Directed multicut is W[1]-hard, even for four terminal pairs

TL;DR

The paper proves that the Multicut problem in directed graphs remains W[1]-hard even with only four terminal pairs, and also establishes W[1]-hardness for Steiner Orientation, resolving key open problems in parameterized complexity.

Contribution

It demonstrates W[1]-hardness of directed Multicut with four terminal pairs and Steiner Orientation, advancing understanding of their computational complexity.

Findings

Multicut in directed graphs is W[1]-hard with four terminal pairs.

Steiner Orientation is W[1]-hard when parameterized by terminal pairs.

Almost completely resolves open problems in parameterized graph separation complexity.

Abstract

We prove that Multicut in directed graphs, parameterized by the size of the cutset, is W[1]-hard and hence unlikely to be fixed-parameter tractable even if restricted to instances with only four terminal pairs. This negative result almost completely resolves one of the central open problems in the area of parameterized complexity of graph separation problems, posted originally by Marx and Razgon [SIAM J. Comput. 43(2):355-388 (2014)], leaving only the case of three terminal pairs open. Our gadget methodology allows us also to prove W[1]-hardness of the Steiner Orientation problem parameterized by the number of terminal pairs, resolving an open problem of Cygan, Kortsarz, and Nutov [SIAM J. Discrete Math. 27(3):1503-1513 (2013)].