[論文レビュー] Information-Based Complexity vs Computational Complexity in Phaseless Polynomial Interpolation

Michał R. Przybyłek, Paweł Siedlecki|arXiv (Cornell University)|Jan 4, 2026

Polynomial and algebraic computation被引用数 0

ひとこと要約

概要: The paper characterizes computational complexity of phaseless polynomial interpolation: reconstruction from 2n−k points is polynomial-time for fixed k, while from (1+c)n+2 points (0≤c<1) is NP-complete; it also shows that a single adaptive query suffices with fewer than 2n+1 points and that evaluation points with unique solutions can be chosen in polynomial time.

ABSTRACT

The authors of ``A note on the complexity of a phaseless polynomial interpolation'' have shown that phaseless polynomial interpolation over $\mathbf{Q}$ is possible with $n+2$ points, where $n$ is the upper-bound on the degree of a polynomial. Nonetheless, their reconstruction algorithm and the method of adaptively choosing evaluation points are exponential time. On the other hand, they have also shown that given $2n+1$ points, the polynomial can be reconstructed in a polynomial time. A conjecture have been put forward, namely that the reconstruction problem from such $n+2$ points is exponential time. Moreover, a question about the number of points sufficient for polynomial time reconstruction have been posed. In this paper, we answer these questions -- we show that (1) reconstruction problem from $2n-k$ for any constant $k$ is polynomial time, (2) reconstruction problem from $(1+c)n+2$ points for any constant $c \in [0, 1)$ is NP-Complete, (3) evaluation points admitting a unique solution can be chosen in polynomial time.

研究の動機と目的

• Clarify the information-theoretic vs computational complexity gap in phaseless polynomial interpolation over Q.
• Determine the exact point threshold where polynomial-time reconstruction becomes possible or NP-hard.
• Develop polynomial-time methods for reconstruction with a fixed small deficiency (k) in the number of points.
• Analyze the role of adaptive query strategies in reducing the required number of evaluation points.

提案手法

• Parameterize the affine space of degree ≤2n 多項式 satisfying the given |p(x_i)| constraints.
• Impose the algebraic condition that the solution must be a perfect square to obtain a polynomial system in k+1 variables.
• Solve the resulting polynomial system using Gröbner bases and LLL-reductions for fixed k.
• Prove NP-completeness via a reduction from Partition for (1+c)n+2 points.
• Show that for fixed k the number of solutions is polynomially bounded (O(n^k)).
• Demonstrate that 2n−k points yield a polynomial-time reconstruction algorithm.

Figure 2 : Intersections of $|p(x)|=|2x^{3}-21x^{2}+85x-126|$ and $|q(x)|=|9x^{2}-63x+114|$ .

実験結果

リサーチクエスチョン

• RQ1 What is the minimal number of phaseless evaluation points required for unique reconstruction (up to a phase) over Q?
• RQ2 Can reconstruction be performed in polynomial time when the number of points is below 2n+1, specifically for m=2n−k with fixed k?
• RQ3 Is reconstruction from (1+c)n+2 points NP-complete for any c in [0,1)?
• RQ4 Can adaptive selection of evaluation points reduce the number of required measurements while preserving tractability?
• RQ5 How does the information-based complexity framework relate to computational complexity in phaseless polynomial interpolation?

主な発見

• Reconstruction from m=2n−k points is solvable in polynomial time for any fixed constant k.
• Reconstruction from (1+c)n+2 points for any constant c∈[0,1) is NP-complete, including the n+2 point case.
• There are only polynomially-many (O(n^k)) solutions when m=2n−k, for fixed k, and all can be found in polynomial time.
• Adaptive strategies allow solving the problem with fewer than 2n+1 points, and a single adaptive query can suffice under certain conditions.
• Polynomial-time reconstruction is achieved by parameterizing an affine space of candidate polynomials and solving a system enforcing the perfect-square constraint via Gröbner bases with LLL reductions.
• The problem is framed within Information-Based Complexity, linking basic information operators with computational models.

Figure 3 : Rational and all real roots for polynomials $a_{4},a_{5},a_{6}$ from Example 2 .