arXiv - CS - Discrete Mathematics最新文献_第5页

Bicriterial Approximation for the Incremental Prize-Collecting Steiner-Tree Problem 增量奖品收集斯坦纳树问题的双标准近似法

arXiv - CS - Discrete Mathematics Pub Date : 2024-07-05 DOI: arxiv-2407.04447

Yann Disser, Svenja M. Griesbach, Max Klimm, Annette Lutz

{"title":"Bicriterial Approximation for the Incremental Prize-Collecting Steiner-Tree Problem","authors":"Yann Disser, Svenja M. Griesbach, Max Klimm, Annette Lutz","doi":"arxiv-2407.04447","DOIUrl":"https://doi.org/arxiv-2407.04447","url":null,"abstract":"We consider an incremental variant of the rooted prize-collecting\u0000Steiner-tree problem with a growing budget constraint. While no incremental\u0000solution exists that simultaneously approximates the optimum for all budgets,\u0000we show that a bicriterial $(alpha,mu)$-approximation is possible, i.e., a\u0000solution that with budget $B+alpha$ for all $B in mathbb{R}_{geq 0}$ is a\u0000multiplicative $mu$-approximation compared to the optimum solution with budget\u0000$B$. For the case that the underlying graph is a tree, we present a\u0000polynomial-time density-greedy algorithm that computes a\u0000$(chi,1)$-approximation, where $chi$ denotes the eccentricity of the root\u0000vertex in the underlying graph, and show that this is best possible. An\u0000adaptation of the density-greedy algorithm for general graphs is\u0000$(gamma,2)$-competitive where $gamma$ is the maximal length of a\u0000vertex-disjoint path starting in the root. While this algorithm does not run in\u0000polynomial time, it can be adapted to a $(gamma,3)$-competitive algorithm that\u0000runs in polynomial time. We further devise a capacity-scaling algorithm that\u0000guarantees a $(3chi,8)$-approximation and, more generally, a\u0000$smash{bigl((4ell - 1)chi, frac{2^{ell +\u00002}}{2^{ell}-1}bigr)}$-approximation for every fixed $ell in mathbb{N}$.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573801","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Flip Dynamics for Sampling Colorings: Improving $(11/6-ε)$ Using a Simple Metric 采样着色的翻转动力学：使用简单度量改进 $(11/6-ε)$

arXiv - CS - Discrete Mathematics Pub Date : 2024-07-05 DOI: arxiv-2407.04870

Charlie Carlson, Eric Vigoda

{"title":"Flip Dynamics for Sampling Colorings: Improving $(11/6-ε)$ Using a Simple Metric","authors":"Charlie Carlson, Eric Vigoda","doi":"arxiv-2407.04870","DOIUrl":"https://doi.org/arxiv-2407.04870","url":null,"abstract":"We present improved bounds for randomly sampling $k$-colorings of graphs with\u0000maximum degree $Delta$; our results hold without any further assumptions on\u0000the graph. The Glauber dynamics is a simple single-site update Markov chain.\u0000Jerrum (1995) proved an optimal $O(nlog{n})$ mixing time bound for Glauber\u0000dynamics whenever $k>2Delta$ where $Delta$ is the maximum degree of the input\u0000graph. This bound was improved by Vigoda (1999) to $k > (11/6)Delta$ using a\u0000\"flip\" dynamics which recolors (small) maximal 2-colored components in each\u0000step. Vigoda's result was the best known for general graphs for 20 years until\u0000Chen et al. (2019) established optimal mixing of the flip dynamics for $k >\u0000(11/6 - epsilon ) Delta$ where $epsilon approx 10^{-5}$. We present the\u0000first substantial improvement over these results. We prove an optimal mixing\u0000time bound of $O(nlog{n})$ for the flip dynamics when $k geq 1.809 Delta$.\u0000This yields, through recent spectral independence results, an optimal\u0000$O(nlog{n})$ mixing time for the Glauber dynamics for the same range of\u0000$k/Delta$ when $Delta=O(1)$. Our proof utilizes path coupling with a simple\u0000weighted Hamming distance for \"unblocked\" neighbors.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573798","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Improved Outerplanarity Bounds for Planar Graphs 平面图的改进外平面性边界

arXiv - CS - Discrete Mathematics Pub Date : 2024-07-05 DOI: arxiv-2407.04282

Therese Biedl, Debajyoti Mondal

引用次数: 0

On the Connectivity of the Flip Graph of Plane Spanning Paths 论平面跨路径翻转图的连通性

arXiv - CS - Discrete Mathematics Pub Date : 2024-07-04 DOI: arxiv-2407.03912

Linda Kleist, Peter Kramer, Christian Rieck

{"title":"On the Connectivity of the Flip Graph of Plane Spanning Paths","authors":"Linda Kleist, Peter Kramer, Christian Rieck","doi":"arxiv-2407.03912","DOIUrl":"https://doi.org/arxiv-2407.03912","url":null,"abstract":"Flip graphs of non-crossing configurations in the plane are widely studied\u0000objects, e.g., flip graph of triangulations, spanning trees, Hamiltonian\u0000cycles, and perfect matchings. Typically, it is an easy exercise to prove\u0000connectivity of a flip graph. In stark contrast, the connectivity of the flip\u0000graph of plane spanning paths on point sets in general position has been an\u0000open problem for more than 16 years. In order to provide new insights, we investigate certain induced subgraphs.\u0000Firstly, we provide tight bounds on the diameter and the radius of the flip\u0000graph of spanning paths on points in convex position with one fixed endpoint.\u0000Secondly, we show that so-called suffix-independent paths induce a connected\u0000subgraph. Consequently, to answer the open problem affirmatively, it suffices\u0000to show that each path can be flipped to some suffix-independent path. Lastly,\u0000we investigate paths where one endpoint is fixed and provide tools to flip to\u0000suffix-independent paths. We show that these tools are strong enough to show\u0000connectivity of the flip graph of plane spanning paths on point sets with at\u0000most two convex layers.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573803","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Algorithmic Results for Weak Roman Domination Problem in Graphs 图中弱罗马支配问题的算法结果

arXiv - CS - Discrete Mathematics Pub Date : 2024-07-04 DOI: arxiv-2407.03812

Kaustav Paul, Ankit Sharma, Arti Pandey

{"title":"Algorithmic Results for Weak Roman Domination Problem in Graphs","authors":"Kaustav Paul, Ankit Sharma, Arti Pandey","doi":"arxiv-2407.03812","DOIUrl":"https://doi.org/arxiv-2407.03812","url":null,"abstract":"Consider a graph $G = (V, E)$ and a function $f: V rightarrow {0, 1, 2}$.\u0000A vertex $u$ with $f(u)=0$ is defined as emph{undefended} by $f$ if it lacks\u0000adjacency to any vertex with a positive $f$-value. The function $f$ is said to\u0000be a emph{Weak Roman Dominating function} (WRD function) if, for every vertex\u0000$u$ with $f(u) = 0$, there exists a neighbour $v$ of $u$ with $f(v) > 0$ and a\u0000new function $f': V rightarrow {0, 1, 2}$ defined in the following way:\u0000$f'(u) = 1$, $f'(v) = f(v) - 1$, and $f'(w) = f(w)$, for all vertices $w$ in\u0000$Vsetminus{u,v}$; so that no vertices are undefended by $f'$. The total\u0000weight of $f$ is equal to $sum_{vin V} f(v)$, and is denoted as $w(f)$. The\u0000emph{Weak Roman Domination Number} denoted by $gamma_r(G)$, represents\u0000$min{w(f)~vert~f$ is a WRD function of $G}$. For a given graph $G$, the\u0000problem of finding a WRD function of weight $gamma_r(G)$ is defined as the\u0000emph{Minimum Weak Roman domination problem}. The problem is already known to\u0000be NP-hard for bipartite and chordal graphs. In this paper, we further study\u0000the algorithmic complexity of the problem. We prove the NP-hardness of the\u0000problem for star convex bipartite graphs and comb convex bipartite graphs,\u0000which are subclasses of bipartite graphs. In addition, we show that for the\u0000bounded degree star convex bipartite graphs, the problem is efficiently\u0000solvable. We also prove the NP-hardness of the problem for split graphs, a\u0000subclass of chordal graphs. On the positive side, we give polynomial-time\u0000algorithms to solve the problem for $P_4$-sparse graphs. Further, we have\u0000presented some approximation results.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"65 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573800","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Minsum Problem for Discrete and Weighted Set Flow on Dynamic Path Network 动态路径网络上离散和加权集流的最小值问题

arXiv - CS - Discrete Mathematics Pub Date : 2024-07-02 DOI: arxiv-2407.02177

Bubai Manna, Bodhayan Roy, Vorapong Suppakitpaisarn

引用次数: 0

Dual Bounded Generation: Polynomial, Second-order Cone and Positive Semidefinite Matrix Inequalities 二元有界生成：多项式、二阶圆锥和正半有限矩阵不等式

arXiv - CS - Discrete Mathematics Pub Date : 2024-07-02 DOI: arxiv-2407.02201

Khaled Elbassioni

引用次数: 0

Linear-Time MaxCut in Multigraphs Parameterized Above the Poljak-Turzík Bound 多图中的线性时间最大切割（MaxCut）参数高于波尔亚克-图尔齐克边界

arXiv - CS - Discrete Mathematics Pub Date : 2024-07-01 DOI: arxiv-2407.01071

Jonas Lill, Kalina Petrova, Simon Weber

{"title":"Linear-Time MaxCut in Multigraphs Parameterized Above the Poljak-Turzík Bound","authors":"Jonas Lill, Kalina Petrova, Simon Weber","doi":"arxiv-2407.01071","DOIUrl":"https://doi.org/arxiv-2407.01071","url":null,"abstract":"MaxCut is a classical NP-complete problem and a crucial building block in\u0000many combinatorial algorithms. The famous Edwards-ErdH{o}s bound states that\u0000any connected graph on n vertices with m edges contains a cut of size at least\u0000$m/2 + (n-1)/4$. Crowston, Jones and Mnich [Algorithmica, 2015] showed that the\u0000MaxCut problem on simple connected graphs admits an FPT algorithm, where the\u0000parameter k is the difference between the desired cut size c and the lower\u0000bound given by the Edwards-ErdH{o}s bound. This was later improved by Etscheid\u0000and Mnich [Algorithmica, 2017] to run in parameterized linear time, i.e.,\u0000$f(k)cdot O(m)$. We improve upon this result in two ways: Firstly, we extend\u0000the algorithm to work also for multigraphs (alternatively, graphs with positive\u0000integer weights). Secondly, we change the parameter; instead of the difference\u0000to the Edwards-ErdH{o}s bound, we use the difference to the Poljak-Turz'ik\u0000bound. The Poljak-Turz'ik bound states that any weighted graph G has a cut of\u0000size at least $w(G)/2 + w_{MSF}(G)/4$, where w(G) denotes the total weight of\u0000G, and $w_{MSF}(G)$ denotes the weight of its minimum spanning forest. In\u0000connected simple graphs the two bounds are equivalent, but for multigraphs the\u0000Poljak-Turz'ik bound can be larger and thus yield a smaller parameter k. Our\u0000algorithm also runs in parameterized linear time, i.e., $f(k)cdot O(m+n)$.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507057","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

My part is bigger than yours -- assessment within a group of peers using the pairwise comparisons method 我的部分比你的大 -- 利用成对比较法在同龄人群体中进行评估

arXiv - CS - Discrete Mathematics Pub Date : 2024-07-01 DOI: arxiv-2407.01843

Konrad Kułakowski, Jacek Szybowski

引用次数: 0

Further Connectivity Results on Plane Spanning Path Reconfiguration 平面跨路径重构的进一步连接结果

arXiv - CS - Discrete Mathematics Pub Date : 2024-06-28 DOI: arxiv-2407.00244

Valentino Boucard, Guilherme D. da Fonseca, Bastien Rivier

引用次数: 0