Revisiting Sparse Dynamic Programming for the 0/1 Knapsack Problem

The 0/1-Knapsack Problem is a classic NP-hard problem. There are two common approaches to obtain the exact solution: branch-and-bound (BB) and dynamic programming (DP). A so-called, "sparse" DP algorithm (SKPDP) that performs fewer operations than the standard algorithm (KPDP) is well known. To the best of our knowledge, there has been no quantitative analysis of the benefits of sparsity. We provide a careful empirical evaluation of SKPDP and observe that for a "large enough" capacity, C, the number of operations performed by SKPDP is invariant with respect to C for many problem instances. This leads to the possibility of an exponential improvement over the conventional KPDP. We experimentally explore SKPDP over a large range of knapsack problem instances and provide a detailed study of the attributes that impact the performance. DP algorithms have a nice regular structure and are amenable to highly parallel implementations. However, due to the dependence structure, parallelizing SKPDP is challenging. We propose two parallelization strategies (fine-grain and coarse-grain) for SKPDP on modern multi-core processors and demonstrate a scalable improvement in the performance. We also compare SKPDP with Branch-and-Bound algorithm.