Heuristic — constructive

Fourier-basis Constructive Solver


Alias	`fourier`
Type	Heuristic — constructive
Complexity	O(K_max · epochs · n · M) per run

Description

Encodes a TSP tour as a closed curve in the complex plane and optimises the Fourier coefficients of that curve with gradient descent. Decoding is a pure argsort: no penalty, no repair step, and no possibility of producing an invalid tour.

Curve representation

The tour is the closed curve

γ(s) = Σ_{k=-K}^{K} c_k · exp(2πi · k · s),   s ∈ [0, 1)

sampled at M evenly-spaced points s_j = j/M. The 2K+1 complex coefficients c_k are the only free variables; gradient descent moves them so the curve passes near each city.

Energy

E(c) = Σ_i  min_j |city_i − γ(s_j)|²   +   λ Σ_k (2πk)² |c_k|²
         attraction                              tension

Attraction: each city pulls its nearest curve sample toward it.
Tension: a smoothness regulariser that weights high-frequency modes proportionally to their squared Fourier frequency, preventing jagged curves that visit no city cleanly.

Coarse-to-fine optimisation loop

initialise c[0] = centroid, c[1] = radius/2, all others = 0
λ ← opts.lambda

for k_active = 1 … K_max:
    basis[k][j] = exp(2πi · ks[k] · j/M)   // pre-computed; constant within stage

    repeat opts.epochs times:
        γ = eval_curve(c, ks, M)
        grad = attraction_gradient + tension_gradient
        for k where |ks[k]| ≤ k_active:
            c[k] -= (lr / n) * grad[k]

    λ *= lambda_decay

Unlocking modes one stage at a time lets the optimiser set the overall loop shape (low modes) before refining local detail (high modes), avoiding the saddle points that occur when all modes compete simultaneously.

Decode (always valid)

s_i = argmin_j |city_i − γ(s_j)|     // nearest curve sample per city
tour = argsort(s_i)                    // sort cities by their curve position

The argsort gives array positions in cities[], which are then mapped to their .id fields — the same pattern used by Christofides. This guarantees a valid Hamiltonian tour regardless of convergence quality.

Options

Field	Default	Range	Description
`k_max`	4	≥ 1	Maximum Fourier mode (number of frequency stages)
`m`	200	≥ 2	Curve sampling resolution (points on γ)
`lambda`	0.05	> 0	Initial tension weight
`lambda_decay`	0.5	(0, 1)	Tension multiplier applied at each stage
`lr`	0.05	> 0	Gradient descent learning rate
`epochs`	400	≥ 1	Gradient steps per k_active stage

epochs follows the same vocabulary as all other solvers in this codebase.

Usage

# standalone
teeline solve fourier -i ./data/tsplib/berlin52.tsp

# as warm-start for 2-opt (recommended)
teeline pipeline --steps=fourier,2opt -i ./data/tsplib/berlin52.tsp

# as warm-start for LK
teeline pipeline --steps=fourier,lk -i ./data/tsplib/berlin52.tsp

Notes

init_tour is ignored: as a purely constructive solver, Fourier does not accept or benefit from a seed tour; the --init-tour pipeline option has no effect on this solver.
f64 internally: all gradient computation uses f64 for numerical stability; coordinates are converted from f32 at input and the final tour is Vec<usize> city IDs.
WASM-compatible: uses num-complex = "0.4" (no_std-compatible, no libm dependency).

Relationship to the Elastic Net

This algorithm is a Fourier-parameterised variant of the Elastic Net (Durbin & Willshaw 1987). Both share the same two-term energy (city attraction + curve smoothness/tension) and optimise via gradient descent. The key differences:

	Elastic Net	This implementation
Curve representation	M explicit node positions	2K+1 Fourier coefficients
Tension term	Sum of squared edge lengths	`λ(2πk)²
Mode schedule	Simultaneous + temperature annealing	Coarse-to-fine frequency unlocking
Tour decode	Explicit node ordering	Argsort of nearest-sample parameter `s_i`

References

Durbin, R. & Willshaw, D. (1987) — "An analogue approach to the travelling salesman problem using an elastic net method", Nature 326, 689–691. DOI: 10.1038/326689a0 (original Elastic Net; this solver is a Fourier-parameterised variant)