On the complexity of Sandwich Problems for $M$-partitions

We present a structural classification of constraint satisfaction problems (CSP) described by reflexive complete $2$-edge-coloured graphs. In particular, this classification extends the structural dichotomy for graph homomorphism problems known as the Hell–Nešetřil theorem (1990). Our classification is also efficient: we can check in polynomial time whether the CSP of a reflexive complete $2$-edge-coloured graph is in P or NP-complete, whereas for arbitrary $2$-edge-coloured graphs, this task is NP-complete. We then apply our main result in the context of matrix partition problems and sandwich problems. Firstly, we obtain one of the few algorithmic solutions to general classes of matrix partition problems. And secondly, we present a P vs. NP-complete classification of sandwich problems for matrix partitions.

💡 Research Summary

The paper tackles the long‑standing challenge of classifying the computational complexity of matrix‑partition problems and their sandwich variants by exploiting a deep connection with constraint satisfaction problems (CSPs) defined on reflexive complete 2‑edge‑coloured graphs. A matrix M with entries in {0,1,*} is encoded as a 2‑coloured graph: 0‑entries become red edges, 1‑entries become blue edges, and *‑entries become both colours. Likewise, any input graph G is transformed into a reflexive complete 2‑coloured graph ν(G) where actual edges are blue and non‑edges are red. Under this encoding, G admits an M‑partition if and only if ν(G) admits a homomorphism to the graph M. Consequently, the M‑partition problem is exactly the homomorphism problem CSP(M) for a reflexive complete 2‑coloured target.

The central theoretical contribution is a structural dichotomy for CSPs whose target is a reflexive complete 2‑edge‑coloured graph H. While for arbitrary 2‑coloured graphs deciding tractability is NP‑complete, the authors show that for the reflexive complete subclass one can decide in polynomial time whether CSP(H) is in P or NP‑complete. The dichotomy hinges on two complementary characterisations:

1. Homogeneous concatenations – The authors define homogeneous sets in 2‑coloured graphs (generalising modular decomposition) and show that if H can be built from a 2‑element seed by repeatedly adding homogeneous sets of size two, then CSP(H) (and its list version) is solvable in polynomial time, indeed in Datalog. This yields the tractable side of the dichotomy (Theorem 16).

2. Hereditary pp‑construction of K₃ – They identify a finite family F of small 2‑coloured graphs such that any H containing a (not necessarily induced) copy of a member of F pp‑constructs the triangle K₃. Since CSP(K₃) is NP‑complete, any such H yields an NP‑complete CSP. The detection of these “forbidden” substructures is performed via algebraic tools: the Siggers power Sig(H) and the p‑cyclic power Cycₚ(H). If Sig(H) satisfies certain identities, H necessarily pp‑constructs K₃; conversely, if H does not pp‑construct K₃, then for every prime p larger than |H| there exists a homomorphism Cycₚ(H)→H.

A further combinatorial notion, alternating components, partitions the vertex set into red‑components (connected via red edges among blue vertices) and blue‑components. The authors prove that if H does not pp‑construct K₃, each alternating component has size at most four (Proposition 39). This bound almost matches the sufficient condition from homogeneous concatenations. The final gap is closed by showing that any H satisfying the size‑four bound but not admitting a homogeneous concatenation fails to admit a homomorphism from Cycₚ(H) for some large prime p, which forces NP‑completeness. Thus Theorem 46 delivers a clean structural dichotomy: H is tractable iff it can be assembled by homogeneous concatenations; otherwise it is NP‑complete. Corollary 47 guarantees that this classification can be decided in polynomial time.

Armed with this dichotomy, the paper resolves two concrete problems:

• Problem 1 (Matrix classification) – For any matrix M, one can efficiently decide whether CSP(M) (hence the M‑partition problem) is polynomial‑time solvable or NP‑complete by checking the structural condition on the associated 2‑coloured graph M.

• Problem 2 (Sandwich problems for matrix partitions) – The sandwich problem asks, given two edge sets E₁⊆E₂ on the same vertex set, whether there exists an intermediate graph whose edge set lies between them and which admits an M‑partition (with or without vertex lists). By translating the intermediate graph into a reflexive complete 2‑coloured graph, the sandwich problem becomes exactly CSP(H) for the same target H=M. Consequently, the same structural dichotomy yields a full P vs. NP‑complete classification for both the ordinary and list sandwich versions. Notably, classic cases such as split‑graph sandwich (polynomial) and stubborn‑partition sandwich (NP‑complete) fall naturally into this framework.

The paper also situates its contribution within the broader CSP landscape. While the Hell–Nešetřil theorem provides a structural dichotomy for uncoloured graph homomorphisms, and algebraic dichotomies exist for general finite‑domain CSPs, those are not efficiently testable. Here, by restricting to reflexive complete 2‑edge‑coloured graphs, the authors achieve both a clean structural classification and an efficient decision procedure, a rare combination in finite‑domain CSP theory. They further discuss connections to smooth digraphs, signed graphs, and list‑homomorphism problems, emphasizing that their result adds a new, practically useful class to the very short list of CSP families with such tractable characterisations.

In summary, the work delivers a novel, efficiently testable structural dichotomy for CSPs on reflexive complete 2‑edge‑coloured graphs, and leverages it to obtain the first broad, algorithmic classification of matrix‑partition problems and their sandwich variants. This bridges algebraic CSP theory, graph homomorphism techniques, and combinatorial matrix‑partition research, opening avenues for extending similar classifications to larger classes of coloured graphs or to other combinatorial optimisation problems.