Eugene M. Luks

Canonical labeling of graphs

We announce an algebraic approach to the problem of assigning <italic>canonical forms</italic> to graphs. We compute canonical forms and the associated canonical labelings (or renumberings) in polynomial time for graphs of bounded valence, in moderately exponential, exp(n<supscrpt>½ + &ogr;(1)</supscrpt>),time for general graphs, in subexponential, n<supscrpt>log n</supscrpt>, time for tournaments and for 2-(&ngr;,&kgr;,λ) block designs with &kgr;,λ bounded and n<supscrpt>log log n</supscrpt> time for λ-planes (symmetric designs) with λ bounded. We prove some related problems NP-hard and indicate some open problems.

Polynomial-time algorithms for permutation groups

A permutation group on n letters may always be represented by a small set of generators, even though its size may be exponential in n. We show that it is practical to use such a representation since many problems such as membership testing, equality testing, and inclusion testing are decidable in polynomial time. In addition, we demonstrate that the normal closure of a subgroup can be computed in polynomial time, and that this proceaure can be used to test a group for solvability. We also describe an approach to computing the intersection of two groups. The procedures and techniques have wide applicability and have recently been used to improve many graph isomorphism algorithms.

Permutation groups in NC

We show that the basic problems of permutation group manipulation admit efficient parallel solutions. Given a permutation group G by a list of generators, we find a set of NC-efficient strong generators in NC. Using this, we show, that the following problems are in NC: membership in G; determining the order of G; finding the center of G; finding a composition series of G along with permutation representations of each composition factor. Moreover, given G, we are able to find the pointwise stabilizer of a set in NC. One consequence is that isomorphism of graphs with bounded multiplicity of eigenvalues is in NC. The analysis of the algorithms depends, in several ways, on consequences of the classification of finite simple groups.

Computational complexity and the classification of finite simple groups

We address the graph isomorphism problem and related fundamental complexity problems of computational group theory. The main results are these: A1. A polynomial time algorithm to test simplicity and find composition factors of a given permutation group (COMP). A2. A polynomial time algorithm to find elements of given prime order p in a permutation group of order divisible by p. A3. A polynomial time reduction of the problem of finding Sylow subgroups of permutation groups (SYLFIND) to finding the intersection of two cosets of permutation groups (INT). As a consequence, one can find Sylow subgroups of solvable groups and of groups with bounded nonabelian composition factors in polynomial time. A4. A polynomial time algorithm to solve SYLFIND for finite simple groups. A5. An ncd/log d algorithm for isomorphism (ISO) of graphs of valency less than d and a consequent improved moderately exponential general graph isomorphism test in exp(c√n log n) steps. A6. A moderately exponential, n,c√n algorithm for INT. Combined with A3, we obtain an nc√n algorithm for SYLFIND as well. All these problems have strong links to each other. ISO easily reduces to INT. A subcase of SYLFIND was solved in polynomial time and applied to bounded valence ISO in [Lul]. Now, SYLFIND is reduced to INT. Interesting special cases of SYLFIND belong to NP ∩ coNP and are not known to have subexponential solutions. All the results stated depend on the classification of finite simple groups. We note that no previous ISO test had no(d) worst case behavior for graphs of valency less than d. It appears that unless there is another radical breakthrough in ISO, independent of the previous one, the simple groups classification is an indispensable tool for further developments.

Computing in solvable matrix groups

The author announces methods for efficient management of solvable matrix groups over finite fields. He shows that solvability and nilpotence can be tested in polynomial-time. Such efficiency seems unlikely for membership-testing, which subsumes the discrete-log problem. However, assuming that the primes in mod G mod (other than the field characteristic) are polynomially-bounded, membership-testing and many other computational problems are in polynomial time. These problems include finding stabilizers of vectors and of subspaces and finding centralizers and intersections of subgroups. An application to solvable permutation groups puts the problem of finding normalizers of subgroups into polynomial time. Some of the results carry over directly to finite matrix groups over algebraic number fields; thus, testing solvability is in polynomial time, as is testing membership and finding Sylow subgroups.<<ETX>>

Computing the composition factors of a permutation group in polynomial time

Given generators for a group of permutations, it is shown that generators for the subgroups in a composition series can be found in polynomial time. The procedure also yields permutation representations of the composition factors.

Parallel algorithms for permutation groups and graph isomorphism

We develop parallel techniques for dealing with permutation group problems. These are most effective on the class of groups with bounded non-abelian composition factors. For this class, we place in NC problems such as membership testing, finding the center and composition factors, and, of particular significance, finding pointwise-set-stabilisers. The last has applications to instances of graph-isomorphism and we show that NC contains isomorphism-testing for vertex-colored graphs with bounded color multiplicity, a problem not long known to be in polynomial time.

Hypergraph isomorphism and structural equivalence of Boolean functions

We show that hypergraph isomorphism can be tested in time O(c”), where n is the sire of the vertex set. In general, input of a hypergraph could require n(2”) space, in which case the isomorphism test is in polynomial time. As a consequence, we put into polynomial time the classic problem of testing whether two Boolean functions, given by truth tables, are related via permutations and complementations of the variables, and therefore have structurally identical network realizations. In fact, the method is parallelizable and we put the problem even into NC. We obtain similarly an NC test of equivalence of truth tables under permutation of variables alone.

Fast management of permutation groups

Novel algorithms for computation in permutation groups are presented. They provide an order-of-magnitude improvement in the worst-case analysis of the basic permutation-group problems, including membership testing and computing the order of the group. For deeper questions about the group, including finding composition factors, an improvement of up to four orders of magnitude is realized. These and other essential investigations are all accomplished in O(n/sup 4/log/sup c/n) time. The approach is distinguished by its recognition and use of the intrinsic structure of the group at hand.<<ETX>>

Computing in quotient groups

We present polynomial-time algorithms for computation in quotient groups G/K of a permutation group G. In effect, these solve, for quotient groups, the problems that are known to be in polynomial-time for permutation groups. Since it is not computationally feasible to represent G/K itself as a permutation group, the methodology for the quotient-group versions of such problems frequently differ markedly from the procedures that have been observed for the K = 1 subcases. Whereas the algorithms for permutation groups may have rested on elementary notions, procedures underlying the extension to quotient groups often utilize deep knowledge of the structure of the group. In some instances, we present algorithms for problems that were not previously known to be in polynomial time, even for permutation groups themselves (K = 1). These problems apparently required access to quotients.