Clobber is a two-player combinatorial game played on a graph with black and white stones. The Solitaire Clobber game is a single-player variant of Clobber that was introduced by Demaine et al. (2004). In the initial position, a black or white stone is placed at each vertex of an undirected graph. A player picks up a stone and clobbers a stone of different color located at an adjacent vertex; the clobbered stone is replaced by the stone that has been moved. Dantas et al. (2020) defined the -correducibility of a graph as follows: a graph is -correducible if for every non-monochromatic initial configuration of stones on and every subset of with , there exists a procedure for moving stones such that is empty of stones. Dantas et al. (2020) showed that a graph is -correducible if and only if it is -connected for and every -connected graph is -correducible for .
We improve their result and show that every -connected graph is -correducible for . Moreover, our connectivity condition is sharp, that is, there are infinitely many -connected graphs that are not -correducible.
{"title":"The Solitaire Clobber game and the correducibility of k-connected graphs","authors":"Tatsuya Fujimori , Shun-ichi Maezawa , Yoshio Okamoto","doi":"10.1016/j.dam.2025.01.009","DOIUrl":"10.1016/j.dam.2025.01.009","url":null,"abstract":"<div><div>Clobber is a two-player combinatorial game played on a graph with black and white stones. The Solitaire Clobber game is a single-player variant of Clobber that was introduced by Demaine et al. (2004). In the initial position, a black or white stone is placed at each vertex of an undirected graph. A player picks up a stone and clobbers a stone of different color located at an adjacent vertex; the clobbered stone is replaced by the stone that has been moved. Dantas et al. (2020) defined the <span><math><mi>k</mi></math></span>-correducibility of a graph as follows: a graph <span><math><mi>G</mi></math></span> is <span><math><mi>k</mi></math></span>-correducible if for every non-monochromatic initial configuration of stones on <span><math><mi>G</mi></math></span> and every subset <span><math><mi>S</mi></math></span> of <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><mrow><mo>|</mo><mi>S</mi><mo>|</mo></mrow><mo>≤</mo><mi>k</mi></mrow></math></span>, there exists a procedure for moving stones such that <span><math><mi>S</mi></math></span> is empty of stones. Dantas et al. (2020) showed that a graph is <span><math><mi>ℓ</mi></math></span>-correducible if and only if it is <span><math><mi>ℓ</mi></math></span>-connected for <span><math><mrow><mi>ℓ</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></math></span> and every <span><math><mfenced><mrow><mi>k</mi><mo>+</mo><mrow><mo>⌈</mo><mfrac><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></mrow></mrow></mfenced></math></span>-connected graph is <span><math><mi>k</mi></math></span>-correducible for <span><math><mrow><mi>k</mi><mo>≥</mo><mn>1</mn></mrow></math></span>.</div><div>We improve their result and show that every <span><math><mi>k</mi></math></span>-connected graph is <span><math><mi>k</mi></math></span>-correducible for <span><math><mrow><mi>k</mi><mo>≥</mo><mn>1</mn></mrow></math></span>. Moreover, our connectivity condition is sharp, that is, there are infinitely many <span><math><mrow><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span>-connected graphs that are not <span><math><mi>k</mi></math></span>-correducible.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"366 ","pages":"Pages 16-22"},"PeriodicalIF":1.0,"publicationDate":"2025-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143161751","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}
Pub Date : 2025-01-17DOI: 10.1016/j.dam.2025.01.015
Nader H. Bshouty , Tsun-Ming Cheung , Gergely Harcos , Hamed Hatami , Anthony Ostuni
Efficiently counting or detecting defective items is a crucial task in various fields ranging from biological testing to quality control to streaming algorithms. The group testing estimation problem concerns estimating the number of defective elements in a collection of total within a given factor. We primarily consider the classical query model, in which a query reveals whether the selected group of elements contains a defective one. We show that any non-adaptive randomized algorithm that estimates the value of within a constant factor requires queries. This confirms that a known upper bound by Bshouty (2019) is tight and resolves a conjecture by Damaschke and Sheikh Muhammad (2010). Additionally, we prove similar matching upper and lower bounds in the threshold query model.
{"title":"A tight lower bound on non-adaptive group testing estimation","authors":"Nader H. Bshouty , Tsun-Ming Cheung , Gergely Harcos , Hamed Hatami , Anthony Ostuni","doi":"10.1016/j.dam.2025.01.015","DOIUrl":"10.1016/j.dam.2025.01.015","url":null,"abstract":"<div><div>Efficiently counting or detecting defective items is a crucial task in various fields ranging from biological testing to quality control to streaming algorithms. The <em>group testing estimation problem</em> concerns estimating the number of defective elements <span><math><mi>d</mi></math></span> in a collection of <span><math><mi>n</mi></math></span> total within a given factor. We primarily consider the classical query model, in which a query reveals whether the selected group of elements contains a defective one. We show that any non-adaptive randomized algorithm that estimates the value of <span><math><mi>d</mi></math></span> within a constant factor requires <span><math><mrow><mi>Ω</mi><mrow><mo>(</mo><mo>log</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> queries. This confirms that a known <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mo>log</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> upper bound by Bshouty (2019) is tight and resolves a conjecture by Damaschke and Sheikh Muhammad (2010). Additionally, we prove similar matching upper and lower bounds in the threshold query model.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"366 ","pages":"Pages 1-15"},"PeriodicalIF":1.0,"publicationDate":"2025-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143161746","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}
Pub Date : 2025-01-16DOI: 10.1016/j.dam.2024.12.030
Thomas M. Kratzke , Douglas B. West
A multi-interval representation of a simple graph assigns each vertex a union of disjoint real intervals so that vertices are adjacent if and only if their assigned sets intersect. The total interval number is the minimum of the total number of intervals used in such a representation of . We present a linear-time algorithm to compute when every block of is a complete graph or a cycle. We apply the algorithm to prove extremal results. For an -vertex cactus (every block is an edge or a cycle), the maximum of is . For an -vertex block graph (every block is a complete graph), the maximum is . In both extremal results there are a few small exceptions.
{"title":"The total interval number of a graph, III: Tree-like graphs","authors":"Thomas M. Kratzke , Douglas B. West","doi":"10.1016/j.dam.2024.12.030","DOIUrl":"10.1016/j.dam.2024.12.030","url":null,"abstract":"<div><div>A <em>multi-interval representation</em> of a simple graph <span><math><mi>G</mi></math></span> assigns each vertex a union of disjoint real intervals so that vertices are adjacent if and only if their assigned sets intersect. The <em>total interval number</em> <span><math><mrow><mi>I</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is the minimum of the total number of intervals used in such a representation of <span><math><mi>G</mi></math></span>. We present a linear-time algorithm to compute <span><math><mrow><mi>I</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> when every block of <span><math><mi>G</mi></math></span> is a complete graph or a cycle. We apply the algorithm to prove extremal results. For an <span><math><mi>n</mi></math></span>-vertex cactus (every block is an edge or a cycle), the maximum of <span><math><mrow><mi>I</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is <span><math><mfenced><mrow><mrow><mo>(</mo><mn>18</mn><mi>n</mi><mo>−</mo><mn>12</mn><mo>)</mo></mrow><mo>/</mo><mn>13</mn></mrow></mfenced></math></span>. For an <span><math><mi>n</mi></math></span>-vertex block graph (every block is a complete graph), the maximum is <span><math><mfenced><mrow><mn>3</mn><mi>n</mi><mo>/</mo><mn>2</mn><mo>−</mo><mn>2</mn></mrow></mfenced></math></span>. In both extremal results there are a few small exceptions.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"365 ","pages":"Pages 163-189"},"PeriodicalIF":1.0,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150131","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}
Processor fault diagnosis is of great significance in fault-tolerant computing of multiprocessor systems. Since interconnection networks are the underlying topology of many multiprocessor systems, the fault tolerance of interconnection networks will directly affect the stability of multiprocessor systems. Non-inclusive diagnosability is a new parameter for fault tolerance evaluation of interconnection networks. In 2023, Guo and Ekinci proposed a new type of network topology called folded hypercube-like networks ( ). This article studies and analyzes the structure of and finds that the family of has a more diverse structure than the hypercube-like networks. Besides, we divide into three categories and solve the non-inclusive diagnosability of , which are , and , respectively.
{"title":"Non-inclusive diagnosability of folded hypercube-like networks","authors":"Nengjin Zhuo , Shumin Zhang , Jou-Ming Chang , Chengfu Ye","doi":"10.1016/j.dam.2025.01.006","DOIUrl":"10.1016/j.dam.2025.01.006","url":null,"abstract":"<div><div>Processor fault diagnosis is of great significance in fault-tolerant computing of multiprocessor systems. Since interconnection networks are the underlying topology of many multiprocessor systems, the fault tolerance of interconnection networks will directly affect the stability of multiprocessor systems. Non-inclusive diagnosability is a new parameter for fault tolerance evaluation of interconnection networks. In 2023, Guo and Ekinci proposed a new type of network topology called folded hypercube-like networks ( <span><math><mrow><mi>F</mi><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span>). This article studies and analyzes the structure of <span><math><mrow><mi>F</mi><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span> and finds that the family of <span><math><mrow><mi>F</mi><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span> has a more diverse structure than the hypercube-like networks. Besides, we divide <span><math><mrow><mi>F</mi><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span> into three categories and solve the non-inclusive diagnosability of <span><math><mrow><mi>F</mi><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span>, which are <span><math><mrow><mn>2</mn><mi>n</mi></mrow></math></span>, <span><math><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></mrow></math></span>, respectively.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"364 ","pages":"Pages 237-246"},"PeriodicalIF":1.0,"publicationDate":"2025-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143142514","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}
Pub Date : 2025-01-14DOI: 10.1016/j.dam.2025.01.008
Bo Ning
The classic Dirac’s 1952 theorem states that every graph on vertices with minimum degree at least contains a Hamilton cycle. A natural problem is whether there is an inductive proof of Dirac’s theorem which avoids Pósa’s rotation technique? In this note, we give a solution to this problem. We left two open problems for further study.
{"title":"An inductive proof of Dirac’s theorem on Hamilton cycles","authors":"Bo Ning","doi":"10.1016/j.dam.2025.01.008","DOIUrl":"10.1016/j.dam.2025.01.008","url":null,"abstract":"<div><div>The classic Dirac’s 1952 theorem states that every graph on <span><math><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow></math></span> vertices with minimum degree at least <span><math><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> contains a Hamilton cycle. A natural problem is whether there is an inductive proof of Dirac’s theorem which avoids Pósa’s rotation technique? In this note, we give a solution to this problem. We left two open problems for further study.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"365 ","pages":"Pages 160-162"},"PeriodicalIF":1.0,"publicationDate":"2025-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150140","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}
Pub Date : 2025-01-13DOI: 10.1016/j.dam.2024.12.029
Xianghui Zhong
In this paper we investigate instances with high integrality ratio of the subtour LP. We determine the instances maximizing the integrality ratio for Rectilinear TSP with up to 10 vertices and for Multidimensional Rectilinear TSP with up to 12 vertices. Based on these instances we give families of instances whose integrality ratio converges to for Rectilinear, Multidimensional Rectilinear and Euclidean TSP that have similar structures.
We also investigate the concept of local optimality with respect to integrality ratio and develop several algorithms to find instances with high integrality ratio. Furthermore, we describe a family of instances that are hard to solve in practice. The currently fastest TSP solver Concorde needs more than two days to solve an instance from the family with 52 vertices.
{"title":"Lower bounds on the integrality ratio of the subtour LP for the traveling salesman problem","authors":"Xianghui Zhong","doi":"10.1016/j.dam.2024.12.029","DOIUrl":"10.1016/j.dam.2024.12.029","url":null,"abstract":"<div><div>In this paper we investigate instances with high integrality ratio of the subtour LP. We determine the instances maximizing the integrality ratio for Rectilinear TSP with up to 10 vertices and for Multidimensional Rectilinear TSP with up to 12 vertices. Based on these instances we give families of instances whose integrality ratio converges to <span><math><mfrac><mrow><mn>4</mn></mrow><mrow><mn>3</mn></mrow></mfrac></math></span> for <span>Rectilinear</span>, <span>Multidimensional Rectilinear</span> and <span>Euclidean TSP</span> that have similar structures.</div><div>We also investigate the concept of local optimality with respect to integrality ratio and develop several algorithms to find instances with high integrality ratio. Furthermore, we describe a family of instances that are hard to solve in practice. The currently fastest TSP solver <span>Concorde</span> needs more than two days to solve an instance from the family with 52 vertices.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"365 ","pages":"Pages 109-129"},"PeriodicalIF":1.0,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150813","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Boolean functions are important primitives in different domains of cryptology, complexity and coding theory, and far beyond in different areas of science and technology. In this paper we connect the tools of cryptology and complexity theory in the domain of resilient Boolean functions. It is well known that the resiliency of a Boolean function and its polynomial degree are directly connected. We first show that borrowing an idea from complexity theory, one can implement resilient Boolean functions on a large number of variables with little amount of circuit. Further, we also look into the search techniques used in the construction of resilient Boolean functions to show the existence and non-existence results of functions with low polynomial degree and high sensitivity on small number of variables. In the process, we settle some previously open problems. Finally, we extend the notion of sensitivity to higher order and present a construction with low polynomial degree and higher order sensitivity exploiting Maiorana-McFarland functions. The questions we raise identify novel combinatorial problems in the domain of Boolean functions.
{"title":"Interplay between resiliency and polynomial degree — Recursive amplification, higher order sensitivity and beyond","authors":"Subhamoy Maitra , Chandra Sekhar Mukherjee , Pantelimon Stănică , Deng Tang","doi":"10.1016/j.dam.2025.01.003","DOIUrl":"10.1016/j.dam.2025.01.003","url":null,"abstract":"<div><div>Boolean functions are important primitives in different domains of cryptology, complexity and coding theory, and far beyond in different areas of science and technology. In this paper we connect the tools of cryptology and complexity theory in the domain of resilient Boolean functions. It is well known that the resiliency of a Boolean function and its polynomial degree are directly connected. We first show that borrowing an idea from complexity theory, one can implement resilient Boolean functions on a large number of variables with little amount of circuit. Further, we also look into the search techniques used in the construction of resilient Boolean functions to show the existence and non-existence results of functions with low polynomial degree and high sensitivity on small number of variables. In the process, we settle some previously open problems. Finally, we extend the notion of sensitivity to higher order and present a construction with low polynomial degree and higher order sensitivity exploiting Maiorana-McFarland functions. The questions we raise identify novel combinatorial problems in the domain of Boolean functions.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"365 ","pages":"Pages 138-159"},"PeriodicalIF":1.0,"publicationDate":"2025-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150142","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}
Pub Date : 2025-01-12DOI: 10.1016/j.dam.2025.01.004
Vladimir I. Benediktovich
More than 50 years ago Chvátal introduced a new graph invariant, which he called graph toughness. From then on a lot of research has been conducted, mainly related to the relationship between toughness conditions and the existence of cyclic structures, in particular, determining whether the graph is Hamiltonian and pancyclic. A pancyclic graph is certainly Hamiltonian, but not conversely. Bondy in 1973, however, suggested the “metaconjecture”that almost any nontrivial condition on a graph which implies that the graph is Hamiltonian also implies that the graph is pancyclic. We confirm the Bondy’s metaconjecture for -tough graphs in the case when in terms of the edge number, the spectral radius and the signless Laplacian spectral radius of the graph.
{"title":"Spectral conditions of pancyclicity for t-tough graphs","authors":"Vladimir I. Benediktovich","doi":"10.1016/j.dam.2025.01.004","DOIUrl":"10.1016/j.dam.2025.01.004","url":null,"abstract":"<div><div>More than 50 years ago Chvátal introduced a new graph invariant, which he called graph toughness. From then on a lot of research has been conducted, mainly related to the relationship between toughness conditions and the existence of cyclic structures, in particular, determining whether the graph is Hamiltonian and pancyclic. A pancyclic graph is certainly Hamiltonian, but not conversely. Bondy in 1973, however, suggested the “metaconjecture”that almost any nontrivial condition on a graph which implies that the graph is Hamiltonian also implies that the graph is pancyclic. We confirm the Bondy’s metaconjecture for <span><math><mi>t</mi></math></span>-tough graphs in the case when <span><math><mrow><mi>t</mi><mo>∈</mo><mrow><mo>{</mo><mn>1</mn><mo>;</mo><mn>2</mn><mo>;</mo><mn>3</mn><mo>}</mo></mrow></mrow></math></span> in terms of the edge number, the spectral radius and the signless Laplacian spectral radius of the graph.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"365 ","pages":"Pages 130-137"},"PeriodicalIF":1.0,"publicationDate":"2025-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150129","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}
Pub Date : 2025-01-10DOI: 10.1016/j.dam.2025.01.002
Xia Li , Weihua Yang , Bo Zhang , Shuang Zhao
Kronk introduced the -path Hamiltonicity of graphs in 1969. A graph is -path Hamiltonian if every path of length not exceeding is contained in a Hamiltonian cycle. We have shown that if is a 2-path of a 2-connected, -regular graph on at most vertices and is connected, then there must exist a Hamiltonian cycle in that contains the 2-path . In this paper, we characterize a class of graphs that illustrate the sharpness of the bound . Additionally, we show that by excluding the class of graphs, both 2-connected, -regular graphs on at most vertices and 3-connected, -regular graphs on at most vertices satisfy that there is a Hamiltonian cycle containing the 2-path if is connected.
{"title":"On regular 2-path Hamiltonian graphs","authors":"Xia Li , Weihua Yang , Bo Zhang , Shuang Zhao","doi":"10.1016/j.dam.2025.01.002","DOIUrl":"10.1016/j.dam.2025.01.002","url":null,"abstract":"<div><div>Kronk introduced the <span><math><mi>l</mi></math></span>-path Hamiltonicity of graphs in 1969. A graph is <span><math><mi>l</mi></math></span>-path Hamiltonian if every path of length not exceeding <span><math><mi>l</mi></math></span> is contained in a Hamiltonian cycle. We have shown that if <span><math><mrow><mi>P</mi><mo>=</mo><mi>u</mi><mi>v</mi><mi>z</mi></mrow></math></span> is a 2-path of a 2-connected, <span><math><mi>k</mi></math></span>-regular graph on at most <span><math><mrow><mn>2</mn><mi>k</mi></mrow></math></span> vertices and <span><math><mrow><mi>G</mi><mo>−</mo><mi>V</mi><mrow><mo>(</mo><mi>P</mi><mo>)</mo></mrow></mrow></math></span> is connected, then there must exist a Hamiltonian cycle in <span><math><mi>G</mi></math></span> that contains the 2-path <span><math><mi>P</mi></math></span>. In this paper, we characterize a class of graphs that illustrate the sharpness of the bound <span><math><mrow><mn>2</mn><mi>k</mi></mrow></math></span>. Additionally, we show that by excluding the class of graphs, both 2-connected, <span><math><mi>k</mi></math></span>-regular graphs on at most <span><math><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></math></span> vertices and 3-connected, <span><math><mi>k</mi></math></span>-regular graphs on at most <span><math><mrow><mn>3</mn><mi>k</mi><mo>−</mo><mn>6</mn></mrow></math></span> vertices satisfy that there is a Hamiltonian cycle containing the 2-path <span><math><mi>P</mi></math></span> if <span><math><mrow><mi>G</mi><mo>−</mo><mi>V</mi><mrow><mo>(</mo><mi>P</mi><mo>)</mo></mrow></mrow></math></span> is connected.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"365 ","pages":"Pages 61-70"},"PeriodicalIF":1.0,"publicationDate":"2025-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150695","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}
Pub Date : 2025-01-10DOI: 10.1016/j.dam.2024.12.026
Minh Hieu Nguyen, Mourad Baiou, Viet Hung Nguyen
This paper deals with a particular case of Bi-Objective Optimization called Bi-Objective Discrete Optimization (BODO), where the feasible set is discrete, and the two objectives take only positive values. Since the feasible set of a BODO problem is discrete and usually finite, it can theoretically be enumerated to identify the Pareto set, which consists of all Pareto-optimal solutions representing different trade-offs between the two objectives. However, in general, this problem is challenging due to two main issues: time complexity, as the number of Pareto-optimal solutions can be exponentially large, and lack of decisiveness. From a practical point of view, the Central Decision Maker (CDM) may be interested in a reduced Pareto set that reflects the CDM’s preferences, which can be obtained by a computationally efficient algorithm.
In this paper, we propose a new criterion for selecting solutions within the Pareto set of BODO. For this purpose, we focus on solutions achieving proportional fairness between two objectives, called generalized Nash Fairness solutions (-NF solutions). The positive parameter , provided by the CDM, reflects the relative importance of the first objective compared to the second one.
We first introduce the -NF solution concept for BODO. We then show that the -NF solution set is a subset of the Pareto set, and this inclusion can be strict. We also propose a recursive Newton-like algorithm for determining the -NF solution set. Finally, an illustrative example of BODO is given.
{"title":"Generalized Nash Fairness Solutions for Bi-Objective Discrete Optimization: Theory and Algorithms","authors":"Minh Hieu Nguyen, Mourad Baiou, Viet Hung Nguyen","doi":"10.1016/j.dam.2024.12.026","DOIUrl":"10.1016/j.dam.2024.12.026","url":null,"abstract":"<div><div>This paper deals with a particular case of Bi-Objective Optimization called <em>Bi-Objective Discrete Optimization</em> (BODO), where the feasible set is discrete, and the two objectives take only positive values. Since the feasible set of a BODO problem is discrete and usually finite, it can theoretically be enumerated to identify the Pareto set, which consists of all Pareto-optimal solutions representing different trade-offs between the two objectives. However, in general, this problem is challenging due to two main issues: time complexity, as the number of Pareto-optimal solutions can be exponentially large, and lack of decisiveness. From a practical point of view, the Central Decision Maker (CDM) may be interested in a reduced Pareto set that reflects the CDM’s preferences, which can be obtained by a computationally efficient algorithm.</div><div>In this paper, we propose a new criterion for selecting solutions within the Pareto set of BODO. For this purpose, we focus on solutions achieving proportional fairness between two objectives, called <em>generalized Nash Fairness solutions</em> (<span><math><mi>ρ</mi></math></span>-NF solutions). The positive parameter <span><math><mi>ρ</mi></math></span>, provided by the CDM, reflects the relative importance of the first objective compared to the second one.</div><div>We first introduce the <span><math><mi>ρ</mi></math></span>-NF solution concept for BODO. We then show that the <span><math><mi>ρ</mi></math></span>-NF solution set is a subset of the Pareto set, and this inclusion can be strict. We also propose a recursive Newton-like algorithm for determining the <span><math><mi>ρ</mi></math></span>-NF solution set. Finally, an illustrative example of BODO is given.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"365 ","pages":"Pages 71-90"},"PeriodicalIF":1.0,"publicationDate":"2025-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150141","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}
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