A Polynomial-Time Randomized Reduction from Tournament Isomorphism to Tournament Asymmetry∗ Pascal Schweitzer RWTH Aachen University, Aachen, Germany
[email protected] 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. 1998 ACM Subject Classification F.2.2 Computations on Discrete Structures, F.1.3 Reducibility and Completeness Keywords and phrases graph isomorphism, asymmetry, tournaments, randomized reductions Digital Object Identifier 10.4230/LIPIcs.ICALP.2017.66 1 Introduction The graph automorphism problem asks whether a given input graph has a non-trivial automorphism. In other words the task is to decide whether a given graph is asymmetric. This computational problem is typically seen in the context of the graph isomorphism problem, which is itself equivalent under polynomial-time Turing reductions to the problem of computing a generating set for all automorphisms of a graph [17].