A finite crisscross method for oriented matroids

AbstractOur paper presents a new finite crisscross method for oriented matroids. Starting from a neither primal nor dual feasible tableau, we reach primal and dual optimal oriented circuits in a finite number of steps if they exist. If there is no optimal tableau then we show that there is no primal feasible circuit or there is no dual feasible cocircuit. So we give a new constructive proof for the general duality theorem (Bland J. Combin. Theory Ser. B 23 (1977), 33–57; Folkman and Lawrence J. Combin. Theory Ser. B 25 (1978), 199–236). Our pivot rule is a generalization of the “anticycling rule” suggested in Bland (op cit; Math. Oper. Res. 2 (1977), 103–107). Finite pivoting rules are given by Edmonds, Fukuda and Todd (Ph.D. dissertation, Univ. of Waterloo, 1982), SIAM Algebraic Discrete Math. 5, No. 4 (1984), 467–485). A general relaxed recursive algorithm was discovered independently by Jensen (Ph.D. thesis, School of OR and IE, Cornell, 1985) which is principally crisscross type. Jensen's is very general and flexible; in fact it can be considered as a family of algorithms. Among the conceivable algorithms in his general family our independently constructed crisscross method is characterized by its extreme simplicity