The paper presents a Branch-and-Cut for solving (0, 1) integer linear programs having a large symmetry group. The group is used for pruning the enumeration tree and for generating cuts. The cuts are non standard, cutting integer feasible solutions but leaving unchanged the optimal value of the problem. Pruning and cut generation are performed by backtracking procedures using a Schreier-Sims table for representing the group. Applications to the generation of covering designs and error correcting codes are presented.

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Using an orderly algorithm, the Steiner triple systems of order 19 are classified; there are 11,084,874,829 pairwise nonisomorphic such designs. For each design, the order of its...

. Macro search is used to derive solutions quickly for large search spaces at the expense of optimality. We present a novel way of building macro tables. Our contribution is twofold: (1) for the first time, we use automatically generated heuristics to find optimal macros, (2) due to the speed-up achieved by (1), we merge consecutive subgoals to reduce the solution lengths. We use the Rubik's Cube to demonstrate our techniques. For this puzzle, a 44% improvement of the average solution length was achieved over macro tables built with previous techniques. 1

l over the generators grows as c l for some constant c>1 depending on G. For groups with abelian subgroups of finite index, we obtain a Las Vegas algorithm for several basic computational tasks, including membership testing and computing a presentation. This generalizes recent work of Beals and Babai, who give a Las Vegas algorithm for the case of finite groups, as well as recent work of Babai, Beals, Cai, Ivanyos, and Luks, who give a deterministic algorithm for the case of abelian groups. # 1999 Academic Press Article ID jcss.1998.1614, available online at http:##www.idealibrary.com on 260 0022-0000#99 #30.00 Copyright # 1999 by Academic Press All rights of reproduction in any form reserved. * Research conducted while visiting IAS and DIMACS and supported in part by an NSF Mathematical Sciences

Abstract. We describe two methods for computing with the elements of untwisted groups of Lie type: using the Steinberg presentation and using highest weight representations. We give algorithms for element arithmetic within the Steinberg presentation. Conversion between this presentation and linear representations is achieved using a new generalisation of row and column reduction. 1.

ly, a constraint is a relation of the form a 1 v 1 + · · · +a n v n k, where v 1 , . . . , v n are variables, a 1 , . . . , a n # Q , k # Z, and is either #, #, or =. Constraints are represented in several different ways, according to what variables and coefficients are allowed: 1. When constraints appear in commands, the variables are arbitrary (within the confines of the language), and the coefficients are in Z. 2. These constraints are internally represented (using the "constraint" class) in the same way, except that Q -coefficients are allowed. This is because some internal operations may result in constraints whose coefficients are not integers. (This can happen when the "incorporate" command is used.) 3. For split linear programming calculations, constraints are first converted to constraints involving v variables and having coefficients in Q (class qvconstraint). 4. Clearing denominators yields constraints having v variables and coefficients in Z (class vconstra...

Janko’s large simple sporadic group J4 was originally constructed by Benson,

In this paper I describe a new method of symmetry breaking using group theory which can represent very large sets of symmetries easily. I also define the class of unique symmetries which can be used to reduce the overhead of performing symmetry breaking during search. This makes it possible to break all symmetry on CSPs with a large number of symmetries.

Abstract. We present an algorithm to solve the orbit-stabilizer problem for a polycyclic group G acting as a subgroup of GL(d, Z) on the elements of Q d. We report on an implementation of our method and use this to observe that the algorithm is practical. 1.