) conjectured that . In 1994, Fu proved that there exist equitable (Δ + 2)-total colorings for all complete r-partite p-balanced graphs of odd order. For the even case, he determined that . Silva, Dantas and Sasaki (2018) verified Wang's conjecture when G is a complete r-partite p-balanced graph, showing that if G has odd order, and if G has even order. In this work we improve this bound by showing that when G is a complete r-partite p-balanced graph with r ≥ 4 even and p even, and for r odd and p even.
{"title":"Equitable Total Chromatic Number of Kr×p for p Even","authors":"Anderson G. da Silva, Simone Dantas, 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":"346 ","pages":"Pages 685-697"},"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}
Pub Date : 2019-08-30DOI: 10.1016/j.entcs.2019.08.059
Allen Ibiapina, Ana Silva
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 G is the set Sb(G) of integers k such that G has a b-coloring with k colors and b(G) = maxSb(G) is the b-chromatic number of G. A graph is b-continous if . 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 G with girth at least 7 contains the integers between 2χ(G) and b(G). 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.
{"title":"Graphs with Girth at Least 8 are b-continuous","authors":"Allen Ibiapina, 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><sub><em>b</em></sub>(<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><sub><em>b</em></sub>(<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":"346 ","pages":"Pages 677-684"},"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}
Pub Date : 2019-08-30DOI: 10.1016/j.entcs.2019.08.063
C.A. Weffort-Santos, C.N. Campos, R.C.S. Schouery
A gap-[k]-vertex-labelling of a simple graph G = (V, E) is a pair (π, cπ) in which π : V (G) → {1, 2, ..., k} is an assignment of labels to the vertices of G and cπ : V (G) → {0, 1, ..., k} is a proper vertex-colouring of G such that, for every v ∈ V (G) of degree at least two, cπ(v) is induced by the largest difference, i.e. the largest gap, between the labels of its neighbours (cases where d(v) = 1 and d(v) = 0 are treated separately). Introduced in 2013 by A. Dehghan et al. [Dehghan, A., M. Sadeghi and A. Ahadi, Algorithmic complexity of proper labeling problems, Theoretical Computer Science 495 (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.
{"title":"On the Complexity of Gap-[2]-vertex-labellings of Subcubic Bipartite Graphs","authors":"C.A. Weffort-Santos, C.N. Campos, 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><sub><em>π</em></sub>) 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><sub><em>π</em></sub> : <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><sub><em>π</em></sub>(<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":"346 ","pages":"Pages 725-734"},"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}
Pub Date : 2019-08-30DOI: 10.1016/j.entcs.2019.08.004
Alexsandro Oliveira Alexandrino , Guilherme Henrique Santos Miranda , Carla Negri Lintzmayer , Zanoni Dias
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.
{"title":"Approximation Algorithms for Sorting Permutations by Length-Weighted Short Rearrangements","authors":"Alexsandro Oliveira Alexandrino , Guilherme Henrique Santos Miranda , Carla Negri Lintzmayer , 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":"346 ","pages":"Pages 29-40"},"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}
Pub Date : 2019-08-30DOI: 10.1016/j.entcs.2019.08.034
Ana Paula dos Santos Dantas , Cid Carvalho de Souza , Zanoni Dias
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.
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