QCSP on Reflexive Tournaments

TL;DR

This paper establishes a complexity classification for the QCSP on reflexive tournaments, showing it is either in NL or NP-hard depending on the structure of the tournament's strongly connected components.

Contribution

It provides a complete dichotomy for the complexity of QCSP on reflexive tournaments based on their strongly connected components.

Findings

QCSP(H) is in NL if both initial and final components are size 1.

QCSP(H) is NP-hard otherwise.

The complexity depends on the structure of the tournament's strongly connected components.

Abstract

We give a complexity dichotomy for the Quantified Constraint Satisfaction Problem QCSP(H) when H is a reflexive tournament. It is well-known that reflexive tournaments can be split into a sequence of strongly connected components H_1,...,H_n so that there exists an edge from every vertex of H_i to every vertex of H_j if and only if i<j. We prove that if H has both its initial and final strongly connected component (possibly equal) of size 1, then QCSP(H) is in NL and otherwise QCSP(H) is NP-hard.