The Maximum Solution Problem on Graphs

We study the complexity of the problem MAX SOL which is a natural optimisation version of the graph homomorphism problem. Given a fixed target graph H with V (H) ⊆ N, and a weight function w : V (G) → Q, an instance of the problem is a graph G and the goal is to find a homomorphism f : G → H which maximises P v∈G f(v) · w(v). MAX SOL can be seen as a restriction of the MIN HOM-problem [Gutin et al., Disc. App. Math., 154 (2006), pp. 881-889] and as a natural generalisation of MAX ONES to larger domains. We present new tools with which we classify the complexity of MAX SOL for irreflexive graphs with degree less than or equal to 2 as well as for small graphs (|V (H)| ≤ 4). We also study an extension of MAX SOL where value lists and arbitrary weights are allowed; somewhat surprisingly, this problem is polynomial-time equivalent to MIN HOM.