Yann Disser, Svenja M. Griesbach, Max Klimm, Annette Lutz
We consider an incremental variant of the rooted prize-collecting Steiner-tree problem with a growing budget constraint. While no incremental solution exists that simultaneously approximates the optimum for all budgets, we show that a bicriterial $(alpha,mu)$-approximation is possible, i.e., a solution that with budget $B+alpha$ for all $B in mathbb{R}_{geq 0}$ is a multiplicative $mu$-approximation compared to the optimum solution with budget $B$. For the case that the underlying graph is a tree, we present a polynomial-time density-greedy algorithm that computes a $(chi,1)$-approximation, where $chi$ denotes the eccentricity of the root vertex in the underlying graph, and show that this is best possible. An adaptation of the density-greedy algorithm for general graphs is $(gamma,2)$-competitive where $gamma$ is the maximal length of a vertex-disjoint path starting in the root. While this algorithm does not run in polynomial time, it can be adapted to a $(gamma,3)$-competitive algorithm that runs in polynomial time. We further devise a capacity-scaling algorithm that guarantees a $(3chi,8)$-approximation and, more generally, a $smash{bigl((4ell - 1)chi, frac{2^{ell + 2}}{2^{ell}-1}bigr)}$-approximation for every fixed $ell in mathbb{N}$.
{"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}
We present improved bounds for randomly sampling $k$-colorings of graphs with maximum degree $Delta$; our results hold without any further assumptions on the graph. The Glauber dynamics is a simple single-site update Markov chain. Jerrum (1995) proved an optimal $O(nlog{n})$ mixing time bound for Glauber dynamics whenever $k>2Delta$ where $Delta$ is the maximum degree of the input graph. This bound was improved by Vigoda (1999) to $k > (11/6)Delta$ using a "flip" dynamics which recolors (small) maximal 2-colored components in each step. Vigoda's result was the best known for general graphs for 20 years until Chen et al. (2019) established optimal mixing of the flip dynamics for $k > (11/6 - epsilon ) Delta$ where $epsilon approx 10^{-5}$. We present the first substantial improvement over these results. We prove an optimal mixing time bound of $O(nlog{n})$ for the flip dynamics when $k geq 1.809 Delta$. This yields, through recent spectral independence results, an optimal $O(nlog{n})$ mixing time for the Glauber dynamics for the same range of $k/Delta$ when $Delta=O(1)$. Our proof utilizes path coupling with a simple weighted Hamming distance for "unblocked" neighbors.
{"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}
In this paper, we study the outerplanarity of planar graphs, i.e., the number of times that we must (in a planar embedding that we can initially freely choose) remove the outerface vertices until the graph is empty. It is well-known that there are $n$-vertex graphs with outerplanarity $tfrac{n}{6}+Theta(1)$, and not difficult to show that the outerplanarity can never be bigger. We give here improved bounds of the form $tfrac{n}{2g}+2g+O(1)$, where $g$ is the fence-girth, i.e., the length of the shortest cycle with vertices on both sides. This parameter $g$ is at least the connectivity of the graph, and often bigger; for example, our results imply that planar bipartite graphs have outerplanarity $tfrac{n}{8}+O(1)$. We also show that the outerplanarity of a planar graph $G$ is at most $tfrac{1}{2}$diam$(G)+O(sqrt{n})$, where diam$(G)$ is the diameter of the graph. All our bounds are tight up to smaller-order terms, and a planar embedding that achieves the outerplanarity bound can be found in linear time.
{"title":"Improved Outerplanarity Bounds for Planar Graphs","authors":"Therese Biedl, Debajyoti Mondal","doi":"arxiv-2407.04282","DOIUrl":"https://doi.org/arxiv-2407.04282","url":null,"abstract":"In this paper, we study the outerplanarity of planar graphs, i.e., the number\u0000of times that we must (in a planar embedding that we can initially freely\u0000choose) remove the outerface vertices until the graph is empty. It is\u0000well-known that there are $n$-vertex graphs with outerplanarity\u0000$tfrac{n}{6}+Theta(1)$, and not difficult to show that the outerplanarity can\u0000never be bigger. We give here improved bounds of the form\u0000$tfrac{n}{2g}+2g+O(1)$, where $g$ is the fence-girth, i.e., the length of the\u0000shortest cycle with vertices on both sides. This parameter $g$ is at least the\u0000connectivity of the graph, and often bigger; for example, our results imply\u0000that planar bipartite graphs have outerplanarity $tfrac{n}{8}+O(1)$. We also\u0000show that the outerplanarity of a planar graph $G$ is at most\u0000$tfrac{1}{2}$diam$(G)+O(sqrt{n})$, where diam$(G)$ is the diameter of the\u0000graph. All our bounds are tight up to smaller-order terms, and a planar\u0000embedding that achieves the outerplanarity bound can be found in linear time.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573802","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}
Flip graphs of non-crossing configurations in the plane are widely studied objects, e.g., flip graph of triangulations, spanning trees, Hamiltonian cycles, and perfect matchings. Typically, it is an easy exercise to prove connectivity of a flip graph. In stark contrast, the connectivity of the flip graph of plane spanning paths on point sets in general position has been an open problem for more than 16 years. In order to provide new insights, we investigate certain induced subgraphs. Firstly, we provide tight bounds on the diameter and the radius of the flip graph of spanning paths on points in convex position with one fixed endpoint. Secondly, we show that so-called suffix-independent paths induce a connected subgraph. Consequently, to answer the open problem affirmatively, it suffices to show that each path can be flipped to some suffix-independent path. Lastly, we investigate paths where one endpoint is fixed and provide tools to flip to suffix-independent paths. We show that these tools are strong enough to show connectivity of the flip graph of plane spanning paths on point sets with at most two convex layers.
{"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}
Consider a graph $G = (V, E)$ and a function $f: V rightarrow {0, 1, 2}$. A vertex $u$ with $f(u)=0$ is defined as emph{undefended} by $f$ if it lacks adjacency to any vertex with a positive $f$-value. The function $f$ is said to be a emph{Weak Roman Dominating function} (WRD function) if, for every vertex $u$ with $f(u) = 0$, there exists a neighbour $v$ of $u$ with $f(v) > 0$ and a new function $f': V rightarrow {0, 1, 2}$ defined in the following way: $f'(u) = 1$, $f'(v) = f(v) - 1$, and $f'(w) = f(w)$, for all vertices $w$ in $Vsetminus{u,v}$; so that no vertices are undefended by $f'$. The total weight of $f$ is equal to $sum_{vin V} f(v)$, and is denoted as $w(f)$. The emph{Weak Roman Domination Number} denoted by $gamma_r(G)$, represents $min{w(f)~vert~f$ is a WRD function of $G}$. For a given graph $G$, the problem of finding a WRD function of weight $gamma_r(G)$ is defined as the emph{Minimum Weak Roman domination problem}. The problem is already known to be NP-hard for bipartite and chordal graphs. In this paper, we further study the algorithmic complexity of the problem. We prove the NP-hardness of the problem for star convex bipartite graphs and comb convex bipartite graphs, which are subclasses of bipartite graphs. In addition, we show that for the bounded degree star convex bipartite graphs, the problem is efficiently solvable. We also prove the NP-hardness of the problem for split graphs, a subclass of chordal graphs. On the positive side, we give polynomial-time algorithms to solve the problem for $P_4$-sparse graphs. Further, we have presented some approximation results.
{"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}
In this research, we examine the minsum flow problem in dynamic path networks where flows are represented as discrete and weighted sets. The minsum flow problem has been widely studied for its relevance in finding evacuation routes during emergencies such as earthquakes. However, previous approaches often assume that individuals are separable and identical, which does not adequately account for the fact that some groups of people, such as families, need to move together and that some groups may be more important than others. To address these limitations, we modify the minsum flow problem to support flows represented as discrete and weighted sets. We also propose a 2-approximation pseudo-polynomial time algorithm to solve this modified problem for path networks with uniform capacity.
{"title":"Minsum Problem for Discrete and Weighted Set Flow on Dynamic Path Network","authors":"Bubai Manna, Bodhayan Roy, Vorapong Suppakitpaisarn","doi":"arxiv-2407.02177","DOIUrl":"https://doi.org/arxiv-2407.02177","url":null,"abstract":"In this research, we examine the minsum flow problem in dynamic path networks\u0000where flows are represented as discrete and weighted sets. The minsum flow\u0000problem has been widely studied for its relevance in finding evacuation routes\u0000during emergencies such as earthquakes. However, previous approaches often\u0000assume that individuals are separable and identical, which does not adequately\u0000account for the fact that some groups of people, such as families, need to move\u0000together and that some groups may be more important than others. To address\u0000these limitations, we modify the minsum flow problem to support flows\u0000represented as discrete and weighted sets. We also propose a 2-approximation\u0000pseudo-polynomial time algorithm to solve this modified problem for path\u0000networks with uniform capacity.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141525979","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}
In the monotone integer dualization problem, we are given two sets of vectors in an integer box such that no vector in the first set is dominated by a vector in the second. The question is to check if the two sets of vectors cover the entire integer box by upward and downward domination, respectively. It is known that the problem is (quasi-)polynomially equivalent to that of enumerating all maximal feasible solutions of a given monotone system of linear/separable/supermodular inequalities over integer vectors. The equivalence is established via showing that the dual family of minimal infeasible vectors has size bounded by a (quasi-)polynomial in the sizes of the family to be generated and the input description. Continuing in this line of work, in this paper, we consider systems of polynomial, second-order cone, and semidefinite inequalities. We give sufficient conditions under which such bounds can be established and highlight some applications.
{"title":"Dual Bounded Generation: Polynomial, Second-order Cone and Positive Semidefinite Matrix Inequalities","authors":"Khaled Elbassioni","doi":"arxiv-2407.02201","DOIUrl":"https://doi.org/arxiv-2407.02201","url":null,"abstract":"In the monotone integer dualization problem, we are given two sets of vectors\u0000in an integer box such that no vector in the first set is dominated by a vector\u0000in the second. The question is to check if the two sets of vectors cover the\u0000entire integer box by upward and downward domination, respectively. It is known\u0000that the problem is (quasi-)polynomially equivalent to that of enumerating all\u0000maximal feasible solutions of a given monotone system of\u0000linear/separable/supermodular inequalities over integer vectors. The\u0000equivalence is established via showing that the dual family of minimal\u0000infeasible vectors has size bounded by a (quasi-)polynomial in the sizes of the\u0000family to be generated and the input description. Continuing in this line of\u0000work, in this paper, we consider systems of polynomial, second-order cone, and\u0000semidefinite inequalities. We give sufficient conditions under which such\u0000bounds can be established and highlight some applications.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"59 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141525977","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}
MaxCut is a classical NP-complete problem and a crucial building block in many combinatorial algorithms. The famous Edwards-ErdH{o}s bound states that any connected graph on n vertices with m edges contains a cut of size at least $m/2 + (n-1)/4$. Crowston, Jones and Mnich [Algorithmica, 2015] showed that the MaxCut problem on simple connected graphs admits an FPT algorithm, where the parameter k is the difference between the desired cut size c and the lower bound given by the Edwards-ErdH{o}s bound. This was later improved by Etscheid and Mnich [Algorithmica, 2017] to run in parameterized linear time, i.e., $f(k)cdot O(m)$. We improve upon this result in two ways: Firstly, we extend the algorithm to work also for multigraphs (alternatively, graphs with positive integer weights). Secondly, we change the parameter; instead of the difference to the Edwards-ErdH{o}s bound, we use the difference to the Poljak-Turz'ik bound. The Poljak-Turz'ik bound states that any weighted graph G has a cut of size at least $w(G)/2 + w_{MSF}(G)/4$, where w(G) denotes the total weight of G, and $w_{MSF}(G)$ denotes the weight of its minimum spanning forest. In connected simple graphs the two bounds are equivalent, but for multigraphs the Poljak-Turz'ik bound can be larger and thus yield a smaller parameter k. Our algorithm also runs in parameterized linear time, i.e., $f(k)cdot O(m+n)$.
MaxCut 是一个经典的 NP-完全问题，也是许多组合算法的重要组成部分。著名的 Edwards-ErdH{o}s 定界指出，n 个顶点上有 m 条边的任何连通图都包含一个大小至少为 $m/2 + (n-1)/4$ 的剪切。Crowston、Jones 和 Mnich [Algorithmica, 2015]的研究表明，简单连通图上的最大剪切（MaxCut）问题允许一种 FPT 算法，其中参数 k 是所需剪切大小 c 与 Edwards-ErdH{o}s 定界给出的下限之间的差值。后来，Etscheidand Mnich [Algorithmica, 2017]对这一算法进行了改进，使其可以在参数化线性时间内运行，即$f(k)cdot O(m)$。我们从两方面改进了这一结果：首先，我们扩展了算法，使其也适用于多图（或者说，具有正整数权重的图）。其次，我们改变了参数；不再使用与 Edwards-ErdH{o}s 边界的差值，而是使用与 Poljak-Turz'ikbound 的差值。Poljak-Turz'ik 约束指出，任何加权图 G 的切口大小至少为 $w(G)/2+w_{MSF}(G)/4$，其中 w(G) 表示 G 的总权重，$w_{MSF}(G)$ 表示其最小生成林的权重。在互不相连的简单图中，这两个边界是等价的，但对于多图，波利亚克-图尔兹（Poljak-Turz'ik）边界可能更大，从而产生更小的参数 k。Oural算法也可以在参数化线性时间内运行，即$f(k)cdot O(m+n)$。
{"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}
A project (e.g. writing a collaborative research paper) is often a group effort. At the end, each contributor identifies his or her contribution, often verbally. The reward, however, is quite often financial in nature. This leads to the question of what (percentage) share in the creation of the paper is due to individual authors. Different authors may have various opinions on the matter, and, even worse, their opinions may have different relevance. In this paper, we present a simple models that allows aggregation of experts' opinions linking the priority of his preference directly to the assessment made by other experts. In this approach, the greater the contribution of a given expert, the greater the importance of his opinion. The presented method can be considered as an attempt to find consensus among a group of peers involved in the same project. Hence, its applications may go beyond the proposed study example of writing a scientific paper.
{"title":"My part is bigger than yours -- assessment within a group of peers using the pairwise comparisons method","authors":"Konrad Kułakowski, Jacek Szybowski","doi":"arxiv-2407.01843","DOIUrl":"https://doi.org/arxiv-2407.01843","url":null,"abstract":"A project (e.g. writing a collaborative research paper) is often a group\u0000effort. At the end, each contributor identifies his or her contribution, often\u0000verbally. The reward, however, is quite often financial in nature. This leads\u0000to the question of what (percentage) share in the creation of the paper is due\u0000to individual authors. Different authors may have various opinions on the\u0000matter, and, even worse, their opinions may have different relevance. In this\u0000paper, we present a simple models that allows aggregation of experts' opinions\u0000linking the priority of his preference directly to the assessment made by other\u0000experts. In this approach, the greater the contribution of a given expert, the\u0000greater the importance of his opinion. The presented method can be considered\u0000as an attempt to find consensus among a group of peers involved in the same\u0000project. Hence, its applications may go beyond the proposed study example of\u0000writing a scientific paper.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"75 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141525978","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}
Valentino Boucard, Guilherme D. da Fonseca, Bastien Rivier
Given a finite set $ S $ of points, we consider the following reconfiguration graph. The vertices are the plane spanning paths of $ S $ and there is an edge between two vertices if the two corresponding paths differ by two edges (one removed, one added). Since 2007, this graph is conjectured to be connected but no proof has been found. In this paper, we prove several results to support the conjecture. Mainly, we show that if all but one point of $ S $ are in convex position, then the graph is connected with diameter at most $ 2 | S | $ and that for $ | S | geq 3 $ every connected component has at least $ 3 $ vertices.
给定一个有限的点集合 $ S $，我们考虑下面的重组图。顶点是 $ S $ 的平面跨越路径，如果两条对应路径相差两条边（一条删除，一条添加），则两个顶点之间有一条边。自 2007 年以来，人们一直猜测这个图是连通的，但没有找到证明。在本文中，我们证明了支持该猜想的几个结果。主要是，我们证明了如果除了一个点之外，$ S $ 的所有点都在凸点上，那么这个图是连通的，直径最多为 $ 2 | S | $，并且对于 $ | S |geq 3 $，每个连通的部分至少有 $ 3 $ 个顶点。
{"title":"Further Connectivity Results on Plane Spanning Path Reconfiguration","authors":"Valentino Boucard, Guilherme D. da Fonseca, Bastien Rivier","doi":"arxiv-2407.00244","DOIUrl":"https://doi.org/arxiv-2407.00244","url":null,"abstract":"Given a finite set $ S $ of points, we consider the following reconfiguration\u0000graph. The vertices are the plane spanning paths of $ S $ and there is an edge\u0000between two vertices if the two corresponding paths differ by two edges (one\u0000removed, one added). Since 2007, this graph is conjectured to be connected but\u0000no proof has been found. In this paper, we prove several results to support the\u0000conjecture. Mainly, we show that if all but one point of $ S $ are in convex\u0000position, then the graph is connected with diameter at most $ 2 | S | $ and\u0000that for $ | S | geq 3 $ every connected component has at least $ 3 $\u0000vertices.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"729 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141525980","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}
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