Reconstructing permutation table to improve the Tabu Search for the PFSP on GPU

Abstract

General-purpose computing on graphics processing unit (GPGPU) has been adopted to accelerate the running of applications which require long execution time in various problem domains. Tabu Search belonging to meta-heuristics optimization has been used to find a suboptimal solution for NP-hard problems within a more reasonable time interval. In this paper, we have investigated in how to improve the performance of Tabu Search algorithm on GPGPU and took the permutation flow shop scheduling problem (PFSP) as the example for our study. In previous approach proposed recently for solving PFSP by Tabu Search on GPU, all the job permutations are stored in global memory to successfully eliminate the occurrences of branch divergence. Nevertheless, the previous algorithm requires a large amount of global memory space, because of a lot of global memory access resulting in system performance degradation. We propose a new approach to address the problem. The main contribution of this paper is an efficient multiple-loop struct to generate most part of the permutation on the fly, which can decrease the size of permutation table and significantly reduce the amount of global memory access. Computational experiments on problems according with benchmark suite for PFSP reveal that the best performance improvement of our approach is about 100%, comparing with the previous work.

Appendix

This appendix argue that the maximum number of block areas is five for each permutation segment table.

The permutation table is constructed as follows. Each permutation is generated by swapping two positions on the parent permutation, resulting in $C_2^n$ child permutations totally. These child permutations are placed into the permutation table by ordering defined in Table 4 .

Table 4 The indices of “From” and “To” for each ordered child permutation

Fig. 19

Conceptual overview of the sth group of child permutations

The two indices, “From” and “To” indicate that two positions on the parent permutation are swapped for one child permutation, where $1\le From\le n-1$ and $2\le From\le n$. Note that “From” is smaller than “To” for any one of the child permutations.

The permutation table can be divided into ($n-1)$ groups from left to right, where each group has the same “From” value. For instance, in the 7$^{\mathrm{th}}$ group, the “From” index of each child permutation is 7. We illustrate the conceptual overview of the s$^{\mathrm{th}}$ group in Fig. 19. There are exactly two shaded cells in each column, representing the two swapped positions.

The permutation table will be divided into segment tables, from the left to the right columns. Each permutation segment table consists of 32 consecutive columns because the size of one warp is 32.

Fig. 20

Three cases for dividing permutation segment tables

First, assume that one permutation segment table falls in only one group of child permutations. There are three cases as shown in Fig. 20. Case 1 is derived when the permutation segment table (PST) begins from the first column of the group, where there are three block areas (BAs). Case 2 is obtained when the PST ends at the final column of the group, where there are five BAs. Case 3 is the remaining cases and there are four BAs. In general, there are five BAs in cases 2. However, if the PST in case 2 is equivalent to the s$^{\mathrm{th}}$ group, there are only two BAs because BA3 and BA5 will not exist and BA4 will be merged into BA2.

Fig. 21

Two cases when one PST is across two groups of child permutations

Next, let us look at the cases when one PST includes multiple groups of child permutations, as shown in Fig. 21. Case 5 contains the last columns in the sth group and the first column in the (s+1)th group, where several rows, between Row sand Rown, have the same values in their own row, respectively. There are four BAs in cases 5. Case 4 demonstrates a general case when one PST is comprised of multiple groups. If there are more than two groups to form a PST, the rows with distinct data will be merged into BA2, resulting in two BAs totally.

According to the above analysis, the maximum number of BAs is five.