Pub Date : 2025-12-01Epub Date: 2025-05-22DOI: 10.1016/j.jcss.2025.103679
Riccardo Dondi , Manuel Lafond
One of the most studied problem in theoretical computer science, Vertex Cover, has been recently considered in the temporal graph framework. Here we study a Vertex Cover variant, called k-TimelineCover. Given a temporal graph k-TimelineCover asks to define an interval for each vertex so that for every temporal edge existing in a timestamp t, at least one of the endpoints has an interval that includes t. The goal is to decide whether it is possible to cover every temporal edge while using vertex intervals of total span at most k. k-TimelineCover has been shown to be NP-hard, but its parameterized complexity has not been fully understood when parameterizing by the span of the solution. We settle this open problem by giving an FPT algorithm that combines two techniques, a modified form of iterative compression and a reduction to Digraph Pair Cut.
{"title":"An FPT algorithm for timeline cover","authors":"Riccardo Dondi , Manuel Lafond","doi":"10.1016/j.jcss.2025.103679","DOIUrl":"10.1016/j.jcss.2025.103679","url":null,"abstract":"<div><div>One of the most studied problem in theoretical computer science, <span>Vertex Cover</span>, has been recently considered in the temporal graph framework. Here we study a <span>Vertex Cover</span> variant, called k-<span>TimelineCover</span>. Given a temporal graph k-<span>TimelineCover</span> asks to define an interval for each vertex so that for every temporal edge existing in a timestamp <em>t</em>, at least one of the endpoints has an interval that includes <em>t</em>. The goal is to decide whether it is possible to cover every temporal edge while using vertex intervals of total span at most <em>k</em>. k-<span>TimelineCover</span> has been shown to be NP-hard, but its parameterized complexity has not been fully understood when parameterizing by the span of the solution. We settle this open problem by giving an FPT algorithm that combines two techniques, a modified form of iterative compression and a reduction to <span>Digraph Pair Cut</span>.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"154 ","pages":"Article 103679"},"PeriodicalIF":1.1,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144134844","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the k-Diameter-Optimally Augmenting Tree Problem we are given a tree T of n vertices embedded in an unknown metric space. An oracle can report the cost of any edge in constant time, and we want to augment T with k shortcuts to minimize the resulting diameter. When , -time algorithms exist for paths and trees. We show that queries cannot provide a better than 10/9-approximation for trees when . For any constant , we design a linear-time -approximation algorithm for paths when , thus establishing a dichotomy between paths and trees for . Our algorithm employs an ad-hoc data structure, which we also use in a linear-time 4-approximation algorithm for trees, and to compute the diameter of (possibly non-metric) graphs with edges in time .
{"title":"Finding diameter-reducing shortcuts in trees","authors":"Davide Bilò , Luciano Gualà , Stefano Leucci , Luca Pepè Sciarria","doi":"10.1016/j.jcss.2025.103658","DOIUrl":"10.1016/j.jcss.2025.103658","url":null,"abstract":"<div><div>In the <em>k-Diameter-Optimally Augmenting Tree Problem</em> we are given a tree <em>T</em> of <em>n</em> vertices embedded in an unknown <em>metric</em> space. An oracle can report the cost of any edge in constant time, and we want to augment <em>T</em> with <em>k</em> shortcuts to minimize the resulting diameter. When <span><math><mi>k</mi><mo>=</mo><mn>1</mn></math></span>, <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span>-time algorithms exist for paths and trees. We show that <span><math><mi>o</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> queries cannot provide a better than 10/9-approximation for trees when <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span>. For any constant <span><math><mi>ε</mi><mo>></mo><mn>0</mn></math></span>, we design a linear-time <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>ε</mi><mo>)</mo></math></span>-approximation algorithm for paths when <span><math><mi>k</mi><mo>=</mo><mi>o</mi><mo>(</mo><msqrt><mrow><mi>log</mi><mo></mo><mi>n</mi></mrow></msqrt><mo>)</mo></math></span>, thus establishing a dichotomy between paths and trees for <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span>. Our algorithm employs an ad-hoc data structure, which we also use in a linear-time 4-approximation algorithm for trees, and to compute the diameter of (possibly non-metric) graphs with <span><math><mi>n</mi><mo>+</mo><mi>k</mi><mo>−</mo><mn>1</mn></math></span> edges in time <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>k</mi><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"153 ","pages":"Article 103658"},"PeriodicalIF":1.1,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143879465","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-09-01Epub Date: 2025-03-22DOI: 10.1016/j.jcss.2025.103656
Ajay D. Kshemkalyani , Gokarna Sharma
Given an undirected, anonymous, port-labeled graph of n memory-less nodes, m edges, and degree Δ, we consider the problem of dispersing robots (or tokens) positioned initially arbitrarily on the nodes of the graph to exactly k different nodes, one on each node. The objective is to simultaneously minimize time and memory requirement at each robot. The best previously known algorithm solves this problem in time storing bits at each robot, where is the number of nodes with multiple robots positioned on them in the initial configuration. In this paper, we present a novel multi-source DFS traversal algorithm solving this problem in time with bits at each robot. The memory complexity of our algorithm is already asymptotically optimal and the time complexity is asymptotically optimal for the graphs of constant degree . The result holds in both synchronous and asynchronous settings.
{"title":"Near-optimal dispersion on arbitrary anonymous graphs","authors":"Ajay D. Kshemkalyani , Gokarna Sharma","doi":"10.1016/j.jcss.2025.103656","DOIUrl":"10.1016/j.jcss.2025.103656","url":null,"abstract":"<div><div>Given an undirected, anonymous, port-labeled graph of <em>n</em> memory-less nodes, <em>m</em> edges, and degree Δ, we consider the problem of dispersing <span><math><mi>k</mi><mo>≤</mo><mi>n</mi></math></span> robots (or tokens) positioned initially arbitrarily on the nodes of the graph to exactly <em>k</em> different nodes, one on each node. The objective is to simultaneously minimize time and memory requirement at each robot. The best previously known algorithm solves this problem in <span><math><mi>O</mi><mo>(</mo><mi>min</mi><mo></mo><mo>{</mo><mi>m</mi><mo>,</mo><mi>k</mi><mi>Δ</mi><mo>}</mo><mo>⋅</mo><mi>log</mi><mo></mo><mi>ℓ</mi><mo>)</mo></math></span> time storing <span><math><mi>O</mi><mo>(</mo><mi>log</mi><mo></mo><mo>(</mo><mi>k</mi><mo>+</mo><mi>Δ</mi><mo>)</mo><mo>)</mo></math></span> bits at each robot, where <span><math><mi>ℓ</mi><mo>≤</mo><mi>k</mi><mo>/</mo><mn>2</mn></math></span> is the number of nodes with multiple robots positioned on them in the initial configuration. In this paper, we present a novel multi-source DFS traversal algorithm solving this problem in <span><math><mi>O</mi><mo>(</mo><mi>min</mi><mo></mo><mo>{</mo><mi>m</mi><mo>,</mo><mi>k</mi><mi>Δ</mi><mo>}</mo><mo>)</mo></math></span> time with <span><math><mi>O</mi><mo>(</mo><mi>log</mi><mo></mo><mo>(</mo><mi>k</mi><mo>+</mo><mi>Δ</mi><mo>)</mo><mo>)</mo></math></span> bits at each robot. The memory complexity of our algorithm is already asymptotically optimal and the time complexity is asymptotically optimal for the graphs of constant degree <span><math><mi>Δ</mi><mo>=</mo><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span>. The result holds in both synchronous and asynchronous settings.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"152 ","pages":"Article 103656"},"PeriodicalIF":1.1,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143739196","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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}
Pub Date : 2025-09-01Epub Date: 2025-03-20DOI: 10.1016/j.jcss.2025.103657
Yufeng Wu , Louxin Zhang
Reconstructing a parsimonious phylogenetic network that displays multiple phylogenetic trees is an important problem in phylogenetics, where the complexity of the inferred networks is measured by reticulation numbers. The reticulation number for a set of trees is defined as the minimum number of reticulations in a phylogenetic network that displays those trees. A mathematical problem is bounding the reticulation number for multiple trees over a fixed number of taxa. While this problem has been extensively studied for two trees, much less is known about the upper bounds on the reticulation numbers for three or more arbitrary trees. In this paper, we present a few non-trivial upper bounds on reticulation numbers for three or more trees.
{"title":"Bounding the number of reticulation events for displaying multiple trees in a phylogenetic network","authors":"Yufeng Wu , Louxin Zhang","doi":"10.1016/j.jcss.2025.103657","DOIUrl":"10.1016/j.jcss.2025.103657","url":null,"abstract":"<div><div>Reconstructing a parsimonious phylogenetic network that displays multiple phylogenetic trees is an important problem in phylogenetics, where the complexity of the inferred networks is measured by reticulation numbers. The reticulation number for a set of trees is defined as the minimum number of reticulations in a phylogenetic network that displays those trees. A mathematical problem is bounding the reticulation number for multiple trees over a fixed number of taxa. While this problem has been extensively studied for two trees, much less is known about the upper bounds on the reticulation numbers for three or more arbitrary trees. In this paper, we present a few non-trivial upper bounds on reticulation numbers for three or more trees.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"152 ","pages":"Article 103657"},"PeriodicalIF":1.1,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143739197","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-09-01Epub Date: 2025-02-21DOI: 10.1016/j.jcss.2025.103637
Takumi Tada, Kazuya Haraguchi
For a set system , we call each a component. A nonempty subset is a removable set (RS) of C if is a component. We say that a set system has subset-disjoint (SD) property if, for any two components with , every minimal RS Y of C satisfies either or . Assuming that an SD set system is implicitly given by an oracle that returns a minimal RS of a component, we provide an algorithm that enumerates all components in linear time/space with respect to and oracle running time/space. We then extend this algorithm to linear-delay enumeration of all 2-edge-connected (or 2-vertex-connected) induced subgraphs in an undirected graph and of all strongly connected subgraphs in a digraph.
{"title":"A linear delay algorithm in SD set system and its application to subgraph enumeration","authors":"Takumi Tada, Kazuya Haraguchi","doi":"10.1016/j.jcss.2025.103637","DOIUrl":"10.1016/j.jcss.2025.103637","url":null,"abstract":"<div><div>For a set system <span><math><mo>(</mo><mi>V</mi><mo>,</mo><mi>C</mi><mo>⊆</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>V</mi></mrow></msup><mo>)</mo></math></span>, we call each <span><math><mi>C</mi><mo>∈</mo><mi>C</mi></math></span> a component. A nonempty subset <span><math><mi>Y</mi><mo>⊊</mo><mi>C</mi></math></span> is a removable set (RS) of <em>C</em> if <span><math><mi>C</mi><mo>∖</mo><mi>Y</mi></math></span> is a component. We say that a set system has subset-disjoint (SD) property if, for any two components <span><math><mi>C</mi><mo>,</mo><msup><mrow><mi>C</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> with <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>⊊</mo><mi>C</mi></math></span>, every minimal RS <em>Y</em> of <em>C</em> satisfies either <span><math><mi>Y</mi><mo>⊆</mo><msup><mrow><mi>C</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> or <span><math><mi>Y</mi><mo>∩</mo><msup><mrow><mi>C</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>=</mo><mo>∅</mo></math></span>. Assuming that an SD set system is implicitly given by an oracle that returns a minimal RS of a component, we provide an algorithm that enumerates all components in linear time/space with respect to <span><math><mo>|</mo><mi>V</mi><mo>|</mo></math></span> and oracle running time/space. We then extend this algorithm to linear-delay enumeration of all 2-edge-connected (or 2-vertex-connected) induced subgraphs in an undirected graph and of all strongly connected subgraphs in a digraph.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"152 ","pages":"Article 103637"},"PeriodicalIF":1.1,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143509232","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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}
Pub Date : 2025-09-01Epub Date: 2025-03-04DOI: 10.1016/j.jcss.2025.103638
Leslie Ann Goldberg, John Lapinskas
In contention resolution, multiple processors are trying to coordinate to send discrete messages through a shared channel with limited communication. If two processors send at the same time, the messages collide and are not transmitted successfully. Queue-free backoff protocols are an important special case — for example, Google Drive and AWS instruct their users to implement binary exponential backoff to handle busy periods. It is a long-standing conjecture of Aldous (1987) [4] that no stable backoff protocols exist for any positive arrival rate of processors. This foundational question remains open; instability is only known in general when the arrival rate of processors is at least 0.42 (Goldberg et al., 2004 [13]). We prove Aldous' conjecture for all backoff protocols outside of a tightly-constrained special case using a new domination technique to get around the main difficulty, which is the strong dependencies between messages.
在争用解决中，多个处理器试图通过有限通信的共享通道协调发送离散消息。如果两个处理器同时发送消息，则消息会发生冲突，无法成功传输。无队列回退协议是一个重要的特例——例如，谷歌Drive和AWS指导他们的用户实现二进制指数回退来处理繁忙时段。Aldous(1987)[4]的一个长期猜想是，对于任何正的处理器到达率，不存在稳定的后退协议。这个基本问题仍然悬而未决；一般来说，只有当处理器到达率至少为0.42时，才知道不稳定性（Goldberg et al., 2004[13]）。我们使用一种新的支配技术证明了除严格约束的特殊情况外所有退退协议的Aldous猜想，以绕过消息之间的强依赖关系这一主要困难。
{"title":"Instability of backoff protocols with arbitrary arrival rates","authors":"Leslie Ann Goldberg, John Lapinskas","doi":"10.1016/j.jcss.2025.103638","DOIUrl":"10.1016/j.jcss.2025.103638","url":null,"abstract":"<div><div>In contention resolution, multiple processors are trying to coordinate to send discrete messages through a shared channel with limited communication. If two processors send at the same time, the messages collide and are not transmitted successfully. Queue-free backoff protocols are an important special case — for example, Google Drive and AWS instruct their users to implement binary exponential backoff to handle busy periods. It is a long-standing conjecture of Aldous (1987) <span><span>[4]</span></span> that no stable backoff protocols exist for any positive arrival rate of processors. This foundational question remains open; instability is only known in general when the arrival rate of processors is at least 0.42 (Goldberg et al., 2004 <span><span>[13]</span></span>). We prove Aldous' conjecture for all backoff protocols outside of a tightly-constrained special case using a new domination technique to get around the main difficulty, which is the strong dependencies between messages.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"152 ","pages":"Article 103638"},"PeriodicalIF":1.1,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143577062","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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}
Pub Date : 2025-09-01Epub Date: 2025-03-19DOI: 10.1016/j.jcss.2025.103655
Martin Frohn , Niels Holtgrefe , Leo van Iersel , Mark Jones , Steven Kelk
Semi-directed networks are partially directed graphs that model evolution where the directed edges represent reticulate evolutionary events. We present an algorithm that reconstructs binary n-leaf semi-directed level-1 networks in time from its quarnets (4-leaf subnetworks). Our method assumes we have direct access to all quarnets, yet uses only an asymptotically optimal number of quarnets. When the network is assumed to contain no triangles, our method instead relies only on four-cycle quarnets and the splits of the other quarnets. A variant of our algorithm works with quartets rather than quarnets and we show that it reconstructs most of a semi-directed level-1 network from an asymptotically optimal of the quartets it displays. Additionally, we provide an time algorithm that reconstructs the tree-of-blobs of any binary n-leaf semi-directed network with unbounded level from splits of its quarnets.
{"title":"Reconstructing semi-directed level-1 networks using few quarnets","authors":"Martin Frohn , Niels Holtgrefe , Leo van Iersel , Mark Jones , Steven Kelk","doi":"10.1016/j.jcss.2025.103655","DOIUrl":"10.1016/j.jcss.2025.103655","url":null,"abstract":"<div><div>Semi-directed networks are partially directed graphs that model evolution where the directed edges represent reticulate evolutionary events. We present an algorithm that reconstructs binary <em>n</em>-leaf semi-directed level-1 networks in <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> time from its quarnets (4-leaf subnetworks). Our method assumes we have direct access to all quarnets, yet uses only an asymptotically optimal number of <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span> quarnets. When the network is assumed to contain no triangles, our method instead relies only on four-cycle quarnets and the splits of the other quarnets. A variant of our algorithm works with quartets rather than quarnets and we show that it reconstructs most of a semi-directed level-1 network from an asymptotically optimal <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span> of the quartets it displays. Additionally, we provide an <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></math></span> time algorithm that reconstructs the tree-of-blobs of any binary <em>n</em>-leaf semi-directed network with unbounded level from <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></math></span> splits of its quarnets.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"152 ","pages":"Article 103655"},"PeriodicalIF":1.1,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143715791","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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}
Pub Date : 2025-08-01Epub Date: 2025-02-05DOI: 10.1016/j.jcss.2025.103634
Wojciech Czerwiński , Maciej Dębski , Tomasz Gogasz , Gordon Hoi , Sanjay Jain , Michał Skrzypczak , Frank Stephan , Christopher Tan
This paper studies the complexity of operations on finite automata and the complexity of their decision problems when the alphabet is unary. Let n denote the number of states of the input automata considered. The following main results are obtained:
(1) Equality and inclusion of NFAs can be decided within time . The previous upper bound was by Chrobak (1986).
(2) One can determine a UFA (unambiguous finite automata) for complement of another UFA or union of two UFAs using at most quasipolynomial number of states. However, for concatenation of two n-state UFAs, the worst case is a UFA having states.
(3) Results when an infinite ω-word given by a UFA or an NFA is a member of a given regular ω-language are obtained.
{"title":"Languages given by finite automata over the unary alphabet","authors":"Wojciech Czerwiński , Maciej Dębski , Tomasz Gogasz , Gordon Hoi , Sanjay Jain , Michał Skrzypczak , Frank Stephan , Christopher Tan","doi":"10.1016/j.jcss.2025.103634","DOIUrl":"10.1016/j.jcss.2025.103634","url":null,"abstract":"<div><div>This paper studies the complexity of operations on finite automata and the complexity of their decision problems when the alphabet is unary. Let <em>n</em> denote the number of states of the input automata considered. The following main results are obtained:</div><div>(1) Equality and inclusion of NFAs can be decided within time <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mo>(</mo><msup><mrow><mo>(</mo><mi>n</mi><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>)</mo></mrow></msup></math></span>. The previous upper bound <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mo>(</mo><msup><mrow><mo>(</mo><mi>n</mi><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>)</mo></mrow></msup></math></span> was by Chrobak (1986).</div><div>(2) One can determine a UFA (unambiguous finite automata) for complement of another UFA or union of two UFAs using at most quasipolynomial number of states. However, for concatenation of two <em>n</em>-state UFAs, the worst case is a UFA having <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>Θ</mi><mo>(</mo><msup><mrow><mo>(</mo><mi>n</mi><msup><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msup><mo></mo><mi>n</mi><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>)</mo></mrow></msup></math></span> states.</div><div>(3) Results when an infinite <em>ω</em>-word given by a UFA or an NFA is a member of a given regular <em>ω</em>-language are obtained.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"151 ","pages":"Article 103634"},"PeriodicalIF":1.1,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143419429","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-01Epub Date: 2025-02-13DOI: 10.1016/j.jcss.2025.103636
Luís Cunha , Gabriel Duarte , Fábio Protti , Loana Nogueira , Uéverton Souza
The Yutsis property of a graph G is the property of partitioning its vertex set into two induced trees. Although recognizing Yutsis graphs is NP-complete even on planar graphs, it is still possible to consider two even more challenging problems: (i) recognizing k-Yutsis graphs, which are graphs that have their vertex sets partitioned into k induced trees, for a fixed ; (ii) determining the tree cover number of a given graph G, i.e., the minimum number of vertex-disjoint induced trees covering all vertices of G. We prove that determining the tree cover number of a split graph G is NP-hard, contrasting with the polynomial-time recognition of k-Yutsis chordal graphs. We also investigate the tree cover number computation and the k-Yutsis graph recognition concerning treewidth and clique-width parameterizations.
{"title":"Induced tree covering and the generalized Yutsis property","authors":"Luís Cunha , Gabriel Duarte , Fábio Protti , Loana Nogueira , Uéverton Souza","doi":"10.1016/j.jcss.2025.103636","DOIUrl":"10.1016/j.jcss.2025.103636","url":null,"abstract":"<div><div>The Yutsis property of a graph <em>G</em> is the property of partitioning its vertex set into two induced trees. Although recognizing Yutsis graphs is NP-complete even on planar graphs, it is still possible to consider two even more challenging problems: (i) recognizing <em>k</em>-Yutsis graphs, which are graphs that have their vertex sets partitioned into <em>k</em> induced trees, for a fixed <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span>; (ii) determining the tree cover number of a given graph <em>G</em>, i.e., the minimum number of vertex-disjoint induced trees covering all vertices of <em>G</em>. We prove that determining the tree cover number of a split graph <em>G</em> is NP-hard, contrasting with the polynomial-time recognition of <em>k</em>-Yutsis chordal graphs. We also investigate the tree cover number computation and the <em>k</em>-Yutsis graph recognition concerning treewidth and clique-width parameterizations.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"151 ","pages":"Article 103636"},"PeriodicalIF":1.1,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143419432","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-01Epub Date: 2025-02-13DOI: 10.1016/j.jcss.2025.103635
Cristina Bazgan , André Nichterlein , Sofia Vazquez Alferez
We analyze the computational complexity of the following computational problems called Bounded-Density Edge Deletion and Bounded-Density Vertex Deletion: Given a graph G, a budget k and a target density , are there k edges (k vertices) whose removal from G results in a graph where the densest subgraph has density at most ? Here, the density of a graph is the number of its edges divided by the number of its vertices. We prove that both problems are polynomial-time solvable on trees and cliques but are NP-complete on planar bipartite graphs and split graphs. From a parameterized point of view, we show that both problems are fixed-parameter tractable with respect to the vertex cover number but W[1]-hard with respect to the solution size. Furthermore, we prove that Bounded-Density Edge Deletion is W[1]-hard with respect to the feedback edge number, demonstrating that the problem remains hard on very sparse graphs.
{"title":"Destroying densest subgraphs is hard","authors":"Cristina Bazgan , André Nichterlein , Sofia Vazquez Alferez","doi":"10.1016/j.jcss.2025.103635","DOIUrl":"10.1016/j.jcss.2025.103635","url":null,"abstract":"<div><div>We analyze the computational complexity of the following computational problems called <span>Bounded-Density Edge Deletion</span> and <span>Bounded-Density Vertex Deletion</span>: Given a graph <em>G</em>, a budget <em>k</em> and a target density <span><math><msub><mrow><mi>τ</mi></mrow><mrow><mi>ρ</mi></mrow></msub></math></span>, are there <em>k</em> edges (<em>k</em> vertices) whose removal from <em>G</em> results in a graph where the densest subgraph has density at most <span><math><msub><mrow><mi>τ</mi></mrow><mrow><mi>ρ</mi></mrow></msub></math></span>? Here, the density of a graph is the number of its edges divided by the number of its vertices. We prove that both problems are polynomial-time solvable on trees and cliques but are NP-complete on planar bipartite graphs and split graphs. From a parameterized point of view, we show that both problems are fixed-parameter tractable with respect to the vertex cover number but W[1]-hard with respect to the solution size. Furthermore, we prove that <span>Bounded-Density Edge Deletion</span> is W[1]-hard with respect to the feedback edge number, demonstrating that the problem remains hard on very sparse graphs.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"151 ","pages":"Article 103635"},"PeriodicalIF":1.1,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143444148","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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}
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