Solving knapsack problems on GPU

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1. What are the contributions in "Solving knapsack problems on gpu" ?

2. What is the effect of the compression on the matrix?

3. how many threads access the global memory?

4. What is the name of the software development kit?

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In this article, the authors propose a parallel implementation of the dynamic programming method for the knapsack problem on NVIDIA GPU.. Furthermore, in order to limit the communication between the CPU and the GPU, a compression technique is presented which decreases significantly the memory occupancy.

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The reduction of the size on the matrix increases with the number of variables, resulting in a more efficient compression and the overhead does not exceed 3% of the overall one.

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the global memory access by all threads is done in one or two transactions if:• threads access:– either 32-bit words, resulting in one 64-byte memory transaction,– or 64-bit words, resulting in one 128-byte memory transaction,– or 128-bit words, resulting in two 128-byte memory transactions;• all 16 words lie in the same segment of size equal to the memory transaction size (or twice the memory transaction size when accessing 128-bit words);• threads access the words in sequence (the kth thread in the half-warp accesses the kth word).

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NVIDIA introduced, in 2006, CUDA, a software development kit that enables users to solve many complex computational problems on their GPU cards.

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return f(C) (the optimal bound of the KP)The high memory requirement are frequently cited as the main drawback of dynamic programming.

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A simple way to reduce the vector m d is then to compress it as follows:for ĉ ∈ {0, ..., rc− lc}, m d cĉ = m dĉ+lc with lc = min{ĉ ∈ {1, ..., C} | m dĉ−1 = 0 and m dĉ 6= 0},rc = max{ĉ ∈ {1, ..., C} | m dĉ−1 6= 232 − 1 and m dĉ = 232 − 1}.

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Then the authors have to retrieve only the values of m dĉ for ĉ ∈ {lc, ..., rc− 1} and the authors process lc and rc directly on the GPU via the algorithm 4.Algorithm 4 (Thread compression on GPU) blocks id: the ID of the belonging block, thread id: the ID of the thread within the belonging block, m d: the input vector, lc: shared variable initiate with the value C, rc: shared variable initiate with the value 0,ĉ:=blocks id * 512 + thread id, if ĉ ≤ 0 or ĉ >

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In section 4 the authors propose a compression method which permits one to reduce significantly the memory occupancy and communication between CPU and GPU.

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Table 1 shows that in the worst case the size of the compressed data (size of M c+ 2. dn/32e) corresponds to only 0.3% of the size of the initial the matrix M ,which leads to a very small memory occupancy as compared with the original dynamic programming algorithm.

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This function creates C − c min + 1 threads for the GPU and groups them into blocks of 512 threads (the maximum size of a block of one dimension), i.e. d(C − c min + 1)/512e blocks.

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Given a set of n items i, with profit pi ∈ N∗+ and weight wi ∈ N∗+, and a knapsack with the capacity C ∈ N∗+, KP can be defined as the following linear integer programming problem:(KP ) max n∑ i=1 pi.xi, s.t. n∑ i=1 wi.xi ≤ C,xi ∈ {0, 1}, i ∈ {1, ..., n}.(2.1)To avoid any trivial solution, the authors assume that: ∀i ∈ {1, ..., n}, wi ≤ C, n∑ i=1 wi > C.Bellman’s dynamic programming [15], presented in 1957, was the first exact algorithm to solve KP.

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All threads carry out on the GPU the procedure described in algorithm 3.Algorithm 3 (Thread processing on GPU) blocks id: the ID of the belonging block, thread id: the ID of the thread within the belonging block, k: the step number of the dynamic programming (k ∈ {1, ..., n}), i := (k + 31)%32: the rest of the division of k + 31 by 32,ĉ:=blocks id * 512 + thread id, if ĉ < c min or ĉ >

Solving knapsack problems on GPU

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Most frequently asked questions

1. What are the contributions in "Solving knapsack problems on gpu" ?

2. What is the effect of the compression on the matrix?

3. how many threads access the global memory?

4. What is the name of the software development kit?

5. What is the main drawback of dynamic programming?

6. how to reduce the vector m d?

7. how many threads have to access the values of m d?

8. What is the main idea of the paper?

9. What is the way to solve a problem with a dynamic programming algorithm?

10. How many threads are created for the GPU?

11. What is the first exact algorithm to solve KP?

12. What is the optimum memory access for the GPU?

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References

Knapsack Problems: Algorithms and Computer Implementations

Solving Large-Scale Zero-One Linear Programming Problems

New trends in exact algorithms for the 0-1 knapsack problem

Dynamic Programming Algorithms for the Zero-One Knapsack Problem*

Heuristics for the 0-1 multidimensional knapsack problem

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