Theoretical Computer Science最新文献

{"title":"Some new results on equitable tree-coloring parameters of graphs","authors":"Bei Niu","doi":"10.1016/j.tcs.2025.115178","DOIUrl":"10.1016/j.tcs.2025.115178","url":null,"abstract":"<div><div>Equitable tree-coloring addresses a structure decomposition problem in communication networks while considering security aspects. Specifically, an equitable tree-<em>t</em>-coloring of a graph is a vertex <em>t</em>-coloring such that each color class induces a forest and the size of any two color classes differs by at most one. This paper demonstrated if <em>G</em> is an equitably tree-<em>t</em>-colorable graph and <em>H</em> is an arbitrary graph with less than 2<em>t</em> vertices (<span><math><mi>n</mi><mo>&lt;</mo><mn>2</mn><mi>t</mi></math></span>), it is possible to construct an equitable tree-<em>t</em>-coloring of the corona product graph <span><math><mi>G</mi><mo>∘</mo><mi>H</mi></math></span> in cubic time. Additionally, the equitable tree-coloring of the extended corona product formed by combining graphs <em>G</em> and <em>H</em> has also been investigated.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1037 ","pages":"Article 115178"},"PeriodicalIF":0.9,"publicationDate":"2025-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143579121","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":"Move-optimal arbitrary pattern formation by mobile robots on rectangular grid using near-optimal spatial area","authors":"Avisek Sharma ,&nbsp;Satakshi Ghosh ,&nbsp;Pritam Goswami ,&nbsp;Buddhadeb Sau","doi":"10.1016/j.tcs.2025.115179","DOIUrl":"10.1016/j.tcs.2025.115179","url":null,"abstract":"<div><div>Arbitrary pattern formation (APF) is a well-studied problem in swarm robotics. To the best of our knowledge, the problem has been considered in two different settings: one in a euclidean plane and another in an infinite grid. This work deals with the problem in an infinite rectangular grid setting. The previous works in literature dealing with the APF problem in an infinite grid had a fundamental issue. These deterministic algorithms use a lot of spatial area in the grid to solve the problem, mainly to maintain the asymmetry of the configuration and avoid any collision. These solution techniques cannot be useful if there is a spatial constraint in the application field. In this work, we consider luminous robots (each robot equipped with a light that can take three colors) to avoid symmetry, but we carefully designed a deterministic algorithm that solves the APF problem using the minimal required spatial area in the grid if the initial pattern is asymmetric. The robots are autonomous, identical, and anonymous, and they operate in Look-Compute-Move cycles under a fully-asynchronous scheduler. The APF algorithm proposed in <span><span>[1]</span></span> by Bose et al. can be modified using luminous robots so that it uses minimal spatial area, but that algorithm is not move-optimal. The algorithm proposed in this paper not only uses minimal spatial area but is also asymptotically move-optimal.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1038 ","pages":"Article 115179"},"PeriodicalIF":0.9,"publicationDate":"2025-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143609682","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":"Solving NP-hard problems on GaTEx graphs: Linear-time algorithms for perfect orderings, cliques, colorings, and independent sets","authors":"Marc Hellmuth ,&nbsp;Guillaume E. Scholz","doi":"10.1016/j.tcs.2025.115157","DOIUrl":"10.1016/j.tcs.2025.115157","url":null,"abstract":"<div><div>The class of <span>Ga</span>lled-<span>T</span>ree <span>Ex</span>plainable (<span>GaTEx</span>) graphs has recently been discovered as a natural generalization of cographs. Cographs are precisely those graphs that can be uniquely represented by a rooted tree where the leaves correspond to the vertices of the graph. As a generalization, <span>GaTEx</span> graphs are precisely those that can be uniquely represented by a particular rooted acyclic network, called a galled-tree.</div><div>This paper explores the use of galled-trees to solve combinatorial problems on <span>GaTEx</span> graphs that are, in general, NP-hard. We demonstrate that finding a maximum clique, an optimal vertex coloring, a perfect order, as well as a maximum independent set in <span>GaTEx</span> graphs can be efficiently done in linear time. The key idea behind the linear-time algorithms is to utilize the galled-trees that explain the <span>GaTEx</span> graphs as a guide for computing the respective cliques, colorings, perfect orders, or independent sets.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1037 ","pages":"Article 115157"},"PeriodicalIF":0.9,"publicationDate":"2025-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143579122","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Kolmogorov-Loveland betting strategies lose the Betting game on open sets","authors":"Tomislav Petrović","doi":"10.1016/j.tcs.2025.115177","DOIUrl":"10.1016/j.tcs.2025.115177","url":null,"abstract":"<div><div>Whether Kolmogorov-Loveland randomness (KLR) is the same as Martin-Löf randomness (MLR) is a major open problem in the study of algorithmic randomness. More general classes of betting strategies than Kolmogorov-Loveland ones have been studied in <span><span>[8]</span></span>, <span><span>[13]</span></span>, <span><span>[12]</span></span>. In each case it was proven that the class induces a notion of randomness equivalent to MLR. In all of those proofs, it was shown that the class contains a finite set of betting strategies such that for any given bound, when betting on a binary sequence contained in an effective open set of small enough measure, at least one of the betting strategies in the set earns capital larger than the bound. We show that the class of Kolmogorov-Loveland betting strategies does not have this property.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1037 ","pages":"Article 115177"},"PeriodicalIF":0.9,"publicationDate":"2025-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143600729","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":"Reconfigurable routing in data center networks","authors":"David C. Kutner,&nbsp;Iain A. Stewart","doi":"10.1016/j.tcs.2025.115154","DOIUrl":"10.1016/j.tcs.2025.115154","url":null,"abstract":"<div><div>A hybrid network is a static (electronic) network that is augmented with optical switches. The Reconfigurable Routing Problem (RRP) in hybrid networks is the problem of finding settings for the optical switches augmenting a static network so as to achieve optimal delivery of some given workload. The problem has previously been studied in various scenarios with both tractability and NP-hardness results obtained. However, the data center and interconnection networks to which the problem is most relevant are almost always such that the static network is highly structured (and often node-symmetric) whereas all previous results assume that the static network can be arbitrary (which makes existing computational hardness results less technologically relevant and also easier to obtain). In this paper, and for the first time, we prove various intractability results for RRP where the underlying static network is highly structured, for example consisting of a hypercube, and also extend some existing tractability results.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1038 ","pages":"Article 115154"},"PeriodicalIF":0.9,"publicationDate":"2025-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143609681","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The exact spanning ratio of the parallelogram Delaunay graph","authors":"Prosenjit Bose ,&nbsp;Jean-Lou De Carufel ,&nbsp;Sandrine Njoo","doi":"10.1016/j.tcs.2025.115152","DOIUrl":"10.1016/j.tcs.2025.115152","url":null,"abstract":"<div><div>Finding the exact spanning ratio of a Delaunay graph has been one of the longstanding open problems in Computational Geometry. Currently there are only four convex shapes for which the exact spanning ratio of their Delaunay graph is known: the equilateral triangle, the square, the regular hexagon and the rectangle. We add a fifth convex shape by proving the exact spanning ratio of the parallelogram Delaunay graph. The worst-case spanning ratio is <em>exactly</em><span><span><span><math><mfrac><mrow><msqrt><mrow><mn>2</mn></mrow></msqrt><msqrt><mrow><mn>1</mn><mo>+</mo><msup><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn><mi>A</mi><mi>cos</mi><mo>⁡</mo><mo>(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo><mo>+</mo><mo>(</mo><mi>A</mi><mo>+</mo><mi>cos</mi><mo>⁡</mo><mo>(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo><mo>)</mo><msqrt><mrow><mn>1</mn><mo>+</mo><msup><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn><mi>A</mi><mi>cos</mi><mo>⁡</mo><mo>(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></mrow></msqrt></mrow></msqrt></mrow><mrow><mi>sin</mi><mo>⁡</mo><mo>(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></mrow></mfrac><mo>,</mo></math></span></span></span> where <em>A</em> is the aspect ratio and <span><math><msub><mrow><mi>θ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> is the non-obtuse angle of the parallelogram.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1037 ","pages":"Article 115152"},"PeriodicalIF":0.9,"publicationDate":"2025-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143600730","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":"Compact zero-knowledge arguments for Blum integers","authors":"Jules Maire ,&nbsp;Damien Vergnaud","doi":"10.1016/j.tcs.2025.115155","DOIUrl":"10.1016/j.tcs.2025.115155","url":null,"abstract":"<div><div>We present a communication-efficient zero-knowledge argument of knowledge for the factorization of Blum integers, a special class of integers of the form <span><math><mi>n</mi><mo>=</mo><mi>p</mi><mi>q</mi></math></span>, where <em>p</em> and <em>q</em> are distinct prime numbers satisfying <span><math><mi>p</mi><mo>≡</mo><mi>q</mi><mo>≡</mo><mn>3</mn><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mn>4</mn></math></span> and <span><math><mi>p</mi><mo>≃</mo><mi>q</mi><mo>≃</mo><msqrt><mrow><mi>n</mi></mrow></msqrt></math></span>. Existing protocols for proving such statements often incur significant communication costs, especially when demonstrating that <em>p</em> and <em>q</em> are of nearly equal size.</div><div>We leverage the MPC-in-the-head paradigm, a cryptographic technique that transforms secure multi-party computation protocols into efficient zero-knowledge proof systems. In our protocol, the prover uses additive sharing of <em>p</em> and <em>q</em> over the integers. This approach simplifies proving the size relationship <span><math><mi>p</mi><mo>≃</mo><mi>q</mi><mo>≃</mo><msqrt><mrow><mi>n</mi></mrow></msqrt></math></span> and the congruence <span><math><mi>p</mi><mo>≡</mo><mi>q</mi><mo>≡</mo><mn>3</mn><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mn>4</mn></math></span> without requiring costly range proofs. To verify the primality of <em>p</em> and <em>q</em>, we employ the Boneh-Franklin biprimality test.</div><div>Our protocol achieves a significant reduction in communication complexity. For a 2048-bit integer <em>n</em> and 128-bit security, we construct an argument as small as 6.3 KB, with prover and verifier computational costs comparable to existing protocols that require over 131 KB.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1038 ","pages":"Article 115155"},"PeriodicalIF":0.9,"publicationDate":"2025-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143600869","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On 1-planar graphs with bounded cop-number","authors":"Prosenjit Bose ,&nbsp;Jean-Lou De Carufel ,&nbsp;Anil Maheshwari ,&nbsp;Karthik Murali","doi":"10.1016/j.tcs.2025.115160","DOIUrl":"10.1016/j.tcs.2025.115160","url":null,"abstract":"<div><div>Cops and Robbers is a type of pursuit-evasion game played on a graph where a set of cops try to capture a single robber. The cops first choose their initial vertex positions, and later the robber chooses a vertex. The cops and robbers make their moves in alternate turns: in the cops' turn, every cop can either choose to move to an adjacent vertex or stay on the same vertex, and likewise the robber in his turn. If the cops can capture the robber in a finite number of rounds, the cops win, otherwise the robber wins. The cop-number of a graph is the minimum number of cops required to catch a robber in the graph. It has long been known that graphs embedded on surfaces (such as planar graphs and toroidal graphs) have a small cop-number. Recently, Durocher et al. <span><span>[21]</span></span> investigated the problem of cop-number for the class of 1-planar graphs, which are graphs that can be embedded in the plane such that each edge is crossed at most once. They showed that unlike planar graphs which require just three cops, 1-planar graphs have an unbounded cop-number. On the positive side, they showed that maximal 1-planar graphs require only three cops by crucially using the fact that the endpoints of every crossing in an embedded maximal 1-planar graph induce a <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>. In this paper, we show that the cop-number remains bounded even under the relaxed condition that the endpoints induce at least three edges. More precisely, let an ×-crossing of an embedded 1-planar graph be a crossing whose endpoints induce a matching; i.e., there is no edge connecting the endpoints apart from the crossing edges themselves. We show that any 1-planar graph that can be embedded without ×-crossings has cop-number at most 21. Moreover, any 1-planar graph that can be embedded with at most <em>γ</em> ×-crossings has cop-number at most <span><math><mi>γ</mi><mo>+</mo><mn>21</mn></math></span>.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1037 ","pages":"Article 115160"},"PeriodicalIF":0.9,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143637331","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Complete decomposition of symmetric tensors in linear time and polylogarithmic precision","authors":"Pascal Koiran ,&nbsp;Subhayan Saha","doi":"10.1016/j.tcs.2025.115159","DOIUrl":"10.1016/j.tcs.2025.115159","url":null,"abstract":"&lt;div&gt;&lt;div&gt;We study symmetric tensor decompositions, i.e., decompositions of the form &lt;span&gt;&lt;math&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⊗&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt; where &lt;em&gt;T&lt;/em&gt; is a symmetric tensor of order 3 and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;. In order to obtain efficient decomposition algorithms, it is necessary to require additional properties from the &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;. In this paper we assume that the &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; are linearly independent. This implies &lt;span&gt;&lt;math&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, i.e., the decomposition of &lt;em&gt;T&lt;/em&gt; is &lt;em&gt;undercomplete&lt;/em&gt;. We will moreover assume that &lt;span&gt;&lt;math&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; (we plan to extend this work to the case &lt;span&gt;&lt;math&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; in a forthcoming paper).&lt;/div&gt;&lt;div&gt;We give a randomized algorithm for the following problem: given &lt;em&gt;T&lt;/em&gt;, an accuracy parameter &lt;em&gt;ε&lt;/em&gt;, and an upper bound &lt;em&gt;B&lt;/em&gt; on the &lt;em&gt;condition number&lt;/em&gt; of the tensor, output vectors &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt; such that &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; (up to permutation and multiplication by phases) with high probability. The main novel features of our algorithm are:&lt;ul&gt;&lt;li&gt;&lt;span&gt;•&lt;/span&gt;&lt;span&gt;&lt;div&gt;We provide the first algorithm for this problem that works in the computation model of finite arithmetic and requires only poly-logarithmic (in &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt;) many bits of precision.&lt;/div&gt;&lt;/span&gt;&lt;/li&gt;&lt;li&gt;&lt;span&gt;•&lt;/span&gt;&lt;span&gt;&lt;div&gt;Moreover, this is also the first algorithm that runs in linear time in the size of the input tensor. It requires &lt;span&gt;&lt;math&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; arithmetic operations for all accuracy parameters &lt;span&gt;&lt;math&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mtext&gt;poly&lt;/mtext&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt;.&lt;/div&gt;&lt;/span&gt;&lt;/li&gt;&lt;/ul&gt;&lt;/div&gt;&lt;div&gt;In order to obtain these results, we rely on a mix of techniques from algorithm design and algorithm analysis. The algorithm is a modified version of simultaneous diagonalisation algorithm for symmetric ","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1037 ","pages":"Article 115159"},"PeriodicalIF":0.9,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143636584","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":"(min⁡,+) matrix and vector products for inputs decomposable into few monotone subsequences","authors":"Andrzej Lingas ,&nbsp;Mia Persson","doi":"10.1016/j.tcs.2025.115158","DOIUrl":"10.1016/j.tcs.2025.115158","url":null,"abstract":"&lt;div&gt;&lt;div&gt;We study the time complexity of computing the &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;min&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; matrix product of two &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; integer matrices in terms of &lt;em&gt;n&lt;/em&gt; and the number of monotone subsequences the rows of the first matrix and the columns of the second matrix can be decomposed into. In particular, we show that if each row of the first matrix can be decomposed into at most &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; monotone subsequences and each column of the second matrix can be decomposed into at most &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; monotone subsequences such that all the subsequences are non-decreasing or all of them are non-increasing then the &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;min&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; product of the matrices can be computed in &lt;span&gt;&lt;math&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2.569&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; time. On the other hand, we observe that if all the rows of the first matrix are non-decreasing and all columns of the second matrix are non-increasing or &lt;em&gt;vice versa&lt;/em&gt; then this case is as hard as the general one. We also present six cases of the restrictions on the input integer matrices under which the problem of computing the &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;min&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; matrix product is equally hard as that of computing the minimum and maximum witnesses of Boolean matrix product.&lt;/div&gt;&lt;div&gt;Similarly, we also study the time complexity of computing the &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;min&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; convolution of two &lt;em&gt;n&lt;/em&gt;-dimensional integer vectors in terms of &lt;em&gt;n&lt;/em&gt; and the number of monotone subsequences the two vectors can be decomposed into. We show that if the first vector can be decomposed into at most &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; monotone subsequences and the second vector can be decomposed into at most &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; subsequences such that all the subsequences of the first vector are non-decreasing and all the subsequences of the second vector are non-increasing or &lt;em&gt;vice versa&lt;/em&gt; then their &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;min&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; convolution can be computed in &lt;span&gt;&lt;math&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;˜&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1.5&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; time. ","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1037 ","pages":"Article 115158"},"PeriodicalIF":0.9,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143637458","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}