[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82638-en":3,"doc-seo-82638-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82638,16904993612988,"Olivia Brown","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Faster Parameterized Broadcasting","Faster Parameterized Broadcasting studies Telephone Broadcast on connected graphs, asking for the minimum number of synchronous rounds needed for a single source to inform all vertices when each informed vertex can transmit to at most one neighbor per round. While Telephone Broadcast remains NP-hard on several restricted graph classes, recent work established fixed-parameter tractability for parameters like vertex cover, vertex integrity, and distance to clique. This paper presents improved parameterized algorithms using Turing reductions to edge-weighted b-matching, achieving faster running times.","Faster Parameterized Broadcasting  \nÉdouard Bonnet \\# Ñ  \nCNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR 5668, Lyon, France Carl Feghali 1 \\# Ñ  \nCNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR 5668, Lyon, France Manolis Vasilakis \\# Ñ  \nUniversité Paris Dauphine – PSL, CNRS UMR7243, LAMSADE, Paris, France  \n~~ Abstract ~~  \nGiven a connected graph G and a source s ∈ V (G), what is the smallest number of rounds necessary for all vertices of G to receive a message initially only held by s, where at each round every informed vertex passes the message to one of its neighbors? This problem is called Telephone Broadcast and is suprisingly hard: it remains NP-hard on cycles intersecting at a single shared vertex, in particular, graphs of pathwidth 2 with a linear feedback vertex set of size 1, as well as on graphs with treedepth at most 6 [Egami et al.; MFCS ’25] . Vertex cover number, vertex integrity, and distance to clique are among the few parameters for which Telephone Broadcast is fixed-parameter tractable. There is a 2O(vc3 ) nO(1)-time algorithm parameterized by vertex cover number vc [Fomin, Fraigniaud, Golovach; TCS ’24], a double-exponential algorithm parameterized by vertex integrity vi, and a 2O (k2 ) nO(1)-time algorithm parameterized by distance to clique k [Egami et al. ; MFCS ’25] .  \nIn this paper, we give improved parameterized algorithms for Telephone Broadcast with running times 2O(vc log vc)nO(1) , 2O(vi2 log vi) nO(1), and 2O (k log k)nO(1) . The main ingredient that makes our algorithms faster is a Turing reduction to edge-weighted b-Matching.  \n2012 ACM Subject Classification Theory of computation → Parameterized complexity and exact algorithms  \nKeywords and phrases Parameterized Complexity, Structural Graph Parameters, Telephone Broadcast  \nFunding Édouard Bonnet: Supported by the ANR project TWIN-WIDTH (ANR-21-CE48-0014-01) . Manolis Vasilakis: Supported by the ANR project S-EX-AP-PE-AL (ANR-21-CE48-0022) and the GDR ROD Mobility Scholarship.  \n 1  Introduction  \nBroadcasting is one of the most basic primitives in communication networks: a message initially held by one source has to reach every vertex of the network. In the classical telephone model [26], the network is represented by a connected graph G, time is divided into synchronous rounds, and in each round every informed vertex may transmit the message to at most one uninformed neighbor. For a source s ∈ V (G), the broadcast time b (G, s) is the minimum number of rounds needed to inform all vertices. If |V(G)| = n, then ⌈log2 n⌉ ⩽ b (G, s) ⩽ n − 1, and both bounds are tight: complete graphs realize the lower bound, while a path whose source is an endpoint realizes the upper bound. Equivalently, a broadcast protocol can be represented by a spanning tree rooted at s together with an ordering of the children of every vertex. Telephone Broadcast then asks, given a connected graph G, a source s, and an integer t, whether b (G, s) ⩽ t.  \n1 We are deeply saddened to report that our colleague and friend, Carl Feghali, recently passed away. We greatly enjoyed working on this project with him.  \n2 Faster Parameterized Broadcasting  \nThe problem is old, natural, and algorithmically challenging. Slater, Cockayne, and Hedetniemi [35] initiated the systematic study of the problem, proving that it is NP-complete in general but polynomial-time solvable on trees. The surrounding literature is extensive; see, for instance, the survey of Hedetniemi, Hedetniemi, and Liestman [26], the survey of Fraigniaud and Lazard [19], and the book of Hromkovic et al. [27] . The computational difficulty already appears on quite restricted graph classes: Jansen and Müller [28] proved NP-hardness on chordal graphs and grid graphs, among others.  \nThe parameterized-complexity study of Telephone Broadcast was initiated by Fomin, Fraigniaud, and Golovach [18] . They gave a 3n nO(1)-time exact algorithm and proved fixed-parameter tractability for feedback edge ","cbCaiaftBbuaCZLq","https://ap.wps.com/l/cbCaiaftBbuaCZLq","pdf",753749,1,19,"English","en",105,"# Introduction\n## Problem definition and classical bounds\n# Contribution\n## Improved parameterized algorithms\n## Turing reduction to b-Matching","[{\"question\":\"What problem does the paper address?\",\"answer\":\"The paper addresses Telephone Broadcast: given a connected graph, a source vertex, and a number of rounds t, it asks whether the broadcast time from the source is at most t.\"},{\"question\":\"Why is Telephone Broadcast computationally difficult?\",\"answer\":\"Telephone Broadcast is NP-hard even on several restricted graph classes such as cycles intersecting at a single shared vertex, including graphs with small structural parameters like certain pathwidth-2 instances and graphs with treedepth at most 6.\"},{\"question\":\"Which parameters does the paper improve algorithms for?\",\"answer\":\"It improves the parameter dependence of positive results for vertex cover number, vertex integrity, and distance to clique, giving faster running times than previously known 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