Exact Cover with light

TL;DR

This paper proposes an optical device leveraging light's parallelism to efficiently determine the existence of an exact cover in a set, by encoding solutions as light paths with delays.

Contribution

It introduces a novel optical approach that generates all solutions simultaneously and identifies exact covers through timing analysis of light signals.

Findings

Successfully models the Exact Cover problem with optical paths.

Enables parallel solution generation using light delays.

Provides a method to detect exact covers via timing at the destination.

Abstract

We suggest a new optical solution for solving the YES/NO version of the Exact Cover problem by using the massive parallelism of light. The idea is to build an optical device which can generate all possible solutions of the problem and then to pick the correct one. In our case the device has a graph-like representation and the light is traversing it by following the routes given by the connections between nodes. The nodes are connected by arcs in a special way which lets us to generate all possible covers (exact or not) of the given set. For selecting the correct solution we assign to each item, from the set to be covered, a special integer number. These numbers will actually represent delays induced to light when it passes through arcs. The solution is represented as a subray arriving at a certain moment in the destination node. This will tell us if an exact cover does exist or not.