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.

💡 Research Summary

The surveyed paper provides a comprehensive overview of the computational complexity landscape surrounding surjective homomorphism problems, positioning them alongside three closely related families: list homomorphism, retraction, and compaction. A surjective homomorphism from an input structure G to a target structure H is a mapping that preserves relational edges while guaranteeing that every vertex of H appears as the image of some vertex of G. This global “onto” requirement distinguishes the problem from the classic homomorphism and constraint satisfaction problems (CSPs), where only local constraints are enforced.

The authors first examine the relationship with list homomorphism. In list homomorphism each vertex of G is equipped with a list of admissible images in H; the decision problem asks whether a homomorphism respecting these lists exists. While a reduction from list homomorphism to surjective homomorphism is straightforward, the converse is not: the surjectivity condition imposes a counting constraint that cannot be captured by local lists alone. Consequently, instances that are tractable for list homomorphism may become NP‑complete when the surjectivity requirement is added.

Next, the paper compares surjective homomorphism to retraction. A retraction is a homomorphism from G onto a substructure H that acts as the identity on H. Surjective homomorphism relaxes the substructure requirement—H need not be a subgraph of G—yet still demands that the image cover all of H. Thus retraction can be seen as a special case of surjective homomorphism where the target is embedded in the source. However, the added global constraint often pushes the problem out of the polynomial‑time tractable zone identified for many retraction instances.

The third family, compaction, asks for a homomorphism whose image is “dense” in H but not necessarily equal to H. The authors point out that even when compaction admits polynomial‑time algorithms (for example on trees or bounded‑degree graphs), the corresponding surjective version can be NP‑complete, illustrating that the requirement of exact coverage fundamentally changes the complexity profile.

A central contribution of the survey is the identification of three concrete open problems that have repeatedly appeared in the literature. The first concerns surjective homomorphisms on bipartite graphs; this case is known to be polynomial‑time solvable, providing a rare positive result. The second problem asks for the complexity of surjective homomorphisms on a restricted class of directed graphs (e.g., orientations of cycles with bounded indegree). The third problem concerns structures of bounded domain size (for instance, three‑element algebras) and asks whether the surjective homomorphism decision problem is in P or NP‑complete. For both the second and third cases, no definitive classification is known, and they constitute boundary instances that resist the standard CSP dichotomy techniques.

The authors systematically apply the algebraic toolkit that has driven recent breakthroughs in CSP classification—core reductions, polymorphism analysis, and the use of Maltsev conditions—to these surjective settings. They find that many of the polymorphisms that guarantee tractability for ordinary CSPs (e.g., majority, Maltsev, or near‑unanimity operations) fail to preserve surjectivity, rendering the existing dichotomy theorems inapplicable. This observation underscores a deeper theoretical obstacle: surjectivity introduces a global counting dimension that does not decompose cleanly under the local algebraic invariants traditionally used in CSP theory.

In the concluding sections, the paper outlines several promising research directions. First, it proposes the development of a new algebraic notion—“surjectivity‑preserving polymorphisms”—that could capture the essential global constraint and potentially lead to a refined dichotomy. Second, it advocates for a parameterized complexity analysis, investigating whether structural parameters such as treewidth, feedback vertex set size, or clique‑width render the problem fixed‑parameter tractable. Third, the authors suggest exploring approximation algorithms and hardness of approximation results, since exact decision may be intractable while good approximations could still be feasible.

Overall, the survey positions surjective homomorphism problems as a frontier in the study of CSP‑like decision problems. By collating known results, highlighting persistent open cases, and diagnosing why existing algebraic methods fall short, the paper provides a clear roadmap for future work aimed at extending the powerful CSP classification machinery to accommodate global surjectivity constraints.