Theoretical Computer Science最新文献

{"title":"On competitive ratio for online uniform facility location problem in random-order model","authors":"Mengzhen Li ,&nbsp;Runjie Miao ,&nbsp;Chenchen Wu ,&nbsp;Dachuan Xu","doi":"10.1016/j.tcs.2026.115878","DOIUrl":"10.1016/j.tcs.2026.115878","url":null,"abstract":"<div><div>We study the online facility location problem, where clients arrive sequentially in a random order and must be assigned to an open facility immediately and irrevocably upon arrival. At the initial stage, the set of facilities is fully known. We present a 8-competitive online algorithm for the uniform facility cost case, providing the first competitive ratio result for this setting. Our algorithm reduces the competitive ratio by 75% compared to the previously known 33-competitive ratio for the nonuniform case. The analysis offers new theoretical insights into online algorithms for the nonuniform case and establishes a foundation for practical applications in decision-making contexts.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1072 ","pages":"Article 115878"},"PeriodicalIF":1.0,"publicationDate":"2026-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147388266","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"More characterizations of morphic words","authors":"Golnaz Badkobeh,&nbsp;Pascal Ochem","doi":"10.1016/j.tcs.2026.115875","DOIUrl":"10.1016/j.tcs.2026.115875","url":null,"abstract":"<div><div>An interesting phenomenon in combinatorics on words is when every recurrent word satisfying some avoidance constraints has the same factor set as a morphic word. An early example is the Hall-Thue word, fixed point of the morphism <span><math><mrow><mn>0</mn><mo>→</mo><mn>012</mn></mrow></math></span>, <span><math><mrow><mn>1</mn><mo>→</mo><mn>02</mn></mrow></math></span>, <span><math><mrow><mn>2</mn><mo>→</mo><mn>1</mn></mrow></math></span>, which is essentially the only ternary word avoiding squares and the factors <span>010</span> and <span>212</span>. We provide some examples of this phenomenon from various contexts.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1072 ","pages":"Article 115875"},"PeriodicalIF":1.0,"publicationDate":"2026-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147388265","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Efficient algorithms for the interval maximum coverage problem","authors":"Vacharapat Mettanant","doi":"10.1016/j.tcs.2026.115802","DOIUrl":"10.1016/j.tcs.2026.115802","url":null,"abstract":"<div><div>This paper studies a special case of the Maximum Interval Multi-Cover (MaxIMC) problem, called the <em>Interval Maximum Coverage Problem</em>. Given a set of points <em>P</em> on the real line, a collection of intervals <em>I</em>, and a budget <em>K</em>, the goal is to select up to <em>K</em> intervals that maximize the number of covered points. While the computational complexity of the general MaxIMC problem with arbitrary coverage requirements remains open, this special case admits efficient polynomial-time solutions. We develop an exact algorithm that improves computational efficiency when the number of intervals is extremely large, and a near-linear-time approximation algorithm for the case where each interval covers exactly <em>r</em> points. We provide formal proofs of correctness, detailed complexity analysis, and experimental results demonstrating the practical efficiency and effectiveness of the proposed algorithms.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1069 ","pages":"Article 115802"},"PeriodicalIF":1.0,"publicationDate":"2026-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146147122","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"SOBRA - Shielding Optimization for BRAchytherapy","authors":"Guillaume Blin ,&nbsp;Adrian Miclăuş ,&nbsp;Sebastian Ordyniak ,&nbsp;Alexandru Popa","doi":"10.1016/j.tcs.2026.115797","DOIUrl":"10.1016/j.tcs.2026.115797","url":null,"abstract":"<div><div>In this paper, we study a combinatorial problem which arises in the development of innovative treatment strategies and equipment using tunable shields in internal radiotherapy. From an algorithmic point of view, the problem is related to circular integer word decomposition into circular binary words under constraints. We consider several variants of the problem, depending on constraints and parameters and present exact, approximation, fixed parameter tractable algorithms and NP-hardness and APX-hardness results.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1069 ","pages":"Article 115797"},"PeriodicalIF":1.0,"publicationDate":"2026-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146147123","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Open shop scheduling problem with a flexible maintenance period: Revisited","authors":"Yuan Yuan ,&nbsp;Xin Han ,&nbsp;Xingwu Liu ,&nbsp;Yong Zhou ,&nbsp;Hao Lu","doi":"10.1016/j.tcs.2026.115794","DOIUrl":"10.1016/j.tcs.2026.115794","url":null,"abstract":"<div><div>This paper addresses the two-machine open shop scheduling problem where one flexible maintenance period is imposed on the second machine. The maintenance period must start within a given time window and has a fixed duration. The objective is to minimize the makespan. We demonstrate that a <span><math><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>ϵ</mi><mo>)</mo><mi>ρ</mi></mrow></math></span>-approximation algorithm can be constructed for the studied problem if there exists a <em>ρ</em>-approximation algorithm for the version with a fixed maintenance period. As a consequence, by applying the polynomial-time approximation scheme (PTAS) for the fixed maintenance period, we derive a PTAS for the problem under consideration, thereby solving an open question in the literature. Furthermore, we propose a 4/3-approximation algorithm with <em>O</em>(<em>n</em>) time complexity, which outperforms the previous 3/2-approximation algorithm presented in the literature.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1069 ","pages":"Article 115794"},"PeriodicalIF":1.0,"publicationDate":"2026-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146191943","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Weakly leveled planarity with bounded span","authors":"Michael A. Bekos ,&nbsp;Giordano Da Lozzo ,&nbsp;Fabrizio Frati ,&nbsp;Siddharth Gupta ,&nbsp;Philipp Kindermann ,&nbsp;Giuseppe Liotta ,&nbsp;Ignaz Rutter ,&nbsp;Ioannis G. Tollis","doi":"10.1016/j.tcs.2026.115811","DOIUrl":"10.1016/j.tcs.2026.115811","url":null,"abstract":"<div><div>This paper studies planar drawings of graphs in which each vertex is represented as a point along a sequence of horizontal lines, called levels, and each edge is either a horizontal segment or a strictly <em>y</em>-monotone curve. A graph is <em>s</em>-span weakly leveled planar if it admits such a drawing where the edges have span at most <em>s</em>; the span of an edge is the number of levels it touches minus one. We investigate the problem of computing <em>s</em>-span weakly leveled planar drawings from both the computational and the combinatorial perspectives. We prove the problem to be para-NP-hard with respect to its natural parameter <em>s</em> and investigate its complexity with respect to widely used structural parameters. We show the existence of a polynomial-size kernel with respect to vertex cover number and prove that the problem is FPT when parameterized by treedepth. We also present upper and lower bounds on the span for various graph classes. Notably, we show that cycle trees, a family of 2-outerplanar graphs generalizing Halin graphs, are Θ(log <em>n</em>)-span weakly leveled planar and 4-span weakly leveled planar when 3-connected. As a byproduct of these combinatorial results, we obtain improved bounds on the edge-length ratio of the graph families under consideration.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1069 ","pages":"Article 115811"},"PeriodicalIF":1.0,"publicationDate":"2026-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146191942","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"k-Submodular and approximately non-k-submodular maximization under p-system and ℓ knapsack constraints","authors":"Hanlu Ye,&nbsp;Heqing Li,&nbsp;Min Li,&nbsp;Yang Zhou,&nbsp;Qian Liu","doi":"10.1016/j.tcs.2026.115791","DOIUrl":"10.1016/j.tcs.2026.115791","url":null,"abstract":"&lt;div&gt;&lt;div&gt;This paper addresses the problem of &lt;em&gt;k&lt;/em&gt;-submodular and approximately non-&lt;em&gt;k&lt;/em&gt;-submodular maximization under &lt;em&gt;p&lt;/em&gt;-system and ℓ knapsack constraints. For monotone &lt;em&gt;k&lt;/em&gt;-submodular functions, we first propose a greedy algorithm, achieving a &lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;ϵ&lt;/mi&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt;-approximation and a &lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;ϵ&lt;/mi&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt;-approximation for non-monotone case, with the &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;ϵ&lt;/mi&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; time complexity, where ϵ′ is a very small positive number. We further introduce an improved algorithm that enhances the approximation ratio to &lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;ϵ&lt;/mi&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;ϵ&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mfrac&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;ϵ&lt;/mi&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;ϵ&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mfrac&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt;, respectively, while reducing the time complexity to &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mfrac&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;msup&gt;&lt;mi&gt;ϵ&lt;/mi&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/msup&gt;&lt;/mfrac&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. For monotone &lt;em&gt;k&lt;/em&gt;-submodular functions with curvature &lt;em&gt;c&lt;/em&gt;, we obtain an approximation result of &lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;ϵ&lt;/mi&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;ϵ&lt;/mi&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mfrac&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt;. Additionally, we provide an approximation guarantee of &lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;min&lt;/mi&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mi&gt;ϵ&lt;/mi&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1069 ","pages":"Article 115791"},"PeriodicalIF":1.0,"publicationDate":"2026-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146191941","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Finding trails in multigraphs with restricted transitions","authors":"Hortensia Galeana-Sánchez ,&nbsp;Carlos Vilchis-Alfaro","doi":"10.1016/j.tcs.2026.115798","DOIUrl":"10.1016/j.tcs.2026.115798","url":null,"abstract":"<div><div>Let <em>G</em> be a multigraph. A transition in <em>G</em> is a pair of adjacent edges. For each vertex <em>x</em> of <em>G</em>, a set <em>T</em>(<em>x</em>) of allowed transition with respect to <em>x</em> is a set of unordered pairs of edges incident to <em>x</em>. A transition system <em>T</em> is a set {<em>T</em>(<em>x</em>): <em>x</em> ∈ <em>V</em>(<em>G</em>)}, where <em>T</em>(<em>x</em>) is a fixed set of allowed transition with respect to <em>x</em>. Given a multigraph <em>G</em> and a transition system <em>T</em>, for each <em>x</em> ∈ <em>V</em>(<em>G</em>), the transition graph of <em>x</em>, denoted by <em>G_x</em>, is a graph such that its vertex set is the set of edges incident to <em>x</em>; and two vertices <em>e</em> and <em>g</em> of <em>G_x</em> are adjacent whenever <em>eg</em> ∈ <em>T</em>(<em>x</em>). A trail <span><math><mrow><mi>P</mi><mo>=</mo><mo>(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>,</mo><msub><mi>e</mi><mn>0</mn></msub><mo>,</mo><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>x</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>e</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>x</mi><mi>k</mi></msub><mo>)</mo></mrow></math></span> in <em>G</em> is <em>T</em>-compatible if for every <span><math><mrow><mi>i</mi><mo>∈</mo><mo>{</mo><mn>0</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>−</mo><mn>2</mn><mo>}</mo></mrow></math></span>, <span><math><mrow><msub><mi>e</mi><mi>i</mi></msub><msub><mi>e</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>∈</mo><mi>T</mi><mrow><mo>(</mo><msub><mi>x</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span>.</div><div>In this paper we deal with the problem of finding <em>T</em>-compatible trails between <em>s</em> and <em>t</em> two given vertices in a multigraph with transition system <em>T</em>. First, we prove that finding a <em>T</em>-compatible trail between two given edges can be done in polynomial time. Consequently, finding a <em>T</em>-compatible <span><math><mrow><mi>s</mi><mo>−</mo><mi>t</mi></mrow></math></span> trail can be done in polynomial time. Moreover, it can be found a shortest <em>T</em>-compatible <span><math><mrow><mi>s</mi><mo>−</mo><mi>t</mi></mrow></math></span> trail and a closed <em>T</em>-compatible trail containing a given vertex in polynomial time. Finally, we study multigraphs with transition systems such that their transition graph is connected or complete multipartite graph. The properly colored setting is a particular case of transition systems where all its transition graphs are complete multipartite graphs.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1069 ","pages":"Article 115798"},"PeriodicalIF":1.0,"publicationDate":"2026-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146147124","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Respecting lower bounds in uniform lower and upper bounded facility location problem","authors":"Neelima Gupta ,&nbsp;Sapna Grover ,&nbsp;Rajni Dabas","doi":"10.1016/j.tcs.2026.115803","DOIUrl":"10.1016/j.tcs.2026.115803","url":null,"abstract":"<div><div>With growing emphasis on e-commerce marketplace platforms where we have a central platform mediating between the seller and the buyer, it becomes important to keep a check on the availability and profitability of the central store. A store serving too few clients can be non-profitable and a store getting too many orders can lead to bad service to the customers which can be detrimental for the business. In this paper, we study this problem, called <em>lower and upper bounded facility location problem</em>, with lower and upper bounds on the number of clients an open facility serves. Approximation algorithms are known for the restricted variants of the problem with only the upper bounds or only the lower bounds. The only work that deals with bounds on both sides violates both the bounds Friggstad et al. [1]. In this paper, we present the first approximation algorithm for the problem that preserves the lower bounds and violates the upper bound by a factor of 5/2 when both, the lower and the upper bounds are uniform. We first give an approximation algorithm violating both the upper and the lower bounds and, then get rid of violation in lower bounds by transforming the problem instance to an instance of upper bounded facility location problem.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1069 ","pages":"Article 115803"},"PeriodicalIF":1.0,"publicationDate":"2026-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146192892","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Generalized code distance through rotated logical states in quantum error correction","authors":"Valentine Nyirahafashimana ,&nbsp;Nurisya Mohd Shah ,&nbsp;Umair Abdul Halim ,&nbsp;Mohamed Othman","doi":"10.1016/j.tcs.2026.115795","DOIUrl":"10.1016/j.tcs.2026.115795","url":null,"abstract":"<div><div>We construct rotated logical states by applying rotation operators to stabilizer states, thereby extending the logical basis and modifying stabilizer generators. These rotation operators affect the effective code distance <em>d_R</em>, which decays exponentially with rotation angles (<em>θ, ϕ</em>), influencing error correction performance. We quantify the scaling behavior of logical error rates under circuit-level noise, comparing standard depolarizing (SD) and superconducting-inspired (SI) noise models with small and large rotations. Our findings show that the rotated code scales as 0.68<em>d_R</em>(0.65<em>d_R</em>) for SD and 0.81<em>d_R</em>(0.77<em>d_R</em>) for SI, with small rotation angles leading to a steeper decay of logical error rates. At a physical error rate (<em>p_phy</em>) of <span><math><msup><mn>10</mn><mrow><mo>−</mo><mn>4</mn></mrow></msup></math></span>, logical errors decrease exponentially with <em>d_R</em>, particularly under SI noise, which exhibits stronger suppression. The threshold error rates for rotated logical states are compared with previous results, demonstrating improved resilience against noise. By extending the logical state basis, rotation-based encoding increases error suppression beyond traditional stabilizer codes, offering a promising approach to advancing quantum error correction.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1068 ","pages":"Article 115795"},"PeriodicalIF":1.0,"publicationDate":"2026-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146190788","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}