Geometry Of The Subset Sum Problem -- Part I

TL;DR

The paper presents a geometric framework for the Subset Sum Problem by embedding subset sums as a universal family of non-decreasing paths in the plane and introducing complementary curve pairs $p_k$ and $q_k$ alongside transformational machinery. It constructs a layered orbital graph $G_0$ and an overarching transformation graph $\mathcal{H}$ to generate all NDPs and shows that all solutions correspond to zero paths or intersections with an orbital line, enabling a polynomial-time algorithm (IHM) that refines and filters a solution graph toward a decision. The work separates combinatorial and relational components of SSP, analyzes the interaction structure via segments, edge interactions, and configuration graphs, and derives bounds on the number of intersection events and path multiplicities that depend on $n$, $m$, and additive structure. If the claims hold, the method yields a universal, geometry-driven approach with provable polynomial-time behavior in the worst-case, ties to counting (FP vs #P), and even implications about quantum-classical complexity containment, while providing a rich geometric picture of SSP's solution space. The practical impact lies in a novel lens for SSP analysis, potential algorithmic pathways for special instance classes, and a framework that links combinatorial, geometric, and complexity-theoretic aspects of subset-sum-like problems.

Abstract

We announce two breakthrough results concerning important questions in the Theory of Computational Complexity. In this expository paper, a systematic and comprehensive geometric characterization of the Subset Sum Problem is presented. We show the existence of a universal geometric structure, comprised of a family of non-decreasing paths in the Cartesian plane, that captures any instance of the problem of size $n$. Inspired by the geometric structure, we provide an unconditional, deterministic and polynomial time algorithm, albeit with fairly high complexity, thereby showing that $\mathcal{P} = \mathcal{NP}$. Furthermore, our algorithm also outputs the number of solutions to the problem in polynomial time, thus leading to $\mathcal{FP} = \mathcal{\#P}$. As a bonus, one important consequence of our results, out of many, is that the quantum-polynomial class $\mathcal{BQP} \subseteq \mathcal{P}$. Not only this, but we show that when multiple solutions exist, they can be placed in certain equivalence classes based on geometric attributes, and be compactly represented by a polynomial sized directed acyclic graph. We show that the Subset Sum Problem has two aspects, namely a combinatorial aspect and a relational aspect, and that it is the latter which is the primary determiner of complexity. We reveal a surprising connection between the size of the elements and their number, and the precise way in which they affect the complexity. In particular, we show that for all instances of the Subset Sum Problem, the complexity is independent of the size of elements, once the difference between consecutive elements exceeds $\lceil{7\log{}n}\rceil$ bits in size. We provide some numerical examples to illustrate the algorithm, and also show how it can be used to estimate some difficult combinatorial quantities such as the number of restricted partitions.

Figures (23)

• Figure 1: Point Reflection Symmetry for complementary curves $p_6$ (blue) and $q_6$ (cyan) of the main example. The symmetric pairs of points are connected by a dotted line.
• Figure 2: Illustration of coverage of subspaces $S_1,S_2,\ldots,S_9$, by complementary curves $p_k$ and $q_k$. In each case the $p_k$ curve is in blue and the $q_k$ curve in cyan. In $S_1$ and $S_2$ the complementary curves are identical and overlap. In $S_3$, the curves $p_3$ and $q_3$ are distinct in one point, and together they cover the space. In $S_4$, the curves $p_4$ and $q_4$ are distinct in more points, and together they cover the space. In $S_5$, the curves $p_5$ and $q_5$ are distinct in several points but do not cover the space. There are several isolated points. This pattern continues for higher subspaces.
• Figure 3: Illustration of the application of $\{\tau\}_{k=4}^9$ on the $p_9$ curve of main example. The $p_9$ curve is shown in blue.
• Figure 4: Illustration of the application of $\{\tau\}_{k=4}^9$ on the $q_9$ curve of main example. The $q_9$ curve is shown in cyan.
• Figure 5: Illustration of the application of two $\tau$ operations in sequence on the $q_9$ curve of main example.

Theorems & Definitions (47)

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