Discrete Applied Mathematics最新文献

{"title":"Infinite grid exploration with synchronous myopic robots without chirality","authors":"Quentin Bramas ,&nbsp;Pascal Lafourcade ,&nbsp;Stéphane Devismes","doi":"10.1016/j.dam.2025.05.031","DOIUrl":"10.1016/j.dam.2025.05.031","url":null,"abstract":"<div><div>In this paper, we consider the exploration of an infinite grid by a swarm of fully-synchronous robots with weak capabilities: they are disoriented, opaque, do not communicate explicitely, have limited visibility, and cannot occupy the same position at the same time.</div><div>Our first result shows that, in this context, minimizing the visibility range and the number of used colors are two orthogonal issues: it is impossible to design a solution to our exploration problem that is optimal <em>w.r.t.</em> both parameters simultaneously. Consequently, we address optimality of these two criteria separately by proposing two algorithms; the former being optimal in terms of visibility range, the latter being optimal in terms of number of used colors. More precisely, the first algorithm solves the problem using eight oblivious robots under visibility range two (this visibility being optimal when considering oblivious robots), and the second algorithm solves the problem under visibility range one using six robots and two colors (which is optimal under this visibility range). Finally, we also tackle the optimality in terms of number of robots. According to the lower bound given in Bramas et al. (2020), we propose an algorithm working with a minimum number of robots (five) under visibility range one. This latter uses twelve colors and also guarantees that nodes are visited infinitely often.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"375 ","pages":"Pages 193-214"},"PeriodicalIF":1.0,"publicationDate":"2025-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144298870","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Fairness in graph-theoretical optimization problems","authors":"Christopher Hojny,&nbsp;Frits Spieksma,&nbsp;Sten Wessel","doi":"10.1016/j.dam.2025.05.026","DOIUrl":"10.1016/j.dam.2025.05.026","url":null,"abstract":"<div><div>There is arbitrariness in optimum solutions of graph-theoretic problems that can give rise to unfairness. Incorporating fairness in such problems, however, can be done in multiple ways. For instance, fairness can be defined on an individual level, for individual vertices or edges of a given graph, or on a group level. In this work, we analyze in detail two individual-fairness measures that are based on finding a probability distribution over the set of solutions. One measure guarantees uniform fairness, i.e., entities have equal chance of being part of the solution when sampling from this probability distribution. The other measure maximizes the minimum probability for every entity of being selected in a solution. In particular, we reveal that computing these individual-fairness measures is in fact equivalent to computing the fractional covering number and the fractional partitioning number of a hypergraph. In addition, we show that for a general class of problems that we classify as independence systems, these two measures coincide. We also analyze group fairness and how this can be combined with the individual-fairness measures. Finally, we establish the computational complexity of determining group-fair solutions for a variant of the matching problem.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"375 ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144290572","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The resistance distance of a dual number weighted graph","authors":"Yu Li,&nbsp;Lizhu Sun,&nbsp;Changjiang Bu","doi":"10.1016/j.dam.2025.05.041","DOIUrl":"10.1016/j.dam.2025.05.041","url":null,"abstract":"<div><div>For 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>, assigning each edge <span><math><mrow><mi>e</mi><mo>∈</mo><mi>E</mi></mrow></math></span> a weight of a dual number <span><math><mrow><mi>w</mi><mrow><mo>(</mo><mi>e</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn><mo>+</mo><msub><mrow><mover><mrow><mi>a</mi></mrow><mrow><mo>̂</mo></mrow></mover></mrow><mrow><mi>e</mi></mrow></msub><mi>ɛ</mi></mrow></math></span>, the weighted graph <span><math><mrow><msup><mrow><mi>G</mi></mrow><mrow><mi>w</mi></mrow></msup><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></mrow></math></span> is called a dual number weighted graph, where <span><math><mrow><mo>−</mo><msub><mrow><mover><mrow><mi>a</mi></mrow><mrow><mo>̂</mo></mrow></mover></mrow><mrow><mi>e</mi></mrow></msub></mrow></math></span> can be regarded as the perturbation of the unit resistor on the edge <span><math><mi>e</mi></math></span> of <span><math><mi>G</mi></math></span>. For a connected dual number weighted graph <span><math><msup><mrow><mi>G</mi></mrow><mrow><mi>w</mi></mrow></msup></math></span>, we give some expressions and block representations of generalized inverses of the Laplacian matrix of <span><math><msup><mrow><mi>G</mi></mrow><mrow><mi>w</mi></mrow></msup></math></span>. And using these results, we derive the explicit formulas of the resistance distance and Kirchhoff index of <span><math><msup><mrow><mi>G</mi></mrow><mrow><mi>w</mi></mrow></msup></math></span>. We give the perturbation bounds for the resistance distance and Kirchhoff index of <span><math><mi>G</mi></math></span>. In particular, when only the edge <span><math><mrow><mi>e</mi><mo>=</mo><mrow><mo>{</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>}</mo></mrow></mrow></math></span> of <span><math><mi>G</mi></math></span> is perturbed, we give the perturbation bounds for the Kirchhoff index and resistance distance between vertices <span><math><mi>i</mi></math></span> and <span><math><mi>j</mi></math></span> of <span><math><mi>G</mi></math></span>, respectively.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"375 ","pages":"Pages 154-165"},"PeriodicalIF":1.0,"publicationDate":"2025-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144289075","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the size of immune sets in the k-PULL infection model","authors":"Josep Fàbrega ,&nbsp;Xavier Marcote ,&nbsp;Xavier Muñoz","doi":"10.1016/j.dam.2025.05.047","DOIUrl":"10.1016/j.dam.2025.05.047","url":null,"abstract":"<div><div>This paper addresses immune sets in graphs, which are subsets of nodes that remain unaffected during the spread of influence, failure, or infection. The specific propagation model examined is the <span><math><mi>k</mi></math></span>-PULL infection rule, also referred to as bootstrap percolation. Studying immune sets offers important insights into the structural vulnerabilities and defensive capabilities of networks. In particular, we establish upper bounds for the size of minimal <span><math><mi>k</mi></math></span>-immune sets in graphs with a given maximum degree. Additionally, we focus on the <span><math><mi>k</mi></math></span>-immune number of a graph, defined as the minimum number of vertices in a <span><math><mi>k</mi></math></span>-immune set, and we derive bounds for this parameter. Lastly, we investigate the <span><math><mi>k</mi></math></span>-immune number of the Cartesian product of two graphs.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"375 ","pages":"Pages 166-177"},"PeriodicalIF":1.0,"publicationDate":"2025-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144289074","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Constant time enumeration of weighted trees","authors":"Mengze Qian,&nbsp;Ryuhei Uehara","doi":"10.1016/j.dam.2025.05.043","DOIUrl":"10.1016/j.dam.2025.05.043","url":null,"abstract":"<div><div>This study specifically addresses the enumeration problem of the class of weighted trees. A weighted tree is a rooted unordered tree, and each vertex in a weighted tree is assigned an integer weight. The task is to enumerate the weighted trees. Given a tree <span><math><mi>T</mi></math></span> and a weight <span><math><mi>w</mi></math></span>, the objective is to enumerate all weighted trees that share the same tree structure as <span><math><mi>T</mi></math></span> and have a total weight of <span><math><mi>w</mi></math></span>. Note that the weight assigned to each vertex must be non-negative. The algorithm utilizes reverse search and enumerates each weighted tree in a constant amortized time. By combining our findings with the enumeration of the class of rooted trees, we can efficiently enumerate every weighted tree with a maximum of <span><math><mi>n</mi></math></span> vertices and a total weight of <span><math><mi>w</mi></math></span> for any given positive inputs <span><math><mi>n</mi></math></span> and <span><math><mi>w</mi></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"375 ","pages":"Pages 129-138"},"PeriodicalIF":1.0,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144272534","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The number of nonisomorphic nonorientable 6-gonal embeddings of complete graphs","authors":"Vladimir P. Korzhik","doi":"10.1016/j.dam.2025.05.040","DOIUrl":"10.1016/j.dam.2025.05.040","url":null,"abstract":"<div><div>We show that the number of nonisomorphic nonorientable 6-gonal embeddings of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, <span><math><mrow><mi>n</mi><mo>≡</mo><mn>0</mn></mrow></math></span> or <span><math><mrow><mn>1</mn><mspace></mspace><mrow><mo>(</mo><mo>mod</mo><mspace></mspace><mn>3</mn><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>n</mi><mo>≥</mo><mn>6</mn></mrow></math></span>, such that the boundary walk of every face passes through some vertices twice is greater than the number of nonisomorphic (both orientable and nonorientable) triangular embeddings of the complete graph.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"375 ","pages":"Pages 122-128"},"PeriodicalIF":1.0,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144272532","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Constructions of 2-rotation symmetric bent functions based on Maiorana–McFarland’s bent function","authors":"Yu Guan,&nbsp;Sihong Su","doi":"10.1016/j.dam.2025.05.032","DOIUrl":"10.1016/j.dam.2025.05.032","url":null,"abstract":"<div><div>Bent functions are maximally nonlinear Boolean functions. They are important functions introduced by Rothaus and studied firstly by Dillon and next by many researchers for more than five decades. Rotation symmetric Boolean functions, introduced by Pieprzyk and Qu, are those Boolean functions which are invariant under the cyclic shifts of inputs. In this paper, we first give two flexible construction methods of bent functions on <span><math><mi>n</mi></math></span> variables by defining two subsets <span><math><mi>T</mi></math></span>’s of <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> to modify the support of Maiorana–McFarland’s bent function <span><math><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>=</mo><mi>x</mi><mi>⋅</mi><mi>π</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>⊕</mo><mi>h</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><mi>n</mi><mo>=</mo><mn>2</mn><mi>m</mi></mrow></math></span>, <span><math><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msubsup></mrow></math></span>, <span><math><mi>π</mi></math></span> is a permutation over <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msubsup></math></span>, and <span><math><mi>h</mi></math></span> is a Boolean function on <span><math><mi>m</mi></math></span> variables. These methods have corresponding constraints on <span><math><mi>π</mi></math></span> and <span><math><mi>h</mi></math></span>. Then, we deduce the dual functions of the newly constructed bent functions. Lastly, we propose the methods of constructing 2-rotation symmetric bent functions by redefining the two subsets <span><math><mi>T</mi></math></span>’s of <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> and by imposing more restrictive constraints on <span><math><mi>π</mi></math></span> and <span><math><mi>h</mi></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"375 ","pages":"Pages 139-153"},"PeriodicalIF":1.0,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144272533","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Online and offline packing cylinders in a cylindrical container","authors":"Rashad Moqa ,&nbsp;Jiongzhi Zheng ,&nbsp;Kun He","doi":"10.1016/j.dam.2025.05.039","DOIUrl":"10.1016/j.dam.2025.05.039","url":null,"abstract":"<div><div>This paper addresses the Single Cylindrical Packing Problem with Cylindrical Items (SCPP-CI), which packs cylindrical items into a cylindrical container to maximize the volume utilization. Two variants of the problem are considered: the online version, where items are coming sequentially and need to be packed immediately once coming, and the offline version, where all items are given at once. A deterministic greedy heuristic algorithm is proposed for both the online and offline problems, using two special operators to obtain dense placements and evaluate the quality of feasible placements. In addition, a top-local search algorithm is introduced that considers future benefits for the offline problem. To evaluate the performance of the proposed algorithms, we first degrade the 3D problem to a corresponding packing problem of circles and then compare our results with those found (1) by the CPLEX solver, and (2) by the known heuristic for 2D benchmark instances. We generate 10 groups of the SCPP-CI problem instances (100 instances in each group) with increasing heterogeneity, and compare the results found by our approach with those found by the CPLEX solver for these instances. Computational results show that both our algorithms significantly outperform the baselines for both 2D and 3D problems, indicating the excellent performance of the proposed approach.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"375 ","pages":"Pages 85-104"},"PeriodicalIF":1.0,"publicationDate":"2025-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144262441","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 of the maximum-impact coloring problem on hypergraphs","authors":"Jessica Singer ,&nbsp;Javier Marenco","doi":"10.1016/j.dam.2025.05.042","DOIUrl":"10.1016/j.dam.2025.05.042","url":null,"abstract":"<div><div>Given 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> and a hypergraph <span><math><mrow><mi>H</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> over the same set of vertices, and a finite color set <span><math><mi>C</mi></math></span>, the <em>maximum-impact coloring problem on hypergraphs</em> asks for a <span><math><mi>C</mi></math></span>-coloring of <span><math><mi>G</mi></math></span> maximizing the number of hyperedges of <span><math><mi>H</mi></math></span> whose vertices are assigned the same color. This problem arises in the context of classroom assignment to courses, in which we need to assign a classroom to each lecture and we wish to assign the same classroom to all lectures from the same course. Since imposing this last concern as a constraint may be too restrictive, we seek to maximize the number of courses such that all of its lectures are assigned to the same classroom. In this work we present an integer programming formulation for this NP-hard problem and we explore the associated polytope. We present three families of facet-inducing inequalities, and we report computational experiments suggesting that the dynamic addition of these inequalities within a branch and cut environment may be effective in practice.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"375 ","pages":"Pages 105-121"},"PeriodicalIF":1.0,"publicationDate":"2025-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144262443","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"l1-embeddability under the edge-gluing operation on some nonbipartite graphs","authors":"Guangfu Wang ,&nbsp;Chenyang Li","doi":"10.1016/j.dam.2025.05.036","DOIUrl":"10.1016/j.dam.2025.05.036","url":null,"abstract":"<div><div>An <span><math><msub><mrow><mi>l</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-graph is a graph whose graphic metric space is isomorphic to a subspace of some <span><math><msub><mrow><mi>l</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-space. In this work, we study the <span><math><msub><mrow><mi>l</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-embeddability of graphs obtained by gluing two nonbipartite graphs along an edge, which are from the complete graphs, cocktail party graphs and half-cubes. We show that only the graph obtained by gluing two complete graphs along an edge is also an <span><math><msub><mrow><mi>l</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-graph, and all the other graphs are not, except the trivial cases. This gives an answer to the problem proposed by Wang and Zhang (2013).</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"375 ","pages":"Pages 79-84"},"PeriodicalIF":1.0,"publicationDate":"2025-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144253618","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}