The Complexity of Surjective Homomorphism Problems -- a Survey

We survey known results about the complexity of surjective homomorphism problems, studied in the context of related problems in the literature such as list homomorphism, retraction and compaction. In comparison with these problems, surjective homomorphism problems seem to be harder to classify and we examine especially three concrete problems that have arisen from the literature, two of which remain of open complexity

A complete and equal computational complexity classification of compaction and retraction to all graphs with at most four vertices and some general results

AbstractA very close relationship between the compaction, retraction, and constraint satisfaction problems has been established earlier providing evidence that it is likely to be difficult to give a complete computational complexity classification of the compaction and retraction problems for reflexive or bipartite graphs. In this paper, we give a complete computational complexity classification of the compaction and retraction problems for all graphs (including partially reflexive graphs) with four or fewer vertices. The complexity classification of both the compaction and retraction problems is found to be the same for each of these graphs. This relates to a long-standing open problem concerning the equivalence of the compaction and retraction problems. The study of the compaction and retraction problems for graphs with at most four vertices has a special interest as it covers a popular open problem in relation to the general open problem. We also give complexity results for some general graphs. The compaction and retraction problems are special graph colouring problems, and can also be viewed as partition problems with certain properties. We describe some practical applications also

The Complexity of Approximately Counting Retractions

Let $G$ be a graph that contains an induced subgraph $H$. A retraction from $G$ to $H$ is a homomorphism from $G$ to $H$ that is the identity function on $H$. Retractions are very well-studied: Given $H$, the complexity of deciding whether there is a retraction from an input graph $G$ to $H$ is completely classified, in the sense that it is known for which $H$ this problem is tractable (assuming $\mathrm{P}\neq \mathrm{NP}$). Similarly, the complexity of (exactly) counting retractions from $G$ to $H$ is classified (assuming $\mathrm{FP}\neq \#\mathrm{P}$). However, almost nothing is known about approximately counting retractions. Our first contribution is to give a complete trichotomy for approximately counting retractions to graphs of girth at least $5$. Our second contribution is to locate the retraction counting problem for each $H$ in the complexity landscape of related approximate counting problems. Interestingly, our results are in contrast to the situation in the exact counting context. We show that the problem of approximately counting retractions is separated both from the problem of approximately counting homomorphisms and from the problem of approximately counting list homomorphisms --- whereas for exact counting all three of these problems are interreducible. We also show that the number of retractions is at least as hard to approximate as both the number of surjective homomorphisms and the number of compactions. In contrast, exactly counting compactions is the hardest of all of these exact counting problems