A Dichotomy for Maximum PCSPs on Graphs

Fix two non-empty loopless graphs $G$ and $H$ such that $G$ maps homomorphically to $H$. The Maximum Promise Constraint Satisfaction Problem parameterised by $G$ and $H$ is the following computational problem, denoted by MaxPCSP($G$, $H$): Given an input (multi)graph $X$ that admits a map to $G$ preserving a $ρ$-fraction of the edges, find a map from $X$ to $H$ that preserves a $ρ$-fraction of the edges. As our main result, we give a complete classification of this problem under Khot’s Unique Games Conjecture: The only tractable cases are when $G$ is bipartite and $H$ contains a triangle. Along the way, we establish several results, including an efficient approximation algorithm for the following problem: Given a (multi)graph $X$ which contains a bipartite subgraph with $ρ$ edges, what is the largest triangle-free subgraph of $X$ that can be found efficiently? We present an SDP-based algorithm that finds one with at least $0.8823 ρ$ edges, thus improving on the subgraph with $0.878 ρ$ edges obtained by the classic Max-Cut algorithm of Goemans and Williamson.

💡 Research Summary

The paper studies the Maximum Promise Constraint Satisfaction Problem (MaxPCSP) for undirected graphs. Given two fixed loop‑less graphs G and H with a homomorphism G → H, an instance is a multigraph X together with the promise that X admits a partial homomorphism to G preserving a ρ‑fraction of its edges. The algorithmic task is to find a map from X to H that also preserves at least a ρ‑fraction of the edges. This “promise” version interpolates between exact CSP (all constraints satisfied) and Max‑CSP (maximising satisfied constraints).

The main contribution is a dichotomy theorem (Theorem 1) proved under the Unique Games Conjecture (UGC). It states that MaxPCSP(G,H) is 1‑approximable (i.e., a perfect preservation of the promised fraction is achievable in polynomial time) if and only if G is bipartite and H contains a triangle (K₃). In all other cases, even achieving a 1‑approximation is NP‑hard assuming UGC. This result refines the earlier conjecture of Brakensiek and Guruswami, which suggested that any bipartite G yields a tractable PCSP; the paper shows that the presence of a triangle in H is also required for 1‑approximation.

To establish the tractable side, the authors design two SDP‑based algorithms. The first (Theorem 4) deals with MaxPCSP(K₂, 𝔊₃), where 𝔊₃ is the disjoint union of all finite triangle‑free graphs. They solve the standard Max‑Cut SDP, then apply a novel randomized rounding that mixes two strategies: (i) select only “long edges” (edges whose SDP vectors have an angle larger than a threshold) and cut them with a random hyperplane; (ii) select a set of “shorter edges” (still above a second threshold) and cut them similarly. The mixing probability is chosen based on geometric quantities of the SDP solution. By exploiting the fact that any three vectors with pairwise angles exceeding 2π/3 cannot form a triangle (Lemma 2), the algorithm guarantees a triangle‑free subgraph containing at least 0.8823 ρ edges, improving on the classic Goemans‑Williamson 0.878 ρ bound for Max‑Cut.

The second tractable case (Theorem 5) shows that MaxPCSP(K₂, K₃) is 1‑approximable. The algorithm combines the Goemans‑Williamson SDP for Max‑Cut with the Frieze‑Jerrum rounding for Max‑3‑Cut, using three independent Gaussian vectors instead of a single random hyperplane. This yields a 3‑coloring that preserves the same fraction of edges as the promised 2‑coloring.

For hardness, the paper reduces from Håstad’s 3‑bit PCP and the UGC‑hardness framework of Khot, Kindler, Mossel, O’Donnell, and Oleszkiewicz. When G is non‑bipartite or H lacks a triangle, the reduction shows that distinguishing instances with a ρ‑preserving G‑map from those without an ⌈α ρ⌉‑preserving H‑map is NP‑hard for any α < 1, establishing the impossibility of 1‑approximation. Moreover, for MaxPCSP(K₂, 𝔊₃) they prove that achieving a (25/26 + ε)‑approximation is NP‑hard, again via a PCP‑based construction.

Overall, the paper delivers a clean structural dichotomy for graph MaxPCSPs under UGC, introduces a stronger SDP rounding technique that beats the long‑standing Goemans‑Williamson ratio for triangle‑free subgraphs, and clarifies the precise role of bipartiteness of G and the presence of a triangle in H in determining approximability. These results advance both the theory of promise CSPs and the practice of designing high‑quality approximation algorithms for graph problems.