A Sub-exponential FPT Algorithm and a Polynomial Kernel for Minimum Directed Bisection on Semicomplete Digraphs - Madathil, Jayakrishnan; Sharma, Roohani; Zehavi, Meirav (Algorithmica, 2021)
Given an n -vertex digraph D and a non-negative integer k , the M inimum D irected B isection problem asks if the vertices of D can be partitioned into two parts, say L and R , such that | L | and | R | differ by at most 1 and the number of arcs from R to L is at most k . This problem is known to be NP-hard even when k = 0 . We investigate the parameterized complexity of this problem on semicomplete digraphs. We show that M inimum D irected B isection admits a sub-exponential time fixed-parameter tractable algorithm on semicomplete digraphs. We also show that M inimum D irected B isection admits a polynomial kernel on semicomplete digraphs. To design the kernel, we use ( n , k , k 2 ) -splitters, which, to the best of our knowledge, have never been used before in the design of kernels. We also prove that M inimum D irected B isection is NP-hard on semicomplete digraphs, but polynomial time solvable on tournaments.