Information and Computation最新文献

{"title":"Dichotomy for Holant* problems with one ternary function on domain size 3","authors":"Jin-Yi Cai ,&nbsp;Pinyan Lu ,&nbsp;Mingji Xia","doi":"10.1016/j.ic.2026.105421","DOIUrl":"10.1016/j.ic.2026.105421","url":null,"abstract":"<div><div>Holant problems are a general framework to study the algorithmic complexity of counting problems. Both counting constraint satisfaction problems and graph homomorphisms are special cases. All previous results of Holant problems are over the Boolean domain.<span><span>³</span></span> In this paper, we give the first dichotomy theorem for Holant problems for domain size &gt;2. We discover unexpected tractable families of counting problems, by giving new polynomial time algorithms. This paper also initiates holographic reductions in domains of size &gt;2. This is our main algorithmic technique, and is used for both tractable families and hardness reductions. The dichotomy theorem is the following: For any complex-valued symmetric function <strong>F</strong> with arity 3 on domain size 3, we give an explicit criterion on <strong>F</strong>, such that if <strong>F</strong> satisfies the criterion then the problem <span><math><msup><mrow><mi>Holant</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>F</mi><mo>)</mo></math></span> is computable in polynomial time; otherwise <span><math><msup><mrow><mi>Holant</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>F</mi><mo>)</mo></math></span> is #P-hard.</div></div>","PeriodicalId":54985,"journal":{"name":"Information and Computation","volume":"309 ","pages":"Article 105421"},"PeriodicalIF":1.0,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146174309","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 Co-Path/Cycle Packing and Co-Path Packing faster than 3k","authors":"Yuxi Liu,&nbsp;Mingyu Xiao","doi":"10.1016/j.ic.2026.105406","DOIUrl":"10.1016/j.ic.2026.105406","url":null,"abstract":"<div><div>The <span>Co-Path/Cycle Packing</span> problem (resp. The <span>Co-Path Packing</span> problem) asks whether we can delete at most <em>k</em> vertices from the input graph such that the remaining graph is a collection of induced paths and cycles (resp. induced paths). These two problems are fundamental graph problems that have important applications in bioinformatics. Although these two problems have been extensively studied in parameterized algorithms, it seems hard to break the running time bound <span><math><msup><mrow><mn>3</mn></mrow><mrow><mi>k</mi></mrow></msup></math></span>. In 2015, Feng et al. provided an <span><math><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msup><mrow><mn>3</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math></span>-time randomized algorithm for both of them. Recently, Tsur showed that they can be solved in <span><math><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msup><mrow><mn>3</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math></span> time deterministically. In this paper, by combining several techniques such as path decomposition, dynamic programming, cut &amp; count, and branch-and-search methods, we show that <span>Co-Path/Cycle Packing</span> can be solved in <span><math><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msup><mrow><mn>2.8192</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math></span> time deterministically and <span>Co-Path Packing</span> can be solved in <span><math><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msup><mrow><mn>2.9241</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math></span> time with failure probability <span><math><mo>≤</mo><mn>1</mn><mo>/</mo><mn>3</mn></math></span>. As a by-product, we also show that the <span>Co-Path Packing</span> problem can be solved in <span><math><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msup><mrow><mn>5</mn></mrow><mrow><mi>p</mi></mrow></msup><mo>)</mo></math></span> time with probability at least 2/3 if a path decomposition of width <em>p</em> is given.</div></div>","PeriodicalId":54985,"journal":{"name":"Information and Computation","volume":"309 ","pages":"Article 105406"},"PeriodicalIF":1.0,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146057316","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":"Arena-independent memory bounds for Nash equilibria in reachability games","authors":"James C.A. Main","doi":"10.1016/j.ic.2026.105425","DOIUrl":"10.1016/j.ic.2026.105425","url":null,"abstract":"<div><div>We study the memory requirements of Nash equilibria in turn-based multiplayer games on possibly <em>infinite graphs</em> with reachability, safety, shortest-path, Büchi and co-Büchi objectives.</div><div>We present constructions for <em>finite-memory</em> Nash equilibria in these games that apply to arbitrary game graphs, bypassing the finite-arena requirement that is central in existing approaches. We show that, for these five types of games, from any Nash equilibrium, we can derive another Nash equilibrium where all strategies are finite-memory such that all objectives satisfied by the outcome of the original equilibrium also are by the outcome of the derived equilibrium, without increasing costs for shortest-path games.</div><div>Furthermore, we provide memory bounds that are <em>independent of the size of the game graph</em> for reachability, safety and shortest-path games. These bounds depend only on the number of players.</div><div>To the best of our knowledge, we provide the first results pertaining to finite-memory constrained Nash equilibria in infinite arenas and the first arena-independent memory bounds for Nash equilibria.</div></div>","PeriodicalId":54985,"journal":{"name":"Information and Computation","volume":"309 ","pages":"Article 105425"},"PeriodicalIF":1.0,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147422948","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":"Fair division with prioritized agents","authors":"Xiaolin Bu ,&nbsp;Zihao Li ,&nbsp;Shengxin Liu ,&nbsp;Jiaxin Song ,&nbsp;Biaoshuai Tao ,&nbsp;Ziqi Yu","doi":"10.1016/j.ic.2026.105407","DOIUrl":"10.1016/j.ic.2026.105407","url":null,"abstract":"<div><div>We study the fair division of indivisible items. Since an envy-free allocation may not exist, a standard relaxation is <em>envy-freeness up to one item (EF1)</em>, where any envy can be eliminated by removing a single item from the envied agent's bundle. In many applications, however, it is desirable to designate a subset of <em>prioritized agents</em> for whom strict envy-freeness toward the remaining agents must be guaranteed, while the overall allocation remains EF1. Such agents may correspond to those who were envious in a previous EF1 allocation or to members of underrepresented groups. Motivated by this, we propose a new fairness notion named <em>envy-freeness with prioritized agents</em> <span>EFprior</span>, and study the existence and the algorithmic aspects of computing an <span>EFprior</span> allocation. For additive valuations, the simple round-robin algorithm suffices to compute an <span>EFprior</span> allocation. In this paper, we mainly focus on general valuations. In particular, we present a polynomial-time algorithm that computes an <span>EFprior</span> allocation with most of the items allocated. When all the items need to be allocated, we also present polynomial-time algorithms for several well-motivated special cases. We finally extend the setting to a general prioritized ordering case, where we are given a full ordering of agents and each agent with a higher priority cannot envy an agent with a lower priority. We propose a generalized fairness notion named <em>envy-freeness under rank r</em> <span><math><msub><mrow><mi>EF</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> and present a polynomial-time algorithm with most of the items allocated.</div></div>","PeriodicalId":54985,"journal":{"name":"Information and Computation","volume":"309 ","pages":"Article 105407"},"PeriodicalIF":1.0,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146057594","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":"Spiking neural P systems with non-gated channels","authors":"Yuzhen Zhao ,&nbsp;Zhen Yang ,&nbsp;Yueguo Luo ,&nbsp;Hui Li ,&nbsp;Wenke Zang","doi":"10.1016/j.ic.2025.105399","DOIUrl":"10.1016/j.ic.2025.105399","url":null,"abstract":"<div><div>Spiking neural P systems (SNP systems) are parallel and distributed computational models mimicking the storage, processing, and transmission of spikes in the human brain nervous system. However, they do not consider the non-gated channels in human brain neurons, which would affect the storage and thus further affect the processing and transmission of spikes. To enhance the SNP systems' ability to process complex information and to make SNP systems more biologically plausible, this work constructs the SNP systems with non-gated channels (NGCSNP systems). In NGCSNP systems, when a neuron is not activated, the spikes in it slowly flow out into the environment through non-gated channels until a steady state is reached. That is, the number of spikes in a neuron nonlinearly varies automatically. In terms of computational power, this work proves the Turing universality of NGCSNP systems by simulating register machines. For the ADD/SUB/FIN/INPUT/deterministic ADD modules of the universal register machine, NGCSNP systems utilize only 7, 6, 3, 5, and 4 neurons, with each neuron containing a maximum of 2, 2, 2, 1, and 1 rules for module simulation, respectively. Compared to the other eight variants, NGCSNP systems use fewer computational resources. Also, this work proves the computational efficiency of NGCSNP systems by solving a classical NP-hard problem: Boolean Satisfiability Problems in linear time via a space-for-time strategy. NGCSNP systems introduce nonlinear features at the neuronal level, enhance the computational functionality as well as stability of the systems, save the computational resources, and provide clues for developing computational models that resemble the human brain.</div></div>","PeriodicalId":54985,"journal":{"name":"Information and Computation","volume":"308 ","pages":"Article 105399"},"PeriodicalIF":1.0,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145840783","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":"Truthful approximation for rank-maximal matchings","authors":"Jinshan Zhang ,&nbsp;Hao Xu ,&nbsp;Feng Wang ,&nbsp;Meng Xi ,&nbsp;Xiaotie Deng ,&nbsp;Jianwei Yin","doi":"10.1016/j.ic.2026.105404","DOIUrl":"10.1016/j.ic.2026.105404","url":null,"abstract":"<div><div>In this work, we investigate truthful mechanisms for the rank-maximal matching problem from an approximation standpoint. Our findings narrow the gap between the upper and lower bounds. We introduce a lexicographically truthful (LT) and nearly Pareto optimal (PO) randomized mechanism with an approximation ratio of <span><math><mfrac><mrow><mn>2</mn><msqrt><mrow><mi>e</mi></mrow></msqrt><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn><msqrt><mrow><mi>e</mi></mrow></msqrt><mo>−</mo><mn>2</mn></mrow></mfrac><mo>≈</mo><mn>1.77</mn></math></span>, an improvement over the previous best result of 2. Key to our algorithm are preservation lemmas that enable us to leverage techniques from online algorithms to analyze the new approximation ratio. Furthermore, we present several hardness results across different settings to enhance our upper bound. Notably, we improve the lower bound for the approximation ratio of our LT and PO mechanism to <span><math><mn>18</mn><mo>/</mo><mn>13</mn><mo>≈</mo><mn>1.38</mn></math></span>. To our knowledge, this is the first instance of a lower bound being established using a linear programming approach in this field.</div></div>","PeriodicalId":54985,"journal":{"name":"Information and Computation","volume":"308 ","pages":"Article 105404"},"PeriodicalIF":1.0,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145976646","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":"Algebraic barriers to halving algorithmic information quantities in correlated strings","authors":"Andrei Romashchenko","doi":"10.1016/j.ic.2025.105396","DOIUrl":"10.1016/j.ic.2025.105396","url":null,"abstract":"<div><div>We study the possibility of scaling down algorithmic information quantities in tuples of correlated strings. In particular, we address a question raised by Alexander Shen: whether, for any triple of strings <span><math><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>)</mo></math></span>, there exists a string <em>z</em> such that each conditional Kolmogorov complexity <span><math><mi>C</mi><mo>(</mo><mi>a</mi><mo>|</mo><mi>z</mi><mo>)</mo><mo>,</mo><mi>C</mi><mo>(</mo><mi>b</mi><mo>|</mo><mi>z</mi><mo>)</mo><mo>,</mo><mi>C</mi><mo>(</mo><mi>c</mi><mo>|</mo><mi>z</mi><mo>)</mo></math></span> is approximately half of the corresponding unconditional Kolmogorov complexity. We give a negative answer to this question by constructing a triple <span><math><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>)</mo></math></span> for which no such string <em>z</em> exists. Moreover, we construct a fully explicit example of such a tuple, inspired by Vladimir V'yugin's perspective on the role of structured and natural objects in algorithmic information theory. Our construction is based on combinatorial properties of incidences in finite projective planes and relies on bounds for point-line incidences over prime fields, obtained using tools from additive combinatorics and algebraic methods, notably results by Bourgain–Katz–Tao and Stevens–De Zeeuw. As an application, we show that this impossibility yields lower bounds on the communication complexity of secret key agreement protocols in certain settings. These results reveal algebraic obstructions to efficient information exchange and highlight a separation in information-theoretic behavior between fields with and without proper subfields.</div></div>","PeriodicalId":54985,"journal":{"name":"Information and Computation","volume":"308 ","pages":"Article 105396"},"PeriodicalIF":1.0,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145738183","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":"Verifying and interpreting neural networks using finite automata","authors":"Marco Sälzer ,&nbsp;Eric Alsmann ,&nbsp;Florian Bruse ,&nbsp;Martin Lange","doi":"10.1016/j.ic.2025.105398","DOIUrl":"10.1016/j.ic.2025.105398","url":null,"abstract":"<div><div>Verifying properties and interpreting the behaviour of deep neural networks (DNN) is an important task given their ubiquitous use in applications, including safety-critical ones, and their black-box nature. We propose an automata-theoretic approach to tackling problems arising in DNN analysis. We show that the input-output behaviour of a DNN can be captured precisely by a (special) weak Büchi automaton and we show how these can be used to address common verification and interpretation tasks of DNN like adversarial robustness or minimum sufficient reasons.</div></div>","PeriodicalId":54985,"journal":{"name":"Information and Computation","volume":"308 ","pages":"Article 105398"},"PeriodicalIF":1.0,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145791115","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":"A perfect matching reciprocity method for embedding multiple hypercubes in an augmented cube: Application to Hamiltonian decomposition","authors":"Da-Wei Yang ,&nbsp;Hongyang Zhang ,&nbsp;Rong-Xia Hao ,&nbsp;Sun-Yuan Hsieh","doi":"10.1016/j.ic.2025.105401","DOIUrl":"10.1016/j.ic.2025.105401","url":null,"abstract":"<div><div>This paper focuses on the embeddability of hypercubes in an important class of Cayley graphs, known as augmented cubes. An <em>n</em>-dimensional augmented cube <span><math><mi>A</mi><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is constructed by augmenting the <em>n</em>-dimensional hypercube <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with additional edges, thus making <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> a spanning subgraph of <span><math><mi>A</mi><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. Dong and Wang (2019) first posed the problem of determining the number of <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>-isomorphic subgraphs in <span><math><mi>A</mi><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, which still remains open. By exploiting the Cayley properties of <span><math><mi>A</mi><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, we establish a lower bound for this number. Furthermore, we develop a method for constructing pairs of <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>-isomorphic subgraphs in <span><math><mi>A</mi><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with the minimum number of common edges. This is accomplished through the use of reciprocal perfect matchings, a technique that also relies on the Cayley property of <span><math><mi>A</mi><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. As an application, we prove that <span><math><mi>A</mi><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> admits <span><math><mi>n</mi><mo>−</mo><mn>1</mn></math></span> edge-disjoint Hamiltonian cycles when <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span> is odd and <span><math><mi>n</mi><mo>−</mo><mn>2</mn></math></span> cycles when <em>n</em> is even, thereby confirming a conjecture by Hung (2015) that <span><math><mi>A</mi><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> admits a Hamiltonian decomposition for odd <em>n</em> with <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span>.</div></div>","PeriodicalId":54985,"journal":{"name":"Information and Computation","volume":"308 ","pages":"Article 105401"},"PeriodicalIF":1.0,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145884338","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":"Improved approximations for the capacitated vehicle routing problem with fixed capacity","authors":"Jingyang Zhao,&nbsp;Mingyu Xiao","doi":"10.1016/j.ic.2026.105405","DOIUrl":"10.1016/j.ic.2026.105405","url":null,"abstract":"<div><div>The Capacitated Vehicle Routing Problem (CVRP) is one of the most extensively studied problems in combinatorial optimization. Based on customer demand, we distinguish three variants of CVRP: unit-demand, splittable, and unsplittable. In this paper, we consider <em>k</em>-CVRP in general metrics and on general graphs, where <em>k</em> is the vehicle capacity. All three versions are APX-hard for any fixed <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span>. Based on an <em>α</em>-approximation algorithm for metric TSP, we propose new approximation algorithms. For <span><math><mi>α</mi><mo>=</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>, we obtain a <span><math><mo>(</mo><mfrac><mrow><mn>5</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>−</mo><mi>Θ</mi><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msqrt><mrow><mi>k</mi></mrow></msqrt></mrow></mfrac><mo>)</mo><mo>)</mo></math></span>-approximation algorithm for the splittable and unit-demand cases, and a <span><math><mo>(</mo><mfrac><mrow><mn>5</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>ln</mi><mo>⁡</mo><mn>2</mn><mo>−</mo><mi>Θ</mi><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msqrt><mrow><mi>k</mi></mrow></msqrt></mrow></mfrac><mo>)</mo><mo>)</mo></math></span>-approximation algorithm for the unsplittable case. Our approximation ratio is better than the previous results when <em>k</em> is less than a fairly large value, approximately <span><math><mn>1.7</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>7</mn></mrow></msup></math></span>.</div><div>For small values of <em>k</em>, we design independent and elegant algorithms with further improvements. For the splittable and unit-demand cases, we improve the approximation ratio from 1.792 to 1.500 for <span><math><mi>k</mi><mo>=</mo><mn>3</mn></math></span>, and from 1.750 to 1.500 for <span><math><mi>k</mi><mo>=</mo><mn>4</mn></math></span>. For the unsplittable case, we improve the approximation ratio from 1.792 to 1.500 for <span><math><mi>k</mi><mo>=</mo><mn>3</mn></math></span>, from 2.051 to 1.750 for <span><math><mi>k</mi><mo>=</mo><mn>4</mn></math></span>, and from 2.249 to 2.157 for <span><math><mi>k</mi><mo>=</mo><mn>5</mn></math></span>. The approximation ratio for <span><math><mi>k</mi><mo>=</mo><mn>3</mn></math></span> surprisingly achieves the same value as in the splittable case. Our techniques, such as EX-ITP – an extension of the classic ITP method, have the potential to improve algorithms for other routing problems as well.</div></div>","PeriodicalId":54985,"journal":{"name":"Information and Computation","volume":"308 ","pages":"Article 105405"},"PeriodicalIF":1.0,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145976580","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}