2026-07-23T01:41:48Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:15385 2024-03-06T10:55:37Z ddc:004 open_access

Refuting FPT Algorithms for Some Parameterized Problems Under Gap-ETH Agrawal, Akanksha Allumalla, Ravi Kiran Dhanekula, Varun Teja Token Jumping Bipartite Graphs Fixed Parameter Intractability Half Graphs Gap-Exponential Time Hypothesis In this article we study a well-known problem, called Bipartite Token Jumping and not-so-well known problem(s), which we call, Half (Induced-) Subgraph, and show that under Gap-ETH, these problems do not admit FPT algorithms. The problem Bipartite Token Jumping takes as input a bipartite graph G and two independent sets S,T in G, where |S| = |T| = k, and the objective is to test if there is a sequence of exactly k-sized independent sets ⟨ I₀, I₁,⋯, I_𝓁 ⟩ in G, such that: i) I₀ = S and I_𝓁 = T, and ii) for every j ∈ [𝓁], I_{j} is obtained from I_{j-1} by replacing a vertex in I_{j-1} by a vertex in V(G) ⧵ I_{j-1}. We show that, assuming Gap-ETH, Bipartite Token Jumping does not admit an FPT algorithm. We note that this result resolves one of the (two) open problems posed by Bartier et al. (ISAAC 2020), under Gap-ETH. Most of the known reductions related to Token Jumping exploit the property given by triangles (i.e., C₃s), to obtain the correctness, and our results refutes FPT algorithm for Bipartite Token Jumping, where the input graph cannot have any triangles. For an integer k ∈ ℕ, the half graph S_{k,k} is the graph with vertex set V(S_{k,k}) = A_k ∪ B_k, where A_k = {a₁,a₂,⋯, a_k} and B_k = {b₁,b₂,⋯, b_k}, and for i,j ∈ [k], {a_i,b_j} ∈ E(T_{k,k}) if and only if j ≥ i. We also study the Half (Induced-)Subgraph problem where we are given a graph G and an integer k, and the goal is to check if G contains S_{k,k} as an (induced-)subgraph. Again under Gap-ETH, we show that Half (Induced-)Subgraph does not admit an FPT algorithm, even when the input is a bipartite graph. We believe that the above problem (and its negative) result maybe of independent interest and could be useful obtaining new fixed parameter intractability results. There are very few reductions known in the literature which refute FPT algorithms for a parameterized problem based on assumptions like Gap-ETH. Thus our technique (and simple reductions) exhibits the potential of such conjectures in obtaining new (and possibly easier) proofs for refuting FPT algorithms for parameterized problems. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Akanksha Agrawal and Ravi Kiran Allumalla and Varun Teja Dhanekula 2021 Is Part Of LIPIcs, Volume 214, 16th International Symposium on Parameterized and Exact Computation (IPEC 2021) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.IPEC.2021.2 urn:nbn:de:0030-drops-153851 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2021.2 eng https://creativecommons.org/licenses/by/4.0/legalcode