Discrete Optimization最新文献

{"title":"A polyhedral study of lifted multicuts","authors":"Bjoern Andres ,&nbsp;Silvia Di Gregorio ,&nbsp;Jannik Irmai ,&nbsp;Jan-Hendrik Lange","doi":"10.1016/j.disopt.2022.100757","DOIUrl":"https://doi.org/10.1016/j.disopt.2022.100757","url":null,"abstract":"<div><p>Fundamental to many applications in data analysis are the decompositions of a graph, i.e. partitions of the node set into component-inducing subsets. One way of encoding decompositions is by multicuts, the subsets of those edges that straddle distinct components. Recently, a lifting of multicuts from a graph <span><math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></mrow></math></span> to an augmented graph <span><math><mrow><mover><mrow><mi>G</mi></mrow><mrow><mo>̂</mo></mrow></mover><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>∪</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> has been proposed in the field of image analysis, with the goal of obtaining a more expressive characterization of graph decompositions in which it is made explicit also for pairs <span><math><mrow><mi>F</mi><mo>⊆</mo><mfenced><mfrac><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mfenced><mo>∖</mo><mi>E</mi></mrow></math></span> of non-neighboring nodes whether these are in the same or distinct components. In this work, we study in detail the polytope in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>E</mi><mo>∪</mo><mi>F</mi></mrow></msup></math></span> whose vertices are precisely the characteristic vectors of multicuts of <span><math><mover><mrow><mi>G</mi></mrow><mrow><mo>̂</mo></mrow></mover></math></span> lifted from <span><math><mi>G</mi></math></span>, connecting it, in particular, to the rich body of prior work on the clique partitioning and multilinear polytope.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"47 ","pages":"Article 100757"},"PeriodicalIF":1.1,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49731914","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":"Constant factor approximation for tracking paths and fault tolerant feedback vertex set","authors":"Václav Blažej,&nbsp;Pratibha Choudhary,&nbsp;Dušan Knop,&nbsp;Jan Matyáš Křišťan,&nbsp;Ondřej Suchý,&nbsp;Tomáš Valla","doi":"10.1016/j.disopt.2022.100756","DOIUrl":"https://doi.org/10.1016/j.disopt.2022.100756","url":null,"abstract":"<div><p>Consider a vertex-weighted graph <span><math><mi>G</mi></math></span> with a source <span><math><mi>s</mi></math></span> and a target <span><math><mi>t</mi></math></span>. <span>Tracking Paths</span> requires finding a minimum weight set of vertices (<em>trackers</em>) such that the sequence of trackers in each path from <span><math><mi>s</mi></math></span> to <span><math><mi>t</mi></math></span> is unique. In this work, we derive a factor 6-approximation algorithm for <span>Tracking Paths</span> in weighted graphs and a factor 4-approximation algorithm if the input is unweighted. This is the first constant factor approximation for this problem. While doing so, we also study approximation of the closely related <em>r</em>-<span>Fault Tolerant Feedback Vertex Set</span> problem. There, for a fixed integer <span><math><mi>r</mi></math></span> and a given vertex-weighted graph <span><math><mi>G</mi></math></span>, the task is to find a minimum weight set of vertices intersecting every cycle of <span><math><mi>G</mi></math></span> in at least <span><math><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></math></span> vertices. We give a factor <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></mrow></math></span> approximation algorithm for <em>r</em>-<span>Fault Tolerant Feedback Vertex Set</span> if <span><math><mi>r</mi></math></span> is a constant.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"47 ","pages":"Article 100756"},"PeriodicalIF":1.1,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49731732","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":"Fractional Decomposition Tree Algorithm: A tool for studying the integrality gap of Integer Programs","authors":"Robert Carr ,&nbsp;Arash Haddadan ,&nbsp;Cynthia A. Phillips","doi":"10.1016/j.disopt.2022.100746","DOIUrl":"https://doi.org/10.1016/j.disopt.2022.100746","url":null,"abstract":"<div><p><span><span>We present a new algorithm, Fractional Decomposition Tree (FDT), for finding a feasible solution for an integer program (IP) where all variables are binary. FDT runs in polynomial time and is guaranteed to find a feasible integer solution provided the integrality gap of an instance’s </span>polyhedron, independent of objective function, is bounded. The algorithm gives a construction for Carr and Vempala’s theorem that any feasible solution to the IP’s linear-programming relaxation, when scaled by the instance integrality gap, dominates a </span>convex combination of feasible solutions. FDT is also a tool for studying the integrality gap of IP formulations. The upper bound on the integrality gap of an FDT solution can be exponentially large. However our experiments demonstrate that FDT can be effective in practice. We study the integrality gap of two problems: optimally augmenting a tree to a 2-edge-connected graph and finding a minimum-cost 2-edge-connected multi-subgraph (2EC). We also give a simplified algorithm, DomToIP, that finds a feasible solution to an IP instance, or concludes that it has unbounded integrality gap. We show that FDT’s speed and approximation quality compare well to that of the original feasibility pump heuristic on moderate-sized instances of the vertex cover problem. For a particular set of hard-to-decompose fractional 2EC solutions, FDT always gave a better integer solution than the Best-of-Many Christofides Algorithm (BOMC).</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"47 ","pages":"Article 100746"},"PeriodicalIF":1.1,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49714804","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 Aα-spectral radius of nonregular graphs (digraphs) and maximum degree (outdegree)","authors":"Qixuan Yuan,&nbsp;Ruifang Liu,&nbsp;Jinjiang Yuan","doi":"10.1016/j.disopt.2022.100758","DOIUrl":"https://doi.org/10.1016/j.disopt.2022.100758","url":null,"abstract":"","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"47 ","pages":"100758"},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49731736","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 p-center problem under locational uncertainty of demand points","authors":"Homa Ataei ,&nbsp;Mansoor Davoodi","doi":"10.1016/j.disopt.2023.100759","DOIUrl":"https://doi.org/10.1016/j.disopt.2023.100759","url":null,"abstract":"","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"47 ","pages":"100759"},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49715309","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 (d−2)-leaky forcing number of Qd and ℓ-leaky forcing number of GP(n,1)","authors":"Rebekah Herrman","doi":"10.1016/j.disopt.2022.100744","DOIUrl":"10.1016/j.disopt.2022.100744","url":null,"abstract":"<div><p>Leaky-forcing is a recently introduced variant of zero forcing that has been studied for families of graphs including paths, cycles, wheels, grids, and trees. In this paper, we extend previous results on the leaky forcing number of the <span><math><mi>d</mi></math></span>-dimensional hypercube, <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span>, to show that the <span><math><mrow><mo>(</mo><mi>d</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow></math></span>-leaky forcing number of <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> is <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>. We also examine a question about the relationship between the size of a minimum <span><math><mi>ℓ</mi></math></span>-leaky-forcing set and a minimum zero forcing set for a graph <span><math><mi>G</mi></math></span>.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"46 ","pages":"Article 100744"},"PeriodicalIF":1.1,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"54146676","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":"Parameterized algorithms for generalizations of Directed Feedback Vertex Set","authors":"Alexander Göke ,&nbsp;Dániel Marx ,&nbsp;Matthias Mnich","doi":"10.1016/j.disopt.2022.100740","DOIUrl":"https://doi.org/10.1016/j.disopt.2022.100740","url":null,"abstract":"<div><p>The <span>Directed Feedback Vertex Set</span> (DFVS) problem takes as input a directed graph <span><math><mi>G</mi></math></span> and seeks a smallest vertex set <span><math><mi>S</mi></math></span> that hits all cycles in <span><math><mi>G</mi></math></span>. This is one of Karp’s 21 <span><math><mi>NP</mi></math></span>-complete problems. Resolving the parameterized complexity status of DFVS was a long-standing open problem until Chen et al. (2008) showed its fixed-parameter tractability via a <span><math><mrow><msup><mrow><mn>4</mn></mrow><mrow><mi>k</mi></mrow></msup><mi>k</mi><mo>!</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>O</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msup></mrow></math></span>-time algorithm, where <span><math><mrow><mi>k</mi><mo>=</mo><mrow><mo>|</mo><mi>S</mi><mo>|</mo></mrow></mrow></math></span>.</p><p>Here we show fixed-parameter tractability of two generalizations of DFVS: </p><ul><li><span>•</span><span><p>Find a smallest vertex set <span><math><mi>S</mi></math></span> such that every strong component of <span><math><mrow><mi>G</mi><mo>−</mo><mi>S</mi></mrow></math></span> has size at most <span><math><mi>s</mi></math></span>: we give an algorithm solving this problem in time <span><math><mrow><msup><mrow><mn>4</mn></mrow><mrow><mi>k</mi></mrow></msup><mrow><mo>(</mo><mi>k</mi><mi>s</mi><mo>+</mo><mi>k</mi><mo>+</mo><mi>s</mi><mo>)</mo></mrow><mo>!</mo><mi>⋅</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>O</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msup></mrow></math></span>. This generalizes an algorithm by Xiao (2017) for the undirected version of the problem.</p></span></li><li><span>•</span><span><p>Find a smallest vertex set <span><math><mi>S</mi></math></span> such that every non-trivial strong component of <span><math><mrow><mi>G</mi><mo>−</mo><mi>S</mi></mrow></math></span> is 1-out-regular: we give an algorithm solving this problem in time <span><math><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></mrow></mrow></msup><mi>⋅</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>O</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msup></mrow></math></span>.</p></span></li></ul> We also solve the corresponding arc versions of these problems by fixed-parameter algorithms.</div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"46 ","pages":"Article 100740"},"PeriodicalIF":1.1,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"92079883","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":"A polyhedral study for the cubic formulation of the unconstrained traveling tournament problem","authors":"Marije R. Siemann,&nbsp;Matthias Walter","doi":"10.1016/j.disopt.2022.100741","DOIUrl":"https://doi.org/10.1016/j.disopt.2022.100741","url":null,"abstract":"<div><p>We consider the unconstrained traveling tournament problem, a sports timetabling problem that minimizes traveling of teams. Since its introduction about 20 years ago, most research was devoted to modeling and reformulation approaches. In this paper we carry out a polyhedral study for the cubic integer programming formulation by establishing the dimension of the integer hull as well as of faces induced by model inequalities. Moreover, we introduce a new class of inequalities and show that they are facet-defining. Finally, we evaluate the impact of these inequalities on the linear programming bounds.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"46 ","pages":"Article 100741"},"PeriodicalIF":1.1,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1572528622000469/pdfft?md5=2782e081b2be6a05c56eac20c9af53c2&pid=1-s2.0-S1572528622000469-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"92079885","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Block-structured integer programming: Can we parameterize without the largest coefficient?","authors":"Hua Chen ,&nbsp;Lin Chen ,&nbsp;Guochuan Zhang","doi":"10.1016/j.disopt.2022.100743","DOIUrl":"https://doi.org/10.1016/j.disopt.2022.100743","url":null,"abstract":"&lt;div&gt;&lt;p&gt;We consider 4-block &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;-fold integer programming, which can be written as &lt;/span&gt;&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;max&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;mi&gt;⋅&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&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;, where the constraint matrix &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is composed of small matrices &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;B&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;D&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; such that the first row of &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, the first column of &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, the main diagonal of &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, and all the other entries are 0. There are &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; copies of &lt;span&gt;&lt;math&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, and &lt;span&gt;&lt;math&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. The special case where &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is known as &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-fold integer programming.&lt;/p&gt;&lt;p&gt;Prior algorithmic results for 4-block &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-fold integer programming and its special cases usually take &lt;span&gt;&lt;math&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, the largest absolute value among entries of &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, as part of the parameters. In this paper, we explore the possibility of solving the problems polynomially when the number of rows and columns of the small matrices are constant. We show that, assuming &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mtext&gt;P&lt;/mtext&gt;&lt;mo&gt;≠&lt;/mo&gt;&lt;mtext&gt;NP&lt;/mtext&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, this is not possible even if &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&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;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. However, this becomes possible if &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; or &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, or more generally if &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"46 ","pages":"Article 100743"},"PeriodicalIF":1.1,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"92079884","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":"Cardinality constrained connected balanced partitions of trees under different criteria","authors":"Roberto Cordone ,&nbsp;Davide Franchi ,&nbsp;Andrea Scozzari","doi":"10.1016/j.disopt.2022.100742","DOIUrl":"https://doi.org/10.1016/j.disopt.2022.100742","url":null,"abstract":"<div><p>In this paper we study the problem of partitioning a tree with <span><math><mi>n</mi></math></span> weighted vertices into <span><math><mi>p</mi></math></span> connected components. For each component, we measure its <em>gap</em>, that is, the difference between the maximum and the minimum weight of its vertices, with the aim of minimizing the sum of such differences. We present an <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msup><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> time and <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msup><mi>p</mi><mo>)</mo></mrow></mrow></math></span> space algorithm for this problem. Then, we generalize it, requiring a minimum of <span><math><mrow><mi>ϵ</mi><mo>≥</mo><mn>1</mn></mrow></math></span> nodes in each connected component, and provide an <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msup><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mi>ϵ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> time and <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msup><mi>p</mi><mi>ϵ</mi><mo>)</mo></mrow></mrow></math></span> space algorithm to solve this new problem version. We provide a refinement of our analysis involving the topology of the tree and an improvement of the algorithms for the special case in which the weights of the vertices have a heap structure. All presented algorithms can be straightforwardly extended to other similar objective functions. Actually, for the problem of minimizing the maximum gap with a minimum number of nodes in each component, we propose an algorithm which is independent of <span><math><mi>ϵ</mi></math></span> and requires <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>log</mo><mi>n</mi><mspace></mspace><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> time and <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>p</mi><mo>)</mo></mrow></mrow></math></span> space.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":"46 ","pages":"Article 100742"},"PeriodicalIF":1.1,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"92024745","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}