The complexity of finding minimum-length generator sequences
Theoretical Computer Science
On the diameter of Cayley graphs of the symmetric group
Journal of Combinatorial Theory Series A
Local expansion of vertex-transitive graphs and random generation in finite groups
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
On the diameter of permutation groups
European Journal of Combinatorics
The hardness of approximate optima in lattices, codes, and systems of linear equations
Journal of Computer and System Sciences - Special issue: papers from the 32nd and 34th annual symposia on foundations of computer science, Oct. 2–4, 1991 and Nov. 3–5, 1993
On the diameter of the symmetric group: polynomial bounds
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Finding optimal solutions to Rubik's cube using pattern databases
AAAI'97/IAAI'97 Proceedings of the fourteenth national conference on artificial intelligence and ninth conference on Innovative applications of artificial intelligence
Product growth and mixing in finite groups
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
On the Query Complexity of Testing Orientations for Being Eulerian
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
Series-parallel orientations preserving the cycle-radius
Information Processing Letters
On the query complexity of testing orientations for being Eulerian
ACM Transactions on Algorithms (TALG)
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We compare the diameter of a graph with the directed diameter of its Eulerian orientations. We obtain positive results under certain symmetry conditions.An Eulerian orientation of a graph is an orientation such that each vertex has the same indegree and outdegree. A graph is vertex-transitive if its vertices are equivalent under automorphisms.We show that the directed diameter of an Eulerian orientation of a finite vertex-transitive graph cannot be much larger than the undirected diameter; our bound on the directed diameter is O (dΔ ln n) where d is the undirected diameter, Δ is the (out)degree of the vertices, and n is the number of vertices. This implies that for Eulerian orientations of vertex-transitive graphs-of bounded degree, the gap between the two diameters is at most quadratic.As a consequence, we are able to compare the word length and the positive word length of elements of a finite group in terms of a given set of generators; we show that the gap is at most nearly quadratic, where the term "nearly" refers to a factor, polylogarithmic in the order of the group.It follows that recent polynomial bounds on the diameter of certain large classes of Cayley graphs of the symmetric group and certain linear groups automatically extend to directed Cayley graphs. The result also shows that the directed and undirected versions of long standing conjectures regarding the diameter of Cayley graphs of various classes of groups, including transitive permutation groups and finite simple groups, are equivalent.We also show that for edge-transitive digraphs, the directed diameter is O(d ln n).On the other hand, if we weaken the condition of vertex-transitivity to regularity (all vertices have the same degree), then the directed diameter is no longer polynomially bounded in terms of the undirected diameter and the maximum degree (and In n = O(d ln Δ)).Our upper bounds on the diameter raise the algorithmic challenge to find paths of the length guaranteed by these results. While for undirected graphs, most (but not all) relevant proofs are algorithmic, our bounds for the directed diameter are obtained via a pigeon-hole argument based on expansion and yield existence only.