Abstract
Given an n-vertex digraph D and a non-negative integer k, the Minimum Directed Bisection problem asks if the vertices of D can be partitioned into two parts, say L and R, such that \({\vert {L} \vert }\) and \({\vert {R} \vert }\) 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 Minimum Directed Bisection admits a sub-exponential time fixed-parameter tractable algorithm on semicomplete digraphs. We also show that Minimum Directed Bisection 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 Minimum Directed Bisection is NP-hard on semicomplete digraphs, but polynomial time solvable on tournaments.
Highlights
A bisection of a graph is a partition of its vertex set into two equal parts
In the Minimum Directed Bisection problem, the input consists of a digraph D and an integer k, and the question is to determine whether the vertices of D can be partitioned into two parts, say L and R, such that ||L| − |R|| ≤ 1, and there are at most k arcs with their tails in R and heads in L
We show that Minimum Directed Bisectio√n on semicomplete digraphs admits a sub-exponential time algorithm that runs in time 2O( k log k)nO(1) (Theorem 21)
Summary
A bisection of a graph is a partition of its vertex set into two (almost) equal parts. Semicomplete digraphs are a slight generalization of tournaments, the flexibility in the definition allowing for the presence of anti-parallel arcs makes them distinctly dissimilar to tournaments when it comes to tractability of algorithmic problems This contrast in behavior is perhaps best illustrated by problems such as Cutwidth and Optimal Linear Arrangement (OLA), which are polynomial time solvable on tournaments, but NP-hard on semicomplete digraphs [4]. We show that while Minimum Directed Bisection is polynomial time solvable on tournaments (Lemma 4), it is NP-hard on semicomplete digraphs (Lemma 16).
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