[Rule] Partition to RSDF

[QingyunQian]

Source: Partition
Target: RobustSemidefiniteFeasibility (RSDF)
Motivation: Gives a direct NP-hardness reduction from Partition to RSDF
Reference:

• Ben-Tal, A. & Nemirovski, A. (1998). Robust Convex Optimization. (Section 3.4.1: direct Partition-to-RSDF style hardness construction)
• Gurvits, L. (2003). Classical deterministic complexity of Edmonds' problem and Quantum Entanglement.
• Gharibian, S. (2009/2018). Strong NP-Hardness of the Quantum Separability Problem. (formal promise definition of RSDF)

Reduction Algorithm

Notation:

• Source (Partition): integer vector a = (a_1, ..., a_l) in Z^l; question: does there exist z in {-1, 1}^l such that a^T z = 0?
• Target (RSDF): symmetric rational matrices B_1, ..., B_k in Q^(l x l), and rationals zeta, eta >= 0, with

g(B_1,...,B_k) = max_{||x||_2=1} sum_{i=1}^k (x^T B_i x)^2.

Promise decision:

• YES if g >= zeta + eta
• NO if g <= zeta - eta

Direct Mapping (Partition -> RSDF):

Set k = l(l-1)/2 + 1.

1. Hypercube coupling matrices (k-1 matrices):
   For each pair 1 <= p < q <= l, define a symmetric matrix B^{pq} whose quadratic form is

   x^T B^{pq} x = sqrt(2) * x_p * x_q.
   

   (Equivalent rationally-scaled variants are acceptable in implementation; only polynomial-time computability and threshold scaling matter.)

2. Partition constraint matrix (last matrix):
   Define

   B^k = I - (a a^T)/(1 + a^T a).
   

3. Thresholds:
   Let the absolute YES optimum be

   r* = 2 - 1/l.
   

   (Scaling note: the original Ben-Tal/Nemirovski derivation writes the maximization on ||xi||_2^2 = l,
   where the optimum is 2l^2 - l. RSDF here uses the unit sphere ||x||_2 = 1, and since the objective
   is quartic, values scale by 1/l^2, yielding r* = (2l^2 - l)/l^2 = 2 - 1/l.)

   Set zeta, eta so that:

   • zeta + eta = r*
   • zeta - eta is below r* by at least the polynomially bounded separation margin from the construction.

Correctness sketch:

• The first k-1 matrices force the maximizer geometry to align with sign-vector structure.
• The last matrix enforces the partition balance condition via the projector-like term using a.
• Therefore:
  • Partition YES => g >= zeta + eta (YES side),
  • Partition NO => g <= zeta - eta (NO side),
    with a polynomially separated promise gap.

Size Overhead

Target metric (code name)	Polynomial (using symbols above)
num_matrices (k)	l(l-1)/2 + 1
matrix_dim (l)	l
bit_size of entries and thresholds	polynomial in input bit-size of a

Validation Method

• On small instances, brute-force Partition (z in {-1,1}^l) and compare with RSDF decision from the constructed matrices
• Numerically evaluate g(B_1,...,B_k) by dense sphere sampling / local optimization as a sanity check
• Check both YES and NO instances satisfy the promised inequalities at zeta +/- eta

Example

Source Partition instance:

a = [1, -1], l = 2
Question: exists z in {-1,1}^2 with a^T z = 0?
YES: z = [1, 1]

Target RSDF image (direct):

l = 2 => k = 2

B_1 from pair (1,2):
B_1 = [[0, sqrt(1/2)],
       [sqrt(1/2), 0]]
so x^T B_1 x = sqrt(2) x_1 x_2.

a^T a = 2, aa^T = [[1, -1],[-1, 1]]
B_2 = I - aa^T/(1 + a^T a)
    = I - (1/3)[[1, -1],[-1, 1]]
    = [[2/3, 1/3],[1/3, 2/3]]

Unit vector x = [1/sqrt(2), 1/sqrt(2)] gives:
(x^T B_1 x)^2 = 0.5
(x^T B_2 x)^2 = 1.0
Max g = 1.5

r* = 2 - 1/l = 1.5
set zeta + eta = 1.5 (and zeta - eta by the construction margin)

This is the direct matrix construction route Partition -> RSDF.