Computational complexity and the classification of finite simple groups

Colored Hypergraph Isomorphism is Fixed Parameter Tractable

“…We adapt ideas from [5,16] which applies the halving technique in combination with dynamic programming. Luks [16] gives a 2 O(n) time algorithm for Hypergraph Isomorphism.…”

“…This latter condition is easy to enforce. However, since the above reduction blows up the size of the vertex set in the bipartite encoding, the Zemlyachenko-Luks-Babai graph isomorphism algorithm [3,5,6,25] that runs in time c √ n log n , where n is the size of the vertex set of the graph, does not yield a similar algorithm for hypergraph isomorphism. We note here that the best known hypergraph isomorphism test due to Luks [16] has running time c n .…”

“…)n O (1) , where the parameter b is the maximum size of the color classes and n is the number of vertices of the input graphs. By using the halving technique as introduced in [5] (see also [16]), the running time can be improved to…”

Computing the composition factors of a permutation group in polynomial time

“…For( 2), one finds anynon-trivial block system (for example, fixing x find, for all y,the unique smallest block containing x and y by taking the component of x in the undirected graph whose edges are givenbythe G-orbit of {x, y}i n the set of all unordered pairs); Akinson [2] first observed this to be in polynomial time. Repeat, considering the action of G on the block system, until G acts primitively.P roblems (3),(4), (5) were shown to be in polynomial time in [8] using, in effect, a version of Sims' algorithm [15]. Problem (6) follows directly from (4); for H < | G,itsuffices to test for membership in H the conjugates of the generators of H by the generators of G.A sobserved in [8], Problem (7) is an easy extension of (6) for,when one of the conjugates fails the membership test, increase H by adding this element to the generating set.…”

“…Completing a cycle, there are applications also in the graph isomorphism problem ( [5], [14]). The main result of this paper was announced in [13, section 4] and a sketch of the algorithm appeared in [5].…”