# A polynomial-time randomized reduction from tournament isomorphism to tournament asymmetry

## TL;DR

This paper introduces a new randomized polynomial-time reduction from tournament isomorphism to tournament automorphism, advancing understanding of the computational complexity of these problems.

## Contribution

It presents the first randomized polynomial-time reduction from tournament isomorphism to automorphism, utilizing a novel technique for extracting characteristic subsets from random samples.

## Key findings

- First such reduction for a combinatorial object without a known polynomial-time isomorphism solution.
- Develops a new technique for characteristic subset extraction from random sources.
- Advances the complexity theory of tournament isomorphism and automorphism problems.

## Abstract

The paper develops a new technique to extract a characteristic subset from a random source that repeatedly samples from a set of elements. Here a characteristic subset is a set that when containing an element contains all elements that have the same probability. With this technique at hand the paper looks at the special case of the tournament isomorphism problem that stands in the way towards a polynomial-time algorithm for the graph isomorphism problem. Noting that there is a reduction from the automorphism (asymmetry) problem to the isomorphism problem, a reduction in the other direction is nevertheless not known and remains a thorny open problem. Applying the new technique, we develop a randomized polynomial-time Turing-reduction from the tournament isomorphism problem to the tournament automorphism problem. This is the first such reduction for any kind of combinatorial object not known to have a polynomial-time solvable isomorphism problem.