Refuting FPT algorithms for some parameterized problems under Gap-ETH

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], Ij is obtained from Ij−1 by replacing a vertex in Ij−1 by a vertex in V (G) \ Ij−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], {ai, bj} ∈ 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.