{"title":"Monochromatic partitioning of colored points by lines","authors":"H. Jowhari, M. Rezapour","doi":"10.2139/ssrn.4020894","DOIUrl":"https://doi.org/10.2139/ssrn.4020894","url":null,"abstract":"","PeriodicalId":13545,"journal":{"name":"Inf. Process. Lett.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83659993","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}
{"title":"An Improved Kernel for the Flip Distance Problem on Simple Convex Polygons","authors":"M. Calvo, S. Kelk","doi":"10.2139/ssrn.4149541","DOIUrl":"https://doi.org/10.2139/ssrn.4149541","url":null,"abstract":"","PeriodicalId":13545,"journal":{"name":"Inf. Process. Lett.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89893891","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}
{"title":"Algorithmic results in Roman dominating functions on graphs","authors":"A. Poureidi, J. Fathali","doi":"10.2139/ssrn.4074851","DOIUrl":"https://doi.org/10.2139/ssrn.4074851","url":null,"abstract":"","PeriodicalId":13545,"journal":{"name":"Inf. Process. Lett.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86383208","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}
Pub Date : 2023-01-01DOI: 10.48550/arXiv.2301.05049
V. Keikha, Maria Saumell
Given an $n$-vertex 1.5D terrain $T$ and a set $A$ of $m
给定一个$n$顶点的1.5D地形$T$和$m
{"title":"On Voronoi visibility maps of 1.5D terrains with multiple viewpoints","authors":"V. Keikha, Maria Saumell","doi":"10.48550/arXiv.2301.05049","DOIUrl":"https://doi.org/10.48550/arXiv.2301.05049","url":null,"abstract":"Given an $n$-vertex 1.5D terrain $T$ and a set $A$ of $m<n$ viewpoints, the Voronoi visibility map $vorvis(T,A)$ is a partitioning of $T$ into regions such that each region is assigned to the closest (in Euclidean distance) visible viewpoint. The colored visibility map $colvis(T,A)$ is a partitioning of $T$ into regions that have the same set of visible viewpoints. In this paper, we propose an algorithm to compute $vorvis(T,A)$ that runs in $O(n+(m^2+k_c)log n)$ time, where $k_c$ and $k_v$ denote the total complexity of $colvis(T,A)$ and $vorvis(T,A)$, respectively. This improves upon a previous algorithm for this problem. We also generalize our algorithm to higher order Voronoi visibility maps, and to Voronoi visibility maps with respect to other distances. Finally, we prove bounds relating $k_v$ to $k_c$, and we show an application of our algorithm to a problem on limited range of sight.","PeriodicalId":13545,"journal":{"name":"Inf. Process. Lett.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75467095","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}
Pub Date : 2023-01-01DOI: 10.48550/arXiv.2208.02862
Anita Dürr
In a recent breakthrough paper, Chi et al. (STOC'22) introduce an $tilde{O}(n^{frac{3 + omega}{2}})$ time algorithm to compute Monotone Min-Plus Product between two square matrices of dimensions $n times n$ and entries bounded by $O(n)$. This greatly improves upon the previous $tilde O(n^{frac{12 + omega}{5}})$ time algorithm and as a consequence improves bounds for its applications. Several other applications involve Monotone Min-Plus Product between rectangular matrices, and even if Chi et al.'s algorithm seems applicable for the rectangular case, the generalization is not straightforward. In this paper we present a generalization of the algorithm of Chi et al. to solve Monotone Min-Plus Product for rectangular matrices with polynomial bounded values. We next use this faster algorithm to improve running times for the following applications of Rectangular Monotone Min-Plus Product: $M$-bounded Single Source Replacement Path, Batch Range Mode, $k$-Dyck Edit Distance and 2-approximation of All Pairs Shortest Path. We also improve the running time for Unweighted Tree Edit Distance using the algorithm by Chi et al.
{"title":"Improved Bounds for Rectangular Monotone Min-Plus Product","authors":"Anita Dürr","doi":"10.48550/arXiv.2208.02862","DOIUrl":"https://doi.org/10.48550/arXiv.2208.02862","url":null,"abstract":"In a recent breakthrough paper, Chi et al. (STOC'22) introduce an $tilde{O}(n^{frac{3 + omega}{2}})$ time algorithm to compute Monotone Min-Plus Product between two square matrices of dimensions $n times n$ and entries bounded by $O(n)$. This greatly improves upon the previous $tilde O(n^{frac{12 + omega}{5}})$ time algorithm and as a consequence improves bounds for its applications. Several other applications involve Monotone Min-Plus Product between rectangular matrices, and even if Chi et al.'s algorithm seems applicable for the rectangular case, the generalization is not straightforward. In this paper we present a generalization of the algorithm of Chi et al. to solve Monotone Min-Plus Product for rectangular matrices with polynomial bounded values. We next use this faster algorithm to improve running times for the following applications of Rectangular Monotone Min-Plus Product: $M$-bounded Single Source Replacement Path, Batch Range Mode, $k$-Dyck Edit Distance and 2-approximation of All Pairs Shortest Path. We also improve the running time for Unweighted Tree Edit Distance using the algorithm by Chi et al.","PeriodicalId":13545,"journal":{"name":"Inf. Process. Lett.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72761358","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}
Pub Date : 2022-11-04DOI: 10.48550/arXiv.2211.02525
Florian Hörsch, Z. Szigeti
We say that a tree $T$ is an $S$-Steiner tree if $S subseteq V(T)$ and a hypergraph is an $S$-Steiner hypertree if it can be trimmed to an $S$-Steiner tree. We prove that it is NP-complete to decide, given a hypergraph $mathcal{H}$ and some $S subseteq V(mathcal{H})$, whether there is a subhypergraph of $mathcal{H}$ which is an $S$-Steiner hypertree. As corollaries, we give two negative results for two Steiner orientation problems in hypergraphs. Firstly, we show that it is NP-complete to decide, given a hypergraph $mathcal{H}$, some $r in V(mathcal{H})$ and some $S subseteq V(mathcal{H})$, whether this hypergraph has an orientation in which every vertex of $S$ is reachable from $r$. Secondly, we show that it is NP-complete to decide, given a hypergraph $mathcal{H}$ and some $S subseteq V(mathcal{H})$, whether this hypergraph has an orientation in which any two vertices in $S$ are mutually reachable from each other. This answers a longstanding open question of the Egerv'ary Research group. We further show that it is NP-complete to decide if a given hypergraph has a well-balanced orientation. On the positive side, we show that the problem of finding a Steiner hypertree and the first orientation problem can be solved in polynomial time if the number of terminals $|S|$ is fixed.
{"title":"Steiner connectivity problems in hypergraphs","authors":"Florian Hörsch, Z. Szigeti","doi":"10.48550/arXiv.2211.02525","DOIUrl":"https://doi.org/10.48550/arXiv.2211.02525","url":null,"abstract":"We say that a tree $T$ is an $S$-Steiner tree if $S subseteq V(T)$ and a hypergraph is an $S$-Steiner hypertree if it can be trimmed to an $S$-Steiner tree. We prove that it is NP-complete to decide, given a hypergraph $mathcal{H}$ and some $S subseteq V(mathcal{H})$, whether there is a subhypergraph of $mathcal{H}$ which is an $S$-Steiner hypertree. As corollaries, we give two negative results for two Steiner orientation problems in hypergraphs. Firstly, we show that it is NP-complete to decide, given a hypergraph $mathcal{H}$, some $r in V(mathcal{H})$ and some $S subseteq V(mathcal{H})$, whether this hypergraph has an orientation in which every vertex of $S$ is reachable from $r$. Secondly, we show that it is NP-complete to decide, given a hypergraph $mathcal{H}$ and some $S subseteq V(mathcal{H})$, whether this hypergraph has an orientation in which any two vertices in $S$ are mutually reachable from each other. This answers a longstanding open question of the Egerv'ary Research group. We further show that it is NP-complete to decide if a given hypergraph has a well-balanced orientation. On the positive side, we show that the problem of finding a Steiner hypertree and the first orientation problem can be solved in polynomial time if the number of terminals $|S|$ is fixed.","PeriodicalId":13545,"journal":{"name":"Inf. Process. Lett.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87284998","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}
{"title":"Bicriteria scheduling on an unbounded parallel-batch machine for minimizing makespan and maximum cost","authors":"Shuguang Li, Zhichao Geng","doi":"10.2139/ssrn.4011492","DOIUrl":"https://doi.org/10.2139/ssrn.4011492","url":null,"abstract":"","PeriodicalId":13545,"journal":{"name":"Inf. Process. Lett.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76032095","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}
Given a graph $G$, we denote by $f(G,u_0,k)$ the number of paths of length $k$ in $G$ starting from $u_0$. In graphs of maximum degree 3, with edge weights $i.i.d.$ with $exp(1)$, we provide a simple proof showing that (under the assumption that $f(G,u_0,k)=omega(1)$) the expected weight of the heaviest path of length $k$ in $G$ starting from $u_0$ is at least begin{align*} (1-o(1))left(k+frac{log_2left(f(G,u_0,k)right)}{2}right), end{align*} and the expected weight of the lightest path of length $k$ in $G$ starting from $u_0$ is at most begin{align*} (1+o(1))left(k-frac{log_2left(f(G,u_0,k)right)}{2}right). end{align*} We demonstrate the immediate implication of this result for Hamilton paths and Hamilton cycles in random cubic graphs, where we show that typically there exist paths and cycles of such weight as well. Finally, we discuss the connection of this result to the question of a longest cycle in the giant component of supercritical $G(n,p)$.
{"title":"Heavy and light paths and Hamilton cycles","authors":"Sahar Diskin, Dor Elboim","doi":"10.2139/ssrn.4273484","DOIUrl":"https://doi.org/10.2139/ssrn.4273484","url":null,"abstract":"Given a graph $G$, we denote by $f(G,u_0,k)$ the number of paths of length $k$ in $G$ starting from $u_0$. In graphs of maximum degree 3, with edge weights $i.i.d.$ with $exp(1)$, we provide a simple proof showing that (under the assumption that $f(G,u_0,k)=omega(1)$) the expected weight of the heaviest path of length $k$ in $G$ starting from $u_0$ is at least begin{align*} (1-o(1))left(k+frac{log_2left(f(G,u_0,k)right)}{2}right), end{align*} and the expected weight of the lightest path of length $k$ in $G$ starting from $u_0$ is at most begin{align*} (1+o(1))left(k-frac{log_2left(f(G,u_0,k)right)}{2}right). end{align*} We demonstrate the immediate implication of this result for Hamilton paths and Hamilton cycles in random cubic graphs, where we show that typically there exist paths and cycles of such weight as well. Finally, we discuss the connection of this result to the question of a longest cycle in the giant component of supercritical $G(n,p)$.","PeriodicalId":13545,"journal":{"name":"Inf. Process. Lett.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82247303","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}
Graph invariants are a useful tool in graph theory. Not only do they encode useful information about the graphs to which they are associated, but complete invariants can be used to distinguish between non-isomorphic graphs. Polynomial invariants for graphs such as the well-known Tutte polynomial have been studied for several years, and recently there has been interest to also define such invariants for phylogenetic networks, a special type of graph that arises in the area of evolutionary biology. Recently Liu gave a complete invariant for (phylogenetic) trees. However, the polynomial invariants defined thus far for phylogenetic networks that are not trees require vertex labels and either contain a large number of variables, or they have exponentially many terms in the number of reticulations. This can make it difficult to compute these polynomials and to use them to analyse unlabelled networks. In this paper, we shall show how to circumvent some of these difficulties for rooted cactuses and cactuses. As well as being important in other areas such as operations research, rooted cactuses contain some common classes of phylogenetic networks such phylogenetic trees and level-1 networks. More specifically, we define a polynomial $F$ that is a complete invariant for the class of rooted cactuses without vertices of indegree 1 and outdegree 1 that has 5 variables, and a polynomial $Q$ that is a complete invariant for the class of rooted cactuses that has 6 variables vince{whose degree can be bounded linearly in terms of the size of the rooted cactus}. We also explain how to extend the $Q$ polynomial to define a complete invariant for leaf-labelled rooted cactuses as well as (unrooted) cactuses.
{"title":"Polynomial invariants for cactuses","authors":"L. Iersel, V. Moulton, Yukihiro Murakami","doi":"10.2139/ssrn.4233802","DOIUrl":"https://doi.org/10.2139/ssrn.4233802","url":null,"abstract":"Graph invariants are a useful tool in graph theory. Not only do they encode useful information about the graphs to which they are associated, but complete invariants can be used to distinguish between non-isomorphic graphs. Polynomial invariants for graphs such as the well-known Tutte polynomial have been studied for several years, and recently there has been interest to also define such invariants for phylogenetic networks, a special type of graph that arises in the area of evolutionary biology. Recently Liu gave a complete invariant for (phylogenetic) trees. However, the polynomial invariants defined thus far for phylogenetic networks that are not trees require vertex labels and either contain a large number of variables, or they have exponentially many terms in the number of reticulations. This can make it difficult to compute these polynomials and to use them to analyse unlabelled networks. In this paper, we shall show how to circumvent some of these difficulties for rooted cactuses and cactuses. As well as being important in other areas such as operations research, rooted cactuses contain some common classes of phylogenetic networks such phylogenetic trees and level-1 networks. More specifically, we define a polynomial $F$ that is a complete invariant for the class of rooted cactuses without vertices of indegree 1 and outdegree 1 that has 5 variables, and a polynomial $Q$ that is a complete invariant for the class of rooted cactuses that has 6 variables vince{whose degree can be bounded linearly in terms of the size of the rooted cactus}. We also explain how to extend the $Q$ polynomial to define a complete invariant for leaf-labelled rooted cactuses as well as (unrooted) cactuses.","PeriodicalId":13545,"journal":{"name":"Inf. Process. Lett.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73708972","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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