Discrete Optimization最新文献

{"title":"Maximizing the Mostar index for bipartite graphs and split graphs","authors":"Štefko Miklavič ,&nbsp;Johannes Pardey ,&nbsp;Dieter Rautenbach ,&nbsp;Florian Werner","doi":"10.1016/j.disopt.2023.100768","DOIUrl":"https://doi.org/10.1016/j.disopt.2023.100768","url":null,"abstract":"<div><p>Došlić et al. defined the Mostar index of a graph <span><math><mi>G</mi></math></span> as <span><math><mrow><munder><mrow><mo>∑</mo></mrow><mrow><mi>u</mi><mi>v</mi><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></munder><mspace></mspace><mrow><mo>|</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow><mo>−</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>v</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow></math></span>, where, for an edge <span><math><mrow><mi>u</mi><mi>v</mi></mrow></math></span> of <span><math><mi>G</mi></math></span>, the term <span><math><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow></mrow></math></span> denotes the number of vertices of <span><math><mi>G</mi></math></span> that have a smaller distance in <span><math><mi>G</mi></math></span> to <span><math><mi>u</mi></math></span> than to <span><math><mi>v</mi></math></span><span>. Contributing to conjectures posed by Došlić et al., we show that the Mostar index of bipartite graphs of order </span><span><math><mi>n</mi></math></span> is at most <span><math><mrow><mfrac><mrow><msqrt><mrow><mn>3</mn></mrow></msqrt></mrow><mrow><mn>18</mn></mrow></mfrac><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></math></span>, and that the Mostar index of split graphs of order <span><math><mi>n</mi></math></span> is at most <span><math><mrow><mfrac><mrow><mn>4</mn></mrow><mrow><mn>27</mn></mrow></mfrac><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></math></span>.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"48 ","pages":"Article 100768"},"PeriodicalIF":1.1,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49809016","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Maximizing the Mostar index for bipartite graphs and split graphs","authors":"Štefko Miklavič ,&nbsp;Johannes Pardey ,&nbsp;Dieter Rautenbach ,&nbsp;Florian Werner","doi":"10.1016/j.disopt.2023.100768","DOIUrl":"https://doi.org/10.1016/j.disopt.2023.100768","url":null,"abstract":"<div><p>Došlić et al. defined the Mostar index of a graph <span><math><mi>G</mi></math></span> as <span><math><mrow><munder><mrow><mo>∑</mo></mrow><mrow><mi>u</mi><mi>v</mi><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></munder><mspace></mspace><mrow><mo>|</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow><mo>−</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>v</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow></math></span>, where, for an edge <span><math><mrow><mi>u</mi><mi>v</mi></mrow></math></span> of <span><math><mi>G</mi></math></span>, the term <span><math><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow></mrow></math></span> denotes the number of vertices of <span><math><mi>G</mi></math></span> that have a smaller distance in <span><math><mi>G</mi></math></span> to <span><math><mi>u</mi></math></span> than to <span><math><mi>v</mi></math></span><span>. Contributing to conjectures posed by Došlić et al., we show that the Mostar index of bipartite graphs of order </span><span><math><mi>n</mi></math></span> is at most <span><math><mrow><mfrac><mrow><msqrt><mrow><mn>3</mn></mrow></msqrt></mrow><mrow><mn>18</mn></mrow></mfrac><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></math></span>, and that the Mostar index of split graphs of order <span><math><mi>n</mi></math></span> is at most <span><math><mrow><mfrac><mrow><mn>4</mn></mrow><mrow><mn>27</mn></mrow></mfrac><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></math></span>.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"48 ","pages":"Article 100768"},"PeriodicalIF":1.1,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49716674","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the Rényi–Ulam game with restricted size queries","authors":"Ádám X. Fraknói ,&nbsp;Dávid Á. Márton ,&nbsp;Dániel G. Simon ,&nbsp;Dániel A. Lenger","doi":"10.1016/j.disopt.2023.100772","DOIUrl":"https://doi.org/10.1016/j.disopt.2023.100772","url":null,"abstract":"<div><p>We investigate the following version of the well-known Rényi–Ulam game. Two players – the Questioner and the Responder – play against each other. The Responder thinks of a number from the set <span><math><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></mrow></math></span>, and the Questioner has to find this number. To do this, he can ask whether a chosen set of at most <span><math><mi>k</mi></math></span> elements contains the thought number. The Responder answers with YES or NO immediately, but during the game, he may lie at most <span><math><mi>ℓ</mi></math></span> times. The minimum number of queries needed for the Questioner to surely find the unknown element is denoted by <span><math><mrow><mi>R</mi><msubsup><mrow><mi>U</mi></mrow><mrow><mi>ℓ</mi></mrow><mrow><mi>k</mi></mrow></msubsup><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span>. First, we develop a highly effective tool that we call Convexity Lemma. By using this lemma, we give a general lower bound of <span><math><mrow><mi>R</mi><msubsup><mrow><mi>U</mi></mrow><mrow><mi>ℓ</mi></mrow><mrow><mi>k</mi></mrow></msubsup><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> and an upper bound which differs from the lower one by at most <span><math><mrow><mn>2</mn><mi>ℓ</mi><mo>+</mo><mn>1</mn></mrow></math></span>. We also give its exact value when <span><math><mi>n</mi></math></span> is sufficiently large compared to <span><math><mi>k</mi></math></span>. With these, we managed to improve and generalize the results obtained by Meng, Lin, and Yang in a 2013 paper about the case <span><math><mrow><mi>ℓ</mi><mo>=</mo><mn>1</mn></mrow></math></span>.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"48 ","pages":"Article 100772"},"PeriodicalIF":1.1,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49809018","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"EPTAS for load balancing problem on parallel machines with a non-renewable resource","authors":"G. Jaykrishnan,&nbsp;Asaf Levin","doi":"10.1016/j.disopt.2023.100775","DOIUrl":"https://doi.org/10.1016/j.disopt.2023.100775","url":null,"abstract":"<div><p>The problem considered is the non-preemptive scheduling of independent jobs that consume a resource (which is non-renewable and replenished regularly) on parallel uniformly related machines. The input defines the speed of machines, size of jobs, the quantity of the resource required by the jobs, the replenished quantities, and replenishment dates of the resource. Every job can start processing only after the required quantity of the resource is allocated to the job. The objective function is a generalization of makespan minimization and minimization of the <span><math><msub><mrow><mi>l</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span><span>-norm of the vector of loads of the machines. We present an EPTAS for this problem. Prior to our work only a PTAS was known in this non-renewable resource settings only for the special case of our problem of makespan minimization on identical machines.</span></p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"48 ","pages":"Article 100775"},"PeriodicalIF":1.1,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49716351","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Secretary and online matching problems with machine learned advice","authors":"Antonios Antoniadis ,&nbsp;Themis Gouleakis ,&nbsp;Pieter Kleer ,&nbsp;Pavel Kolev","doi":"10.1016/j.disopt.2023.100778","DOIUrl":"https://doi.org/10.1016/j.disopt.2023.100778","url":null,"abstract":"<div><p>The classic analysis of online algorithms, due to its worst-case nature, can be quite pessimistic when the input instance at hand is far from worst-case. In contrast, machine learning approaches shine in exploiting patterns in past inputs in order to predict the future. However, such predictions, although usually accurate, can be arbitrarily poor. Inspired by a recent line of work, we augment three well-known online settings with machine learned predictions about the future, and develop algorithms that take these predictions into account. In particular, we study the following online selection problems: (i) the classic secretary problem, (ii) online bipartite matching and (iii) the graphic matroid secretary problem. Our algorithms still come with a worst-case performance guarantee in the case that predictions are subpar while obtaining an improved competitive ratio (over the best-known classic online algorithm for each problem) when the predictions are sufficiently accurate. For each algorithm, we establish a trade-off between the competitive ratios obtained in the two respective cases.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"48 ","pages":"Article 100778"},"PeriodicalIF":1.1,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49713319","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The complexity of 2-vertex-connected orientation in mixed graphs","authors":"Florian Hörsch ,&nbsp;Zoltán Szigeti","doi":"10.1016/j.disopt.2023.100774","DOIUrl":"https://doi.org/10.1016/j.disopt.2023.100774","url":null,"abstract":"<div><p>We consider two possible extensions of a theorem of Thomassen characterizing the graphs admitting a 2-vertex-connected orientation. First, we show that the problem of deciding whether a mixed graph has a 2-vertex-connected orientation is NP-hard. This answers a question of Bang-Jensen, Huang and Zhu. For the second part, we call a directed graph <span><math><mrow><mi>D</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>A</mi><mo>)</mo></mrow></mrow></math></span>\u0000<span><math><mrow><mn>2</mn><mi>T</mi></mrow></math></span>-connected for some <span><math><mrow><mi>T</mi><mo>⊆</mo><mi>V</mi></mrow></math></span> if <span><math><mi>D</mi></math></span> is 2-arc-connected and <span><math><mrow><mi>D</mi><mo>−</mo><mi>v</mi></mrow></math></span> is strongly connected for all <span><math><mrow><mi>v</mi><mo>∈</mo><mi>T</mi></mrow></math></span>. We deduce a characterization of the graphs admitting a <span><math><mrow><mn>2</mn><mi>T</mi></mrow></math></span>-connected orientation from the theorem of Thomassen.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"48 ","pages":"Article 100774"},"PeriodicalIF":1.1,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49716350","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Reachability in choice networks","authors":"Piotr Wojciechowski ,&nbsp;K. Subramani ,&nbsp;Alvaro Velasquez","doi":"10.1016/j.disopt.2023.100761","DOIUrl":"https://doi.org/10.1016/j.disopt.2023.100761","url":null,"abstract":"&lt;div&gt;&lt;p&gt;In this paper, we investigate the problem of determining &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; reachability in &lt;strong&gt;choice networks&lt;/strong&gt;. In the traditional &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; reachability problem, we are given a weighted network tuple &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;〈&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;〉&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, with the goal of checking if there exists a path from &lt;span&gt;&lt;math&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; to &lt;span&gt;&lt;math&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; in &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. In an optional choice network, we are given a choice set &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;⊆&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, in addition to the network tuple &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. In the &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; reachability problem in choice networks (OCR&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;), the goal is to find whether there exists a path from vertex &lt;span&gt;&lt;math&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; to vertex &lt;span&gt;&lt;math&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, with the caveat that at most one edge from each edge-pair &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is used in the path. OCR&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; finds applications in a number of domains, including &lt;strong&gt;routing in wireless networks&lt;/strong&gt; and &lt;strong&gt;sensor placement&lt;/strong&gt;. We analyze the computational complexities of the OCR&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; problem and its variants from a number of algorithmic perspectives. We show that the problem is &lt;strong&gt;NP-complete&lt;/strong&gt; in directed acyclic graphs with bounded pathwidth. Additionally, we show that its optimization version is &lt;strong&gt;NPO PB-complete&lt;/strong&gt;. Additionally, we show that the problem is fixed-parameter tractable in the cardinality of the choice set &lt;span&gt;&lt;math&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. In particular, we show that the problem can be solved in time &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. We also consider weighted versions of the OCR&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; problem and detail their computational complexities; in particular, the optimization version of the &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;W&lt;/mi&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; problem is &lt;strong&gt;NPO-complete&lt;/strong&gt;. While similar results have been obtained for related problems, our results improve ","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"48 ","pages":"Article 100761"},"PeriodicalIF":1.1,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49716672","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Graphs with equal Grundy domination and independence number","authors":"Gábor Bacsó ,&nbsp;Boštjan Brešar ,&nbsp;Kirsti Kuenzel ,&nbsp;Douglas F. Rall","doi":"10.1016/j.disopt.2023.100777","DOIUrl":"https://doi.org/10.1016/j.disopt.2023.100777","url":null,"abstract":"<div><p>The Grundy domination number, <span><math><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>gr</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, of a graph <span><math><mi>G</mi></math></span> is the maximum length of a sequence <span><math><mrow><mo>(</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow></math></span> of vertices in <span><math><mi>G</mi></math></span> such that for every <span><math><mrow><mi>i</mi><mo>∈</mo><mrow><mo>{</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>}</mo></mrow></mrow></math></span>, the closed neighborhood <span><math><mrow><mi>N</mi><mrow><mo>[</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>]</mo></mrow></mrow></math></span> contains a vertex that does not belong to any closed neighborhood <span><math><mrow><mi>N</mi><mrow><mo>[</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>]</mo></mrow></mrow></math></span>, where <span><math><mrow><mi>j</mi><mo>&lt;</mo><mi>i</mi></mrow></math></span>. It is well known that the Grundy domination number of any graph <span><math><mi>G</mi></math></span> is greater than or equal to the upper domination number <span><math><mrow><mi>Γ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, which is in turn greater than or equal to the independence number <span><math><mrow><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. In this paper, we initiate the study of the class of graphs <span><math><mi>G</mi></math></span> with <span><math><mrow><mi>Γ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>γ</mi></mrow><mrow><mi>gr</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and its subclass consisting of graphs <span><math><mi>G</mi></math></span> with <span><math><mrow><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>γ</mi></mrow><mrow><mi>gr</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. We characterize the latter class of graphs among all twin-free connected graphs, provide a number of properties of these graphs, and prove that the hypercubes are members of this class. In addition, we give several necessary conditions for graphs <span><math><mi>G</mi></math></span> with <span><math><mrow><mi>Γ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>γ</mi></mrow><mrow><mi>gr</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and present large families of such graphs.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"48 ","pages":"Article 100777"},"PeriodicalIF":1.1,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49734399","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"EPTAS for load balancing problem on parallel machines with a non-renewable resource","authors":"G. Jaykrishnan,&nbsp;Asaf Levin","doi":"10.1016/j.disopt.2023.100775","DOIUrl":"https://doi.org/10.1016/j.disopt.2023.100775","url":null,"abstract":"<div><p>The problem considered is the non-preemptive scheduling of independent jobs that consume a resource (which is non-renewable and replenished regularly) on parallel uniformly related machines. The input defines the speed of machines, size of jobs, the quantity of the resource required by the jobs, the replenished quantities, and replenishment dates of the resource. Every job can start processing only after the required quantity of the resource is allocated to the job. The objective function is a generalization of makespan minimization and minimization of the <span><math><msub><mrow><mi>l</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span><span>-norm of the vector of loads of the machines. We present an EPTAS for this problem. Prior to our work only a PTAS was known in this non-renewable resource settings only for the special case of our problem of makespan minimization on identical machines.</span></p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"48 ","pages":"Article 100775"},"PeriodicalIF":1.1,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49809021","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Approximating single- and multi-objective nonlinear sum and product knapsack problems","authors":"Jan Boeckmann ,&nbsp;Clemens Thielen ,&nbsp;Ulrich Pferschy","doi":"10.1016/j.disopt.2023.100771","DOIUrl":"https://doi.org/10.1016/j.disopt.2023.100771","url":null,"abstract":"<div><p>We present a fully polynomial-time approximation scheme (FPTAS) for a very general version of the well-known knapsack problem. This generalization covers, with few exceptions, all versions of knapsack problems that have been studied in the literature so far and allows for an objective function consisting of sums or products of possibly nonlinear, separable item profits, while the knapsack constraint states an upper bound on the sum of possibly nonlinear, separable item weights. Moreover, we extend our FPTAS to a multi-objective fully polynomial-time approximation scheme (MFPTAS) for the multi-objective version of the problem.</p><p>As applications of our general algorithms, we obtain the first FPTAS for the recently-introduced 0–1 time-bomb knapsack problem as well as FPTASs for a variety of robust knapsack problems. Moreover, we extend our FPTAS to the minimization version of our general problem, which, in particular, allows us to explicitly state an FPTAS for the classical minimization knapsack problem, which has been missing in the literature so far.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"48 ","pages":"Article 100771"},"PeriodicalIF":1.1,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49716337","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}