Table 4.

Reducing the number of optimal time-feasible solutions by bounding k

Dataset	Costvector
	〈−1,1,1,1〉			〈0,1,2,1〉			〈0,2,3,1〉
	k / k ′	# A C / # A C ′	# T / # T ′	k / k ′	# A C / # A C ′	# T / # T ′	k / k ′	# A C / # A C ′	# T / # T ′
EC	3/3	2/2	2/2	3/3	18/16	18/16	3/3	16/16	16/16
GL	4/4	2/2	2/2	4/4	2/2	2/2	4/4	2/2	2/2
SC	6/6	1/1	1/1	6/6	1/1	1/1	6/6	1/1	1/1
RP	9/9	2/2	3/3	8/8	2/2	8/8	2/2	2/2	3/3
PMP	6/6	2/2	2/2	6/6	2/2	2/2	5/5	11/4	11/4
PML	5/5	2/2	2/2	5/5	2/2	2/2	3/3	18/6	18/6
PP	4/4	144/96	144/96	4/4	72/48	72/48	4/4	72/48	72/48
FD	9/10	576/240	944/512	9/9	276/4	408/8	−	−	−
COG2085	14/14	82560/9408	109056/9408	14/14	37088/4032	37568/4032	14/14	46656/5184	46656/5184
COG4965	16/16	31448/15744	44800/22400	16/16	640/320	640/320	13/16	5120/2560	6528/3328

For some datasets, the number of optimal time-feasible solutions may be huge when k is unbounded. In some cases, however, by introducing a bound on k we can greatly reduce the number of time-feasible solutions while keeping their optimality. For all datasets whose number of acyclic solutions is positive for unbounded k, we identified k _start (minimum k whose optimal cost is equal to the optimal cost obtained for unbounded k) and we searched for the minimum k ^′≥k _start whose number of acyclic solutions is non zero. We executed this procedure for every pair (dataset, cost vector) for which the number of optimal acyclic solutions is positive. In the first column, the values for k=k _start and k ^′ are given. # A C/# A C ^′ denotes the number of optimal acyclic solutions for the case when the switches are unbounded and the case when they are bounded by k ^′, respectively. The same relation is shown for the total number of optimal solutions in the column # T/# T ^′.