Pub Date : 2024-12-27DOI: 10.1016/j.ipl.2024.106554
Umang Bhaskar , A.R. Sricharan , Rohit Vaish
We study the problem of fair cake-cutting where each agent receives a connected piece of the cake. A division of the cake is deemed fair if it is equitable, which means that all agents derive the same value from their assigned piece. Prior work has established the existence of a connected equitable division for agents with nonnegative valuations using various techniques. We provide a simple proof of this result using Sperner's lemma. Our proof extends known existence results for connected equitable divisions to significantly more general classes of valuations, including nonnegative valuations with externalities, as well as several interesting subclasses of general (possibly negative) valuations.
{"title":"Connected equitable cake division via Sperner's lemma","authors":"Umang Bhaskar , A.R. Sricharan , Rohit Vaish","doi":"10.1016/j.ipl.2024.106554","DOIUrl":"10.1016/j.ipl.2024.106554","url":null,"abstract":"<div><div>We study the problem of fair cake-cutting where each agent receives a connected piece of the cake. A division of the cake is deemed fair if it is <em>equitable</em>, which means that all agents derive the same value from their assigned piece. Prior work has established the existence of a connected equitable division for agents with nonnegative valuations using various techniques. We provide a simple proof of this result using Sperner's lemma. Our proof extends known existence results for connected equitable divisions to significantly more general classes of valuations, including nonnegative valuations with externalities, as well as several interesting subclasses of general (possibly negative) valuations.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"189 ","pages":"Article 106554"},"PeriodicalIF":0.7,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099049","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-20DOI: 10.1016/j.ipl.2024.106555
Hassene Aissi , Mourad Baiou , Francisco Barahona
Consider an undirected graph with positive integer edge weights. Subramanian [11] established an upper bound of on the number of minimum weight cycles. We present a new algorithm to enumerate all minimum weight cycles with a complexity of . Using this algorithm, we derive the following upper bounds for the number of minimum weight cycles: if the minimum weight is even, the bound is , and if it is odd, the bound is . Notably, we improve Subramanian's bound by an order of magnitude when the minimum weight of a cycle is odd. Additionally, we demonstrate that these bounds are asymptotically tight.
{"title":"New bounds for the number of lightest cycles in undirected graphs","authors":"Hassene Aissi , Mourad Baiou , Francisco Barahona","doi":"10.1016/j.ipl.2024.106555","DOIUrl":"10.1016/j.ipl.2024.106555","url":null,"abstract":"<div><div>Consider an undirected graph <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span> with positive integer edge weights. Subramanian <span><span>[11]</span></span> established an upper bound of <span><math><mo>|</mo><mi>V</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>4</mn></mrow></msup><mo>/</mo><mn>6</mn></math></span> on the number of minimum weight cycles. We present a new algorithm to enumerate all minimum weight cycles with a complexity of <span><math><mi>O</mi><mo>(</mo><mo>|</mo><mi>V</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>3</mn></mrow></msup><mo>(</mo><mo>|</mo><mi>E</mi><mo>|</mo><mo>+</mo><mo>|</mo><mi>V</mi><mo>|</mo><mi>log</mi><mo></mo><mo>|</mo><mi>V</mi><mo>|</mo><mo>)</mo><mo>)</mo></math></span>. Using this algorithm, we derive the following upper bounds for the number of minimum weight cycles: if the minimum weight is even, the bound is <span><math><mo>|</mo><mi>V</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>4</mn></mrow></msup><mo>/</mo><mn>4</mn></math></span>, and if it is odd, the bound is <span><math><mo>|</mo><mi>V</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>3</mn></mrow></msup><mo>/</mo><mn>2</mn></math></span>. Notably, we improve Subramanian's bound by an order of magnitude when the minimum weight of a cycle is odd. Additionally, we demonstrate that these bounds are asymptotically tight.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"189 ","pages":"Article 106555"},"PeriodicalIF":0.7,"publicationDate":"2024-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099048","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-18DOI: 10.1016/j.ipl.2024.106553
Shravas Rao
A matrix satisfies the restricted isometry property if is approximately equal to for all k-sparse vectors x. We give a construction of RIP matrices with the optimal rows using bits of randomness. The main technical ingredient is an extension of the Hanson-Wright inequality to ε-biased distributions.
{"title":"Satisfying the restricted isometry property with the optimal number of rows and slightly less randomness","authors":"Shravas Rao","doi":"10.1016/j.ipl.2024.106553","DOIUrl":"10.1016/j.ipl.2024.106553","url":null,"abstract":"<div><div>A matrix <span><math><mi>Φ</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>Q</mi><mo>×</mo><mi>N</mi></mrow></msup></math></span> satisfies the restricted isometry property if <span><math><msubsup><mrow><mo>‖</mo><mi>Φ</mi><mi>x</mi><mo>‖</mo></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></math></span> is approximately equal to <span><math><msubsup><mrow><mo>‖</mo><mi>x</mi><mo>‖</mo></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></math></span> for all <em>k</em>-sparse vectors <em>x</em>. We give a construction of RIP matrices with the optimal <span><math><mi>Q</mi><mo>=</mo><mi>O</mi><mo>(</mo><mi>k</mi><mi>log</mi><mo></mo><mo>(</mo><mi>N</mi><mo>/</mo><mi>k</mi><mo>)</mo><mo>)</mo></math></span> rows using <span><math><mi>O</mi><mo>(</mo><mi>k</mi><mi>log</mi><mo></mo><mo>(</mo><mi>N</mi><mo>/</mo><mi>k</mi><mo>)</mo><mi>log</mi><mo></mo><mo>(</mo><mi>k</mi><mo>)</mo><mo>)</mo></math></span> bits of randomness. The main technical ingredient is an extension of the Hanson-Wright inequality to <em>ε</em>-biased distributions.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"189 ","pages":"Article 106553"},"PeriodicalIF":0.7,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099047","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-17DOI: 10.1016/j.ipl.2024.106552
Arka Ray , Sai Sandeep
The Geometric Bin Packing (GBP) problem is a generalization of Bin Packing where the input is a set of d-dimensional rectangles, and the goal is to pack them into d-dimensional unit cubes efficiently. It is NP-hard to obtain a PTAS for the problem, even when . For general d, the best-known approximation algorithm has an approximation guarantee that is exponential in d. In contrast, the best hardness of approximation is still a small constant inapproximability from the case when . In this paper, we show that the problem cannot be approximated within a factor unless .
Recently, d-dimensional Vector Bin Packing, a problem closely related to the GBP, was shown to be hard to approximate within a factor when d is a fixed constant, using a notion of Packing Dimension of set families. In this paper, we introduce a geometric analog of it, the Geometric Packing Dimension of set families. While we fall short of obtaining similar inapproximability results for the Geometric Bin Packing problem when d is fixed, we prove a couple of key properties of the Geometric Packing Dimension which highlight fundamental differences between Geometric Bin Packing and Vector Bin Packing.
{"title":"Improved hardness of approximation for Geometric Bin Packing","authors":"Arka Ray , Sai Sandeep","doi":"10.1016/j.ipl.2024.106552","DOIUrl":"10.1016/j.ipl.2024.106552","url":null,"abstract":"<div><div>The Geometric Bin Packing (GBP) problem is a generalization of Bin Packing where the input is a set of <em>d</em>-dimensional rectangles, and the goal is to pack them into <em>d</em>-dimensional unit cubes efficiently. It is NP-hard to obtain a PTAS for the problem, even when <span><math><mi>d</mi><mo>=</mo><mn>2</mn></math></span>. For general <em>d</em>, the best-known approximation algorithm has an approximation guarantee that is exponential in <em>d</em>. In contrast, the best hardness of approximation is still a small constant inapproximability from the case when <span><math><mi>d</mi><mo>=</mo><mn>2</mn></math></span>. In this paper, we show that the problem cannot be approximated within a <span><math><msup><mrow><mi>d</mi></mrow><mrow><mn>1</mn><mo>−</mo><mi>ϵ</mi></mrow></msup></math></span> factor unless <span><math><mtext>NP</mtext><mo>=</mo><mtext>P</mtext></math></span>.</div><div>Recently, <em>d</em>-dimensional Vector Bin Packing, a problem closely related to the GBP, was shown to be hard to approximate within a <span><math><mi>Ω</mi><mo>(</mo><mi>log</mi><mo></mo><mi>d</mi><mo>)</mo></math></span> factor when <em>d</em> is a fixed constant, using a notion of Packing Dimension of set families. In this paper, we introduce a geometric analog of it, the Geometric Packing Dimension of set families. While we fall short of obtaining similar inapproximability results for the Geometric Bin Packing problem when <em>d</em> is fixed, we prove a couple of key properties of the Geometric Packing Dimension which highlight fundamental differences between Geometric Bin Packing and Vector Bin Packing.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"189 ","pages":"Article 106552"},"PeriodicalIF":0.7,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099046","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-10DOI: 10.1016/j.ipl.2024.106551
David Flores-Peñaloza , Mario A. Lopez , Nestaly Marín , David Orden
Let P be a k-colored set of n points in the plane, . We study the problem of deciding if P contains a subset of four points of different colors such that its Rectilinear Convex Hull has positive area. We show this problem to be equivalent to deciding if there exists a point c in the plane such that each of the open quadrants defined by c contains a point of P, each of them having a different color. We provide an -time algorithm for this problem, where the hidden constant does not depend on k; then, we prove that this problem has time complexity in the algebraic computation tree model. No general position assumptions for P are required.
{"title":"An efficient algorithm for identifying rainbow ortho-convex 4-sets in k-colored point sets","authors":"David Flores-Peñaloza , Mario A. Lopez , Nestaly Marín , David Orden","doi":"10.1016/j.ipl.2024.106551","DOIUrl":"10.1016/j.ipl.2024.106551","url":null,"abstract":"<div><div>Let <em>P</em> be a <em>k</em>-colored set of <em>n</em> points in the plane, <span><math><mn>4</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></math></span>. We study the problem of deciding if <em>P</em> contains a subset of four points of different colors such that its Rectilinear Convex Hull has positive area. We show this problem to be equivalent to deciding if there exists a point <em>c</em> in the plane such that each of the open quadrants defined by <em>c</em> contains a point of <em>P</em>, each of them having a different color. We provide an <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span>-time algorithm for this problem, where the hidden constant does not depend on <em>k</em>; then, we prove that this problem has time complexity <span><math><mi>Ω</mi><mo>(</mo><mi>n</mi><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span> in the algebraic computation tree model. No general position assumptions for <em>P</em> are required.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"189 ","pages":"Article 106551"},"PeriodicalIF":0.7,"publicationDate":"2024-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099044","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-09DOI: 10.1016/j.ipl.2024.106550
Adam J. Przeździecki
We present a very simple algorithm for computing Pfaffians which uses no division operations. Essentially, it amounts to iterating matrix multiplication and truncation.
Its complexity, for a matrix, is , where is the cost of matrix multiplication. In case of a sparse matrix, is the cost of the dense-sparse matrix multiplication.
The algorithm is an adaptation of the Bird algorithm for determinants. We show how to extract, with practically no additional work, the characteristic polynomial and the Pfaffian characteristic polynomial from these algorithms.
{"title":"A simple division-free algorithm for computing Pfaffians","authors":"Adam J. Przeździecki","doi":"10.1016/j.ipl.2024.106550","DOIUrl":"10.1016/j.ipl.2024.106550","url":null,"abstract":"<div><div>We present a very simple algorithm for computing Pfaffians which uses no division operations. Essentially, it amounts to iterating matrix multiplication and truncation.</div><div>Its complexity, for a <span><math><mn>2</mn><mi>n</mi><mo>×</mo><mn>2</mn><mi>n</mi></math></span> matrix, is <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>M</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo></math></span>, where <span><math><mi>M</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> is the cost of matrix multiplication. In case of a sparse matrix, <span><math><mi>M</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> is the cost of the dense-sparse matrix multiplication.</div><div>The algorithm is an adaptation of the Bird algorithm for determinants. We show how to extract, with practically no additional work, the characteristic polynomial and the Pfaffian characteristic polynomial from these algorithms.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"189 ","pages":"Article 106550"},"PeriodicalIF":0.7,"publicationDate":"2024-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099045","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-27DOI: 10.1016/j.ipl.2024.106549
Naoyuki Kamiyama
The topic of this paper is the stable matching problem in a bipartite graph. Super-stability is one of the stability concepts in the stable matching problem with ties. It is known that there may not exist a super-stable matching, and the existence of a super-stable matching can be checked in polynomial time. In this paper, we consider the problem of modifying an instance of the stable matching problem with ties by deleting some bounded number of agents in such a way that there exists a super-stable matching in the modified instance. First, we consider the setting where we are allowed to delete agents on only one side. We prove that, in this setting, our problem can be solved in polynomial time. Interestingly, this result is obtained by carefully observing the existing algorithm for checking the existence of a super-stable matching. Next, we consider the setting where we are given an upper bound on the number of deleted agents for each side, and we are allowed to delete agents on both sides. We prove that, in this setting, our problem is NP-complete.
{"title":"Modifying an instance of the super-stable matching problem","authors":"Naoyuki Kamiyama","doi":"10.1016/j.ipl.2024.106549","DOIUrl":"10.1016/j.ipl.2024.106549","url":null,"abstract":"<div><div>The topic of this paper is the stable matching problem in a bipartite graph. Super-stability is one of the stability concepts in the stable matching problem with ties. It is known that there may not exist a super-stable matching, and the existence of a super-stable matching can be checked in polynomial time. In this paper, we consider the problem of modifying an instance of the stable matching problem with ties by deleting some bounded number of agents in such a way that there exists a super-stable matching in the modified instance. First, we consider the setting where we are allowed to delete agents on only one side. We prove that, in this setting, our problem can be solved in polynomial time. Interestingly, this result is obtained by carefully observing the existing algorithm for checking the existence of a super-stable matching. Next, we consider the setting where we are given an upper bound on the number of deleted agents for each side, and we are allowed to delete agents on both sides. We prove that, in this setting, our problem is NP-complete.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"189 ","pages":"Article 106549"},"PeriodicalIF":0.7,"publicationDate":"2024-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142748594","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-22DOI: 10.1016/j.ipl.2024.106542
Chris R. Giannella
Because many dissimilarity functions behave differently in low versus high-dimensional spaces, the behavior of high-dimensional nearest neighbor search has been studied extensively. One line of research involves the characterization of nearest neighbor queries as unstable if their query points have nearly identical dissimilarity with most points in the dataset. This research has shown that, for various data distributions and dissimilarity functions, the probability of query instability approaches one. Previous work in Information Processing Letters by C. Giannella in 2021 explicated this phenomenon for centered Gaussian data and Euclidean distance. This paper addresses the problem of characterizing query instability behavior over centered Gaussian data and a fundamentally different dissimilarity function, cosine dissimilarity. Conditions are provided on the covariance matrices and dataset size function guaranteeing that the probability of query instability goes to one. Furthermore, conditions are provided under which the instability probability is bounded away from one.
{"title":"Instability results for cosine-dissimilarity-based nearest neighbor search on high dimensional Gaussian data","authors":"Chris R. Giannella","doi":"10.1016/j.ipl.2024.106542","DOIUrl":"10.1016/j.ipl.2024.106542","url":null,"abstract":"<div><div>Because many dissimilarity functions behave differently in low versus high-dimensional spaces, the behavior of high-dimensional nearest neighbor search has been studied extensively. One line of research involves the characterization of nearest neighbor queries as unstable if their query points have nearly identical dissimilarity with most points in the dataset. This research has shown that, for various data distributions and dissimilarity functions, the probability of query instability approaches one. Previous work in <em>Information Processing Letters</em> by C. Giannella in 2021 explicated this phenomenon for centered Gaussian data and Euclidean distance. This paper addresses the problem of characterizing query instability behavior over centered Gaussian data and a fundamentally different dissimilarity function, cosine dissimilarity. Conditions are provided on the covariance matrices and dataset size function guaranteeing that the probability of query instability goes to one. Furthermore, conditions are provided under which the instability probability is bounded away from one.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"189 ","pages":"Article 106542"},"PeriodicalIF":0.7,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698622","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-12DOI: 10.1016/j.ipl.2024.106541
Peter Kostolányi
The class of all finitely generated semigroups with a deterministic context-free word problem is shown to be closed under free products, answering a question of T. Brough, A. J. Cain, and M. Pfeiffer. On the other hand, it is proved that the class of all finitely generated monoids with a deterministic context-free word problem is not closed under monoid free products.
证明了具有确定性无上下文词问题的所有有限生成半群的类在自由积下是封闭的，回答了 T. Brough、A. J. Cain 和 M. Pfeiffer 的一个问题。另一方面，证明了具有确定性无上下文词问题的所有有限生成单元的类在单元自由积下不是封闭的。
{"title":"Free products of semigroups and monoids with a deterministic context-free word problem","authors":"Peter Kostolányi","doi":"10.1016/j.ipl.2024.106541","DOIUrl":"10.1016/j.ipl.2024.106541","url":null,"abstract":"<div><div>The class of all finitely generated semigroups with a deterministic context-free word problem is shown to be closed under free products, answering a question of T. Brough, A. J. Cain, and M. Pfeiffer. On the other hand, it is proved that the class of all finitely generated monoids with a deterministic context-free word problem is not closed under monoid free products.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"188 ","pages":"Article 106541"},"PeriodicalIF":0.7,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142653220","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Covering a set of segments in a plane with vehicles of limited autonomy is a problem of practical interest. The limited battery endurance imposes periodical visits to a static base station. Typically, two optimization problems are considered: minimize the number of tours, and minimize the total traveled distance. In a general setting, the problems are NP-hard and in this letter, we study the one-dimensional version. For covering segments on a line, we design efficient solutions for both optimization problems. First, we design a greedy algorithm that is optimal for the first task, and for both tasks when only one segment is considered. Being n and m the number of segments and tours of an optimal solution, respectively, our algorithm runs in time. For the second criterion, our solution is based on Dynamic Programming and runs in time.
用自主能力有限的车辆在平面上覆盖一组区段是一个具有实际意义的问题。由于电池续航时间有限，因此需要定期访问静态基站。通常，需要考虑两个优化问题：最小化巡视次数和最小化总行程。在一般情况下，这两个问题都很难解决，在这封信中，我们研究的是一维问题。对于线路上的覆盖线段，我们为这两个优化问题设计了高效的解决方案。首先，我们设计了一种贪婪算法，该算法对第一项任务和只考虑一个线段时的两项任务都是最优的。由于 n 和 m 分别为最优解的线段数和游程数，我们的算法运行时间为 O(m+n)。对于第二个标准，我们的解决方案基于动态编程，运行时间为 O(n2m)。
{"title":"Covering segments on a line with drones","authors":"Sergey Bereg , José-Miguel Díaz-Báñez , Alina Kasiuk , Miguel-Angel Pérez-Cutiño , Fabio Rodríguez","doi":"10.1016/j.ipl.2024.106540","DOIUrl":"10.1016/j.ipl.2024.106540","url":null,"abstract":"<div><div>Covering a set of segments in a plane with vehicles of limited autonomy is a problem of practical interest. The limited battery endurance imposes periodical visits to a static base station. Typically, two optimization problems are considered: minimize the number of tours, and minimize the total traveled distance. In a general setting, the problems are NP-hard and in this letter, we study the one-dimensional version. For covering segments on a line, we design efficient solutions for both optimization problems. First, we design a greedy algorithm that is optimal for the first task, and for both tasks when only one segment is considered. Being <em>n</em> and <em>m</em> the number of segments and tours of an optimal solution, respectively, our algorithm runs in <span><math><mi>O</mi><mo>(</mo><mi>m</mi><mo>+</mo><mi>n</mi><mo>)</mo></math></span> time. For the second criterion, our solution is based on Dynamic Programming and runs in <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>m</mi><mo>)</mo></math></span> time.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"188 ","pages":"Article 106540"},"PeriodicalIF":0.7,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142592824","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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