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Joined 2026-09-21 Message
Incoming links: Cycle cover of a directed graph
Past exam of the mathematics course of the University of Cambridge 2023 iii Paper 124 4 i Solution 2026-09-28
Interpret the permanent of a matrix as the total weight of the cycle covers of its weighted directed graph. Valiant's reduction builds a graph from three constant-size components.
- A Variable gadget in Valiant's permanent reduction has exactly two relevant cycle-cover states, representing true and false, and exposes occurrence ports consistent with the selected state.
- A Clause gadget in Valiant's permanent reduction has external ports for its three literals and contributes a fixed total weight exactly when at least one selected literal satisfies the clause.
- An Exclusive-or gadget in Valiant's permanent reduction joins occurrence ports while allowing exactly one of two corresponding external edges. Its edges have weights in ; unwanted cycle covers occur in sign-reversing pairs and cancel.
Wire one variable port to every literal occurrence and one clause port to each literal. The balanced-occurrence hypothesis lets the true and false tracks of each variable be paired without extra weighting. A routine gadget case analysis shows that cycle covers surviving cancellation correspond to satisfying assignments, each with the same fixed multiplicity and sign. A small normalization gadget removes that fixed factor, or it can be tracked explicitly. The construction has constant size per variable, clause, and occurrence, so it is a polynomial-time counting reduction from Balanced number 3-SAT to the permanent of a matrix.
Valiant's permanent reduction 2026-09-28
Valiant's permanent reduction encodes satisfying assignments as cycle covers so that their weighted sum is the permanent of a matrix. Local gadgets enforce variable consistency and clause satisfaction, while signed contributions cancel invalid covers.