Pub Date : 2024-10-15DOI: 10.1016/j.dam.2024.09.029
Elaine M. Eschen , Chính T. Hoàng , R. Sritharan
A graph is quasi-triangulated if each of its induced subgraphs has a vertex that is either simplicial (its neighbours form a clique) or co-simplicial (its non-neighbours form an independent set) in the induced subgraph. The problem of characterizing quasi-triangulated graphs by minimal forbidden induced subgraphs is open. We show that a graph is quasi-triangulated if it does not contain as induced subgraph any of the graphs , , , fence, or their complements. The fence is obtained from a by substituting two adjacent vertices for each end-vertex of the and two non-adjacent vertices for each interior vertex of the . We also provide an efficient algorithm to recognize the subclass , for fixed , of quasi-triangulated graphs studied in the literature.
{"title":"On some classes of quasi-triangulated graphs","authors":"Elaine M. Eschen , Chính T. Hoàng , R. Sritharan","doi":"10.1016/j.dam.2024.09.029","DOIUrl":"10.1016/j.dam.2024.09.029","url":null,"abstract":"<div><div>A graph is quasi-triangulated if each of its induced subgraphs has a vertex that is either simplicial (its neighbours form a clique) or co-simplicial (its non-neighbours form an independent set) in the induced subgraph. The problem of characterizing quasi-triangulated graphs by minimal forbidden induced subgraphs is open. We show that a graph is quasi-triangulated if it does not contain as induced subgraph any of the graphs <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span>, <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span>, <span><math><mrow><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>+</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math></span>, <em>fence</em>, or their complements. The fence is obtained from a <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> by substituting two adjacent vertices for each end-vertex of the <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> and two non-adjacent vertices for each interior vertex of the <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>. We also provide an efficient algorithm to recognize the subclass <span><math><msub><mrow><mi>Q</mi><mi>T</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, for fixed <span><math><mrow><mi>k</mi><mo>≥</mo><mn>0</mn></mrow></math></span>, of quasi-triangulated graphs studied in the literature.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"359 ","pages":"Pages 419-424"},"PeriodicalIF":1.0,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142434230","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 : 2024-10-15DOI: 10.1016/j.dam.2024.09.035
Zakir Deniz
A vertex coloring of a graph is said to be a 2-distance coloring if any two vertices at distance at most 2 from each other receive different colors, and the least number of colors for which admits a 2-distance coloring is known as the 2-distance chromatic number of . When is a planar graph with girth at least 6 and maximum degree , we prove that . This improves the best known bound for 2-distance coloring of planar graphs with girth six.
{"title":"An improved bound for 2-distance coloring of planar graphs with girth six","authors":"Zakir Deniz","doi":"10.1016/j.dam.2024.09.035","DOIUrl":"10.1016/j.dam.2024.09.035","url":null,"abstract":"<div><div>A vertex coloring of a graph <span><math><mi>G</mi></math></span> is said to be a 2-distance coloring if any two vertices at distance at most 2 from each other receive different colors, and the least number of colors for which <span><math><mi>G</mi></math></span> admits a 2-distance coloring is known as the 2-distance chromatic number <span><math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> of <span><math><mi>G</mi></math></span>. When <span><math><mi>G</mi></math></span> is a planar graph with girth at least 6 and maximum degree <span><math><mrow><mi>Δ</mi><mo>≥</mo><mn>6</mn></mrow></math></span>, we prove that <span><math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mi>Δ</mi><mo>+</mo><mn>4</mn></mrow></math></span>. This improves the best known bound for 2-distance coloring of planar graphs with girth six.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"361 ","pages":"Pages 121-135"},"PeriodicalIF":1.0,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142432749","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}
In this paper, we introduce saturation and semisaturation functions of sequences, and we prove a number of fundamental results about these functions. Given a forbidden sequence with distinct letters, we say that a sequence on a given alphabet is -saturated if is -sparse, -free, and adding any letter from the alphabet to an arbitrary position in violates -sparsity or induces a copy of . We say that is -semisaturated if is -sparse and adding any letter from the alphabet to violates -sparsity or induces a new copy of . Let the saturation function denote the minimum possible length of a -saturated sequence on an alphabet of size , and let the semisaturation function denote the minimum possible length of a -semisaturated sequence on an alphabet of size . For alternating sequences, we determine both the saturation function and the semisaturation function up to a constant multiplicative factor. We show for every sequence that the semisaturation function is always either or . For the saturation function, we show that every sequence has either or . For every sequence with 2 distinct letters, we show that the saturation function is always either or .
在本文中，我们介绍了序列的饱和函数和半饱和函数，并证明了有关这些函数的一些基本结果。给定一个包含 r 个不同字母的禁止序列 u，如果 s 是 r-稀疏的、无 u 的，并且在 s 的任意位置添加字母表中的任意一个字母都会违反 r-稀疏性或诱导出 u 的副本，我们就说给定字母表上的序列 s 是 u 饱和的；如果 s 是 r-稀疏的，并且在 s 中添加字母表中的任意一个字母都会违反 r-稀疏性或诱导出 u 的新副本，我们就说 s 是 u 半饱和的。让饱和函数 Sat(u,n) 表示大小为 n 的字母表上 u 饱和序列的最小可能长度，让半饱和函数 Ssat(u,n) 表示大小为 n 的字母表上 u 半饱和序列的最小可能长度。我们证明，对于每个序列，半饱和函数总是 O(1) 或 Θ(n)。对于饱和函数，我们证明每个序列 u 要么 Sat(u,n)≥n 要么 Sat(u,n)=O(1)。对于每个有 2 个不同字母的序列，我们证明饱和函数总是 O(1) 或 Θ(n)。
{"title":"Sequence saturation","authors":"Anand , Jesse Geneson , Suchir Kaustav , Shen-Fu Tsai","doi":"10.1016/j.dam.2024.09.034","DOIUrl":"10.1016/j.dam.2024.09.034","url":null,"abstract":"<div><div>In this paper, we introduce saturation and semisaturation functions of sequences, and we prove a number of fundamental results about these functions. Given a forbidden sequence <span><math><mi>u</mi></math></span> with <span><math><mi>r</mi></math></span> distinct letters, we say that a sequence <span><math><mi>s</mi></math></span> on a given alphabet is <span><math><mi>u</mi></math></span>-saturated if <span><math><mi>s</mi></math></span> is <span><math><mi>r</mi></math></span>-sparse, <span><math><mi>u</mi></math></span>-free, and adding any letter from the alphabet to an arbitrary position in <span><math><mi>s</mi></math></span> violates <span><math><mi>r</mi></math></span>-sparsity or induces a copy of <span><math><mi>u</mi></math></span>. We say that <span><math><mi>s</mi></math></span> is <span><math><mi>u</mi></math></span>-semisaturated if <span><math><mi>s</mi></math></span> is <span><math><mi>r</mi></math></span>-sparse and adding any letter from the alphabet to <span><math><mi>s</mi></math></span> violates <span><math><mi>r</mi></math></span>-sparsity or induces a new copy of <span><math><mi>u</mi></math></span>. Let the saturation function <span><math><mrow><mo>Sat</mo><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> denote the minimum possible length of a <span><math><mi>u</mi></math></span>-saturated sequence on an alphabet of size <span><math><mi>n</mi></math></span>, and let the semisaturation function <span><math><mrow><mo>Ssat</mo><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> denote the minimum possible length of a <span><math><mi>u</mi></math></span>-semisaturated sequence on an alphabet of size <span><math><mi>n</mi></math></span>. For alternating sequences, we determine both the saturation function and the semisaturation function up to a constant multiplicative factor. We show for every sequence that the semisaturation function is always either <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span> or <span><math><mrow><mi>Θ</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span>. For the saturation function, we show that every sequence <span><math><mi>u</mi></math></span> has either <span><math><mrow><mo>Sat</mo><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>n</mi><mo>)</mo></mrow><mo>≥</mo><mi>n</mi></mrow></math></span> or <span><math><mrow><mo>Sat</mo><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><mi>O</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>. For every sequence with 2 distinct letters, we show that the saturation function is always either <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span> or <span><math><mrow><mi>Θ</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"360 ","pages":"Pages 382-393"},"PeriodicalIF":1.0,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142424739","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 : 2024-10-08DOI: 10.1016/j.dam.2024.09.028
Igor Razgon
<div><div>In this paper we study syntactic branching programs of bounded repetition representing CNFs of bounded treewidth. For this purpose we introduce two new structural graph parameters <span><math><mi>d</mi></math></span>-pathwidth and clique preserving <span><math><mi>d</mi></math></span>-pathwidth denoted by <span><math><mrow><mi>p</mi><msub><mrow><mi>w</mi></mrow><mrow><mi>d</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>c</mi><mi>p</mi><msub><mrow><mi>w</mi></mrow><mrow><mi>d</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> where <span><math><mi>G</mi></math></span> is a graph. We show that <span><math><mrow><mi>c</mi><mi>p</mi><msub><mrow><mi>w</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mi>O</mi><mrow><mo>(</mo><mi>t</mi><mi>w</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mi>Δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> where <span><math><mrow><mi>t</mi><mi>w</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>Δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> are, respectively the treewidth and maximal degree of <span><math><mi>G</mi></math></span>. Using this upper bound, we demonstrate that each CNF <span><math><mi>ψ</mi></math></span> can be represented as a conjunction of two OBDDs (quite a restricted class of read-twice branching programs) of size <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mrow><mo>(</mo><mi>Δ</mi><mrow><mo>(</mo><mi>ψ</mi><mo>)</mo></mrow><mi>⋅</mi><mi>t</mi><mi>w</mi><msup><mrow><mrow><mo>(</mo><mi>ψ</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></msup></math></span> where <span><math><mrow><mi>t</mi><mi>w</mi><mrow><mo>(</mo><mi>ψ</mi><mo>)</mo></mrow></mrow></math></span> is the treewidth of the primal graph of <span><math><mi>ψ</mi></math></span> and each variable occurs in <span><math><mi>ψ</mi></math></span> at most <span><math><mrow><mi>Δ</mi><mrow><mo>(</mo><mi>ψ</mi><mo>)</mo></mrow></mrow></math></span> times.</div><div>Next, we use <span><math><mi>d</mi></math></span>-pathwidth to obtain lower bounds for monotone branching programs. In particular, we consider the monotone version of syntactic nondeterministic read <span><math><mi>d</mi></math></span> times branching programs (just forbidding negative literals as edge labels) and introduce a further restriction that each computational path can be partitioned into at most <span><math><mi>d</mi></math></span> read-once subpaths. We call the resulting model separable monotone read <span><math><mi>d</mi></math></span> times branching programs and abbreviate them <span><math><mi>d</mi></math></span>-SMNBPs. For each graph <span><math><mi>G</mi></math></span> without isolated vertices, we introduce a CNF <span><math><mrow><mi>ψ</mi><mrow><mo>(</mo><mi>G</mi><
在本文中，我们研究了代表有界树宽 CNF 的有界重复句法分支程序。为此，我们引入了两个新的结构图参数 d-pathwidth 和保簇 d-pathwidth，分别用 pwd(G) 和 cpwd(G) 表示，其中 G 是一个图。我们证明 cpw2(G)≤O(tw(G)Δ(G)) 其中 tw(G) 和 Δ(G) 分别是 G 的树宽和最大度数。利用这一上限，我们证明了每个 CNF ψ 都可以表示为两个大小为 2O(Δ(ψ)⋅tw(ψ)2) 的 OBDDs（相当有限的一类读两次分支程序）的连接，其中 tw(ψ) 是 ψ 的基元图的树宽，每个变量在 ψ 中出现的次数最多为 Δ(ψ)。接下来，我们使用 d 路径宽度来获得单调分支程序的下界。具体来说，我们考虑了语法非确定性读取 d 次分支程序的单调版本（只是禁止负字面作为边标签），并引入了进一步的限制条件，即每条计算路径最多可以划分为 d 个只读一次的子路径。我们将由此产生的模型称为可分离单调读取 d 次分支程序，并简称为 d-SMNBPs。对于每个没有孤立顶点的图 G，我们引入一个 CNF ψ(G)，对于 G 的每条边 e={u,v}，其分句为 (u∨e∨v)。我们将证明表示 ψ(G)的 d-SMNBP 大小至少为 Ω(cpwd(G)) ，其中 c=(8/7)1/12 。我们利用这个 "通用 "下界，为一类 "具体 "的 CNF ψ(Kn)求得指数下界。我们特别证明，对于每个 0<a<1，表示 ψ(Kn)的 na-SMNBP 的大小至少为 cnb，其中 b 是一个任意常数，使得 a+b<1。从 ψ(Kn)可以用一个多元大小的 n-SMNBP 表示的意义上讲，这个下界是严密的。
{"title":"The splitting power of branching programs of bounded repetition and CNFs of bounded width","authors":"Igor Razgon","doi":"10.1016/j.dam.2024.09.028","DOIUrl":"10.1016/j.dam.2024.09.028","url":null,"abstract":"<div><div>In this paper we study syntactic branching programs of bounded repetition representing CNFs of bounded treewidth. For this purpose we introduce two new structural graph parameters <span><math><mi>d</mi></math></span>-pathwidth and clique preserving <span><math><mi>d</mi></math></span>-pathwidth denoted by <span><math><mrow><mi>p</mi><msub><mrow><mi>w</mi></mrow><mrow><mi>d</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>c</mi><mi>p</mi><msub><mrow><mi>w</mi></mrow><mrow><mi>d</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> where <span><math><mi>G</mi></math></span> is a graph. We show that <span><math><mrow><mi>c</mi><mi>p</mi><msub><mrow><mi>w</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mi>O</mi><mrow><mo>(</mo><mi>t</mi><mi>w</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mi>Δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> where <span><math><mrow><mi>t</mi><mi>w</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>Δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> are, respectively the treewidth and maximal degree of <span><math><mi>G</mi></math></span>. Using this upper bound, we demonstrate that each CNF <span><math><mi>ψ</mi></math></span> can be represented as a conjunction of two OBDDs (quite a restricted class of read-twice branching programs) of size <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mrow><mo>(</mo><mi>Δ</mi><mrow><mo>(</mo><mi>ψ</mi><mo>)</mo></mrow><mi>⋅</mi><mi>t</mi><mi>w</mi><msup><mrow><mrow><mo>(</mo><mi>ψ</mi><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></msup></math></span> where <span><math><mrow><mi>t</mi><mi>w</mi><mrow><mo>(</mo><mi>ψ</mi><mo>)</mo></mrow></mrow></math></span> is the treewidth of the primal graph of <span><math><mi>ψ</mi></math></span> and each variable occurs in <span><math><mi>ψ</mi></math></span> at most <span><math><mrow><mi>Δ</mi><mrow><mo>(</mo><mi>ψ</mi><mo>)</mo></mrow></mrow></math></span> times.</div><div>Next, we use <span><math><mi>d</mi></math></span>-pathwidth to obtain lower bounds for monotone branching programs. In particular, we consider the monotone version of syntactic nondeterministic read <span><math><mi>d</mi></math></span> times branching programs (just forbidding negative literals as edge labels) and introduce a further restriction that each computational path can be partitioned into at most <span><math><mi>d</mi></math></span> read-once subpaths. We call the resulting model separable monotone read <span><math><mi>d</mi></math></span> times branching programs and abbreviate them <span><math><mi>d</mi></math></span>-SMNBPs. For each graph <span><math><mi>G</mi></math></span> without isolated vertices, we introduce a CNF <span><math><mrow><mi>ψ</mi><mrow><mo>(</mo><mi>G</mi><","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"360 ","pages":"Pages 366-381"},"PeriodicalIF":1.0,"publicationDate":"2024-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142424797","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 : 2024-10-08DOI: 10.1016/j.dam.2024.09.027
Manu Basavaraju , Erik Jan van Leeuwen , Reza Saei
An induced matching in a graph is a set of edges whose endpoints induce a 1-regular subgraph. It is known that every graph on vertices has at most maximal induced matchings, and this bound is the best possible as any disjoint union of complete graphs forms an extremal graph. It is also known that the bound drops to when the graphs are restricted to the class of triangle-free (-free) graphs. The extremal graphs, in this case, are known to be the disjoint unions of copies of . Along the same line, we study the maximum number of maximal induced matchings when the graphs are restricted to -free graphs and -free graphs. We show that every -free graph on vertices has at most maximal induced matchings and the bound is the best possible obtained by any disjoint union of copies of . When the graphs are restricted to -free graphs, the upper bound drops to , and it is achieved by the disjoint union of copies of the wheel graph .
{"title":"Maximal induced matchings in K4-free and K5-free graphs","authors":"Manu Basavaraju , Erik Jan van Leeuwen , Reza Saei","doi":"10.1016/j.dam.2024.09.027","DOIUrl":"10.1016/j.dam.2024.09.027","url":null,"abstract":"<div><div>An induced matching in a graph is a set of edges whose endpoints induce a 1-regular subgraph. It is known that every graph on <span><math><mi>n</mi></math></span> vertices has at most <span><math><mrow><mn>1</mn><msup><mrow><mn>0</mn></mrow><mrow><mi>n</mi><mo>/</mo><mn>5</mn></mrow></msup><mo>≈</mo><mn>1</mn><mo>.</mo><mn>584</mn><msup><mrow><mn>9</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span> maximal induced matchings, and this bound is the best possible as any disjoint union of complete graphs <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span> forms an extremal graph. It is also known that the bound drops to <span><math><mrow><msup><mrow><mn>3</mn></mrow><mrow><mi>n</mi><mo>/</mo><mn>3</mn></mrow></msup><mo>≈</mo><mn>1</mn><mo>.</mo><mn>442</mn><msup><mrow><mn>3</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span> when the graphs are restricted to the class of triangle-free (<span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>-free) graphs. The extremal graphs, in this case, are known to be the disjoint unions of copies of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn><mo>,</mo><mn>3</mn></mrow></msub></math></span>. Along the same line, we study the maximum number of maximal induced matchings when the graphs are restricted to <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span>-free graphs and <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-free graphs. We show that every <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span>-free graph on <span><math><mi>n</mi></math></span> vertices has at most <span><math><mrow><msup><mrow><mn>6</mn></mrow><mrow><mi>n</mi><mo>/</mo><mn>4</mn></mrow></msup><mo>≈</mo><mn>1</mn><mo>.</mo><mn>565</mn><msup><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span> maximal induced matchings and the bound is the best possible obtained by any disjoint union of copies of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>. When the graphs are restricted to <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-free graphs, the upper bound drops to <span><math><mrow><msup><mrow><mn>8</mn></mrow><mrow><mi>n</mi><mo>/</mo><mn>5</mn></mrow></msup><mo>≈</mo><mn>1</mn><mo>.</mo><mn>515</mn><msup><mrow><mn>8</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span>, and it is achieved by the disjoint union of copies of the wheel graph <span><math><msub><mrow><mi>W</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"359 ","pages":"Pages 407-418"},"PeriodicalIF":1.0,"publicationDate":"2024-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142417386","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}
Pub Date : 2024-10-07DOI: 10.1016/j.dam.2024.09.026
Yasir Ahmad , Xiang-Feng Pan , Umar Ali , Zhuo Diao
Random polynomio chains adopt a geometrically guided framework for analyzing resistance distance, which involves mathematical techniques to comprehend electrical resistance and optimize communication routes within networks. By determining the expected values of resistance distance-based indices, we can understand the typical or average performance of the network in terms of electrical resistance and communication efficiency. In this study, the closed-form formulae for the expected values of the Kirchhoff and additive degree-Kirchhoff indices for the random spiro-polynomio chains are determined. Furthermore, we compute the average values of Kirchhoff and additive degree-Kirchhoff indices for the spiro-polynomio chains with polynomios.
随机多项式链采用几何指导框架来分析电阻距离，其中涉及数学技术来理解网络内的电阻和优化通信线路。通过确定基于电阻距离指数的预期值，我们可以了解网络在电阻和通信效率方面的典型或平均性能。在本研究中，我们确定了随机螺-波里诺米奥链的基尔霍夫指数和加度-基尔霍夫指数预期值的闭式公式。此外，我们还计算了具有 n 个多项式的螺-波里诺米奥链的基尔霍夫指数和加度-基尔霍夫指数的平均值。
{"title":"Computing the expected value of invariants based on resistance distance for random spiro-polynomio chains","authors":"Yasir Ahmad , Xiang-Feng Pan , Umar Ali , Zhuo Diao","doi":"10.1016/j.dam.2024.09.026","DOIUrl":"10.1016/j.dam.2024.09.026","url":null,"abstract":"<div><div>Random polynomio chains adopt a geometrically guided framework for analyzing resistance distance, which involves mathematical techniques to comprehend electrical resistance and optimize communication routes within networks. By determining the expected values of resistance distance-based indices, we can understand the typical or average performance of the network in terms of electrical resistance and communication efficiency. In this study, the closed-form formulae for the expected values of the Kirchhoff and additive degree-Kirchhoff indices for the random spiro-polynomio chains are determined. Furthermore, we compute the average values of Kirchhoff and additive degree-Kirchhoff indices for the spiro-polynomio chains with <span><math><mi>n</mi></math></span> polynomios.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"361 ","pages":"Pages 111-120"},"PeriodicalIF":1.0,"publicationDate":"2024-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142422390","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 : 2024-10-07DOI: 10.1016/j.dam.2024.09.031
M.A. Reyes , C. Dalfó , M.A. Fiol , A. Messegué
The -token graph of a graph is the graph whose vertices are the -subsets of vertices from , two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices in . In this paper, we propose a general method to find the spectrum and eigenspaces of the -token graph of a cycle . The method is based on the theory of lift graphs and the recently introduced theory of over-lifts. In the case of , we use continuous fractions to derive the spectrum and eigenspaces of the 2-token graph of .
图 G 的 k 标记图 Fk(G) 是指其顶点是来自 G 的 k 个顶点子集的图，只要其中两个顶点的对称差是 G 中的一对相邻顶点，那么这两个顶点就是相邻的。本文提出了一种求循环 Cn 的 k 标记图 Fk(Cn) 的谱和特征空间的一般方法。该方法基于提升图理论和最近引入的过提升理论。在 k=2 的情况下，我们使用连续分数来推导 Cn 的 2oken 图的谱和特征空间。
{"title":"A general method to find the spectrum and eigenspaces of the k-token graph of a cycle, and 2-token through continuous fractions","authors":"M.A. Reyes , C. Dalfó , M.A. Fiol , A. Messegué","doi":"10.1016/j.dam.2024.09.031","DOIUrl":"10.1016/j.dam.2024.09.031","url":null,"abstract":"<div><div>The <span><math><mi>k</mi></math></span>-token graph <span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>k</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 graph whose vertices are the <span><math><mi>k</mi></math></span>-subsets of vertices from <span><math><mi>G</mi></math></span>, two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices in <span><math><mi>G</mi></math></span>. In this paper, we propose a general method to find the spectrum and eigenspaces of the <span><math><mi>k</mi></math></span>-token graph <span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> of a cycle <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. The method is based on the theory of lift graphs and the recently introduced theory of over-lifts. In the case of <span><math><mrow><mi>k</mi><mo>=</mo><mn>2</mn></mrow></math></span>, we use continuous fractions to derive the spectrum and eigenspaces of the 2-token graph of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"360 ","pages":"Pages 353-365"},"PeriodicalIF":1.0,"publicationDate":"2024-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142424796","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}
Pub Date : 2024-10-04DOI: 10.1016/j.dam.2024.09.017
Sean McGuinness
<div><div>A well-known <em>symmetric exchange</em> property in matroid theory states that for any two bases <span><math><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> of a matroid and any subset <span><math><mrow><mi>X</mi><mo>⊆</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math></span>, there is a subset <span><math><mrow><mi>Y</mi><mo>⊆</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span> for which <span><math><mrow><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>−</mo><mi>X</mi><mo>+</mo><mi>Y</mi></mrow></math></span> and <span><math><mrow><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>−</mo><mi>Y</mi><mo>+</mo><mi>X</mi></mrow></math></span> are bases. There have been a number of proposed strengthenings of this property. As a strengthening of a well-known conjecture of Gabow, Cordovil and Moreira, Kotlar and Ziv (2012) postulated that for any two bases <span><math><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and any subset <span><math><mrow><mi>X</mi><mo>⊆</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math></span>, there is a subset <span><math><mrow><mi>Y</mi><mo>⊆</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span> and orderings <span><math><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≺</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>≺</mo><mo>⋯</mo><mo>≺</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></math></span> and <span><math><mrow><msub><mrow><mi>y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≺</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>≺</mo><mo>⋯</mo><mo>≺</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></math></span> of <span><math><mi>X</mi></math></span> and <span><math><mi>Y</mi></math></span>, respectively, such that for <span><math><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi></mrow></math></span>, <span><math><mrow><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>−</mo><mrow><mo>{</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow><mo>+</mo><mrow><mo>{</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow></mrow></math></span> and <span><math><mrow><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>−</mo><mrow><mo>{</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi><
矩阵理论中有一个著名的对称交换性质，即对于矩阵的任意两个基 B1 和 B2 以及任意子集 X⊆B1，都有一个子集 Y⊆B2，其中 B1-X+Y 和 B2-Y+X 是基。对这一性质有许多强化建议。作为对 Gabow、Cordovil 和 Moreira 一个著名猜想的加强，Kotlar 和 Ziv（2012 年）假设，对于任意两个基 B1 和 B2 以及任意子集 X⊆B1、有一个子集 Y⊆B2，以及 X 和 Y 的排序 x1≺x2≺⋯≺xk 和 y1≺y2≺⋯≺yk，使得对于 i=1，...,k，B1-{x1,...,xi}+{y1,...,yi}和 B2-{y1,...,yi}+{x1,...,xi}是基；也就是说，存在一个子集 Y⊆B2，对于这个子集，X 与 Y 是可以序列交换的。这个问题的研究进展非常有限；迄今为止，只有当 |X|≤2 时，这个猜想才被验证。在本文中，我们证明了对于在 Z2 以外的有限域上可表示的矩阵，只要 |X|≤ln(n)，其中 n 是矩阵的秩，那么在随机选择 B1、B2 和 X 时，猜想很有可能是真的。
{"title":"Serial exchanges in random bases","authors":"Sean McGuinness","doi":"10.1016/j.dam.2024.09.017","DOIUrl":"10.1016/j.dam.2024.09.017","url":null,"abstract":"<div><div>A well-known <em>symmetric exchange</em> property in matroid theory states that for any two bases <span><math><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> of a matroid and any subset <span><math><mrow><mi>X</mi><mo>⊆</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math></span>, there is a subset <span><math><mrow><mi>Y</mi><mo>⊆</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span> for which <span><math><mrow><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>−</mo><mi>X</mi><mo>+</mo><mi>Y</mi></mrow></math></span> and <span><math><mrow><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>−</mo><mi>Y</mi><mo>+</mo><mi>X</mi></mrow></math></span> are bases. There have been a number of proposed strengthenings of this property. As a strengthening of a well-known conjecture of Gabow, Cordovil and Moreira, Kotlar and Ziv (2012) postulated that for any two bases <span><math><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and any subset <span><math><mrow><mi>X</mi><mo>⊆</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math></span>, there is a subset <span><math><mrow><mi>Y</mi><mo>⊆</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span> and orderings <span><math><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≺</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>≺</mo><mo>⋯</mo><mo>≺</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></math></span> and <span><math><mrow><msub><mrow><mi>y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≺</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>≺</mo><mo>⋯</mo><mo>≺</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></math></span> of <span><math><mi>X</mi></math></span> and <span><math><mi>Y</mi></math></span>, respectively, such that for <span><math><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi></mrow></math></span>, <span><math><mrow><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>−</mo><mrow><mo>{</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow><mo>+</mo><mrow><mo>{</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow></mrow></math></span> and <span><math><mrow><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>−</mo><mrow><mo>{</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi><","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"361 ","pages":"Pages 103-110"},"PeriodicalIF":1.0,"publicationDate":"2024-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142422335","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}
In mathematics, a dissection of a square (or rectangle) into non-congruent rectangles is a Mondrian partition. If all the rectangles have the same area, it is called a perfect Mondrian partition. In this paper, we present a computational result by which we can affirm that there is no perfect Mondrian partition of a length square for . Using the same algorithm we have been able to establish that there is no perfect Mondrian partition of a rectangle for .
在数学中，将正方形（或矩形）分割成不相等的矩形就是蒙德里安分割。如果所有矩形的面积相同，则称为完美的蒙德里安分割。在本文中，我们提出了一个计算结果，通过这个结果，我们可以肯定在 n≤1000 时，长度为 n 的正方形不存在完美的蒙德里安分割。使用同样的算法，我们还能确定在 n,m≤400 时，n×m 矩形不存在完美的蒙德里安分割。
{"title":"There is no perfect Mondrian partition for squares of side lengths less than 1001","authors":"Natalia García-Colín , Dimitri Leemans , Mia Müßig , Érika Roldán","doi":"10.1016/j.dam.2024.09.021","DOIUrl":"10.1016/j.dam.2024.09.021","url":null,"abstract":"<div><div>In mathematics, a dissection of a square (or rectangle) into non-congruent rectangles is a Mondrian partition. If all the rectangles have the same area, it is called a perfect Mondrian partition. In this paper, we present a computational result by which we can affirm that there is no perfect Mondrian partition of a length <span><math><mi>n</mi></math></span> square for <span><math><mrow><mi>n</mi><mo>≤</mo><mn>1000</mn></mrow></math></span>. Using the same algorithm we have been able to establish that there is no perfect Mondrian partition of a <span><math><mrow><mi>n</mi><mo>×</mo><mi>m</mi></mrow></math></span> rectangle for <span><math><mrow><mi>n</mi><mo>,</mo><mi>m</mi><mo>≤</mo><mn>400</mn></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"359 ","pages":"Pages 400-406"},"PeriodicalIF":1.0,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142417388","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 : 2024-10-02DOI: 10.1016/j.dam.2024.09.020
Hongyi Jiang , Samitha Samaranayake
Motivated by a transit line planning problem in transportation systems, we investigate the following capacitated assignment problem under a budget constraint. Our model involves bins and items. Each bin has a utilization cost and an -dimensional capacity vector. Each item has an -dimensional binary weight vector , where the 1s in (if any) appear in consecutive positions, and its assignment to bin yields a reward . The objective is to maximize total rewards through an assignment that satisfies three constraints: (i) the total weights of assigned items do not violate any bin’s capacity; (ii) each item is assigned to at most one open bin; and (iii) the overall utilization costs remain within a total budget .
We propose the first randomized rounding algorithm with a constant approximation ratio for this problem. We then apply our framework to the motivating transit line planning problem, presenting corresponding models and conducting numerical experiments using real-world data. Our results demonstrate significant improvements over previous approaches in addressing this critical transportation challenge.
受交通系统中运输线规划问题的启发，我们研究了以下预算约束下的容纳分配问题。我们的模型涉及 L 个仓和 P 个物品。每个仓 l 有一个使用成本 cl 和一个 nl 维容量向量。每个物品 p 都有一个 nl 维的二进制权重向量 rlp，其中 rlp 中的 1（如果有的话）出现在连续的位置上，将其分配到货仓 l 会产生奖励 vlp。我们的目标是通过满足以下三个约束条件的分配来实现总奖励最大化：(i) 分配项目的总权重不违反任何垃圾箱的容量；(ii) 每个项目最多分配到一个开放的垃圾箱；(iii) 总使用成本保持在总预算 B 的范围内。然后，我们将我们的框架应用于交通线路规划问题，提出了相应的模型，并使用真实世界的数据进行了数值实验。我们的研究结果表明，在解决这一关键的交通挑战方面，我们的方法比以前的方法有了显著的改进。
{"title":"Approximation algorithm for generalized budgeted assignment problems and applications in transportation systems","authors":"Hongyi Jiang , Samitha Samaranayake","doi":"10.1016/j.dam.2024.09.020","DOIUrl":"10.1016/j.dam.2024.09.020","url":null,"abstract":"<div><div>Motivated by a transit line planning problem in transportation systems, we investigate the following capacitated assignment problem under a budget constraint. Our model involves <span><math><mi>L</mi></math></span> bins and <span><math><mi>P</mi></math></span> items. Each bin <span><math><mi>l</mi></math></span> has a utilization cost <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>l</mi></mrow></msub></math></span> and an <span><math><msub><mrow><mi>n</mi></mrow><mrow><mi>l</mi></mrow></msub></math></span>-dimensional capacity vector. Each item <span><math><mi>p</mi></math></span> has an <span><math><msub><mrow><mi>n</mi></mrow><mrow><mi>l</mi></mrow></msub></math></span>-dimensional binary weight vector <span><math><msub><mrow><mi>r</mi></mrow><mrow><mi>l</mi><mi>p</mi></mrow></msub></math></span>, where the 1s in <span><math><msub><mrow><mi>r</mi></mrow><mrow><mi>l</mi><mi>p</mi></mrow></msub></math></span> (if any) appear in consecutive positions, and its assignment to bin <span><math><mi>l</mi></math></span> yields a reward <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>l</mi><mi>p</mi></mrow></msub></math></span>. The objective is to maximize total rewards through an assignment that satisfies three constraints: (i) the total weights of assigned items do not violate any bin’s capacity; (ii) each item is assigned to at most one open bin; and (iii) the overall utilization costs remain within a total budget <span><math><mi>B</mi></math></span>.</div><div>We propose the first randomized rounding algorithm with a constant approximation ratio for this problem. We then apply our framework to the motivating transit line planning problem, presenting corresponding models and conducting numerical experiments using real-world data. Our results demonstrate significant improvements over previous approaches in addressing this critical transportation challenge.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"359 ","pages":"Pages 383-399"},"PeriodicalIF":1.0,"publicationDate":"2024-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142417387","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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