Geometry Of The Subset Sum Problem -- Part I

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.