1-in-3 vs. Not-All-Equal: Dichotomy of a broken promise

TL;DR

This work classifies the tractability frontier for the promise problem $PCSP(1-in-3, NAE)$ under rainbow-free symmetry when weakening the promise from either the left or the right. By encoding inputs via the digraph-based template $\widehat{\mathbf{G}}$ and leveraging polymorphism minions, it establishes two complementary dichotomies: breaking the promise from the right yields polynomial-time solvability exactly when the underlying digraph $\mathbf{G}$ contains a short directed cycle (length at most $3$), while breaking from the left yields tractability exactly when $\mathbf{G}$ is balanced; otherwise both directions are hardness results. A key technical contribution shows that $PCSP(\widehat{D}_{2k}, NAE)$ is -hard for all $k\ge1$, and that appropriate reductions transfer hardness from cycle templates to general bipartite graphs with the same net-length cycles. The results advance the understanding of PCSPs by contrasting local versus global structural properties (short cycles vs balancedness) that govern tractability, and they hint at broader classifications beyond the Boolean rainbow-free regime.

Abstract

The 1-in-3 and Not-All-Equal satisfiability problems for Boolean CNF formulas are two well-known NP-hard problems. In contrast, the promise 1-in-3 vs. Not-All-Equal problem can be solved in polynomial time. In the present work, we investigate this constraint satisfaction problem in a regime where the promise is weakened from either side by a rainbow-free structure, and establish a complexity dichotomy for the resulting class of computational problems.

Figures (1)

• Figure 1: Patterns from the proof of Proposition \ref{['thm:cycleHard']}.

Theorems & Definitions (48)

• Theorem 1
• Theorem 2
• Theorem 3: BWZ21
• Theorem 4
• Theorem 5: notation_old
• Lemma 6
• Theorem 7: BG21:sicomp
• Theorem 7
• Proposition 7
• proof