Being a significant construct in a wide range of combinatorial problems, the k-subset sum problem (k-SSP) computes k-element subsets, out of an n-element set, satisfying a user-defined aggregation value. In this paper, we formulate the k-subset sum problem as a search (optimization) problem over the space of integers associated with combination elements. And by using rigorous computational experiments using the search space over more than 1014 integer numbers, we show that our approach is effective and efficient: it is feasible to find any combination with a user-defined sum within 104 function evaluations by using a gradient-free optimization algorithm. Our scheme opens the door to further advance the understanding of combinatorial problems by improved/tailored gradient-free optimization algorithms based on enumerative encoding. Also, our approach realizes the practical building block for combinatorial problems in planning and operations research using k-SSP concepts.

On k-subset sum using enumerative encoding

An improved balanced algorithm for the subset-sum problem

Addressing the subset sum problem is relevant to study resource management problems efficiently. In this paper, we study a new scheme to sample solutions for the subset sum problem based on swarm-based optimization algorithms with distinct forms of selection pressure, the balance of exploration-exploitation, the multimodality considerations, and a search space defined by numbers associated with subsets of fixed size. Our experiments show that it is feasible to find optimal subsets with few number of fitness evaluations, and that Particle Swarm Optimization with Fitness Euclidean Ratio converges faster to the global optima with zero variability over independent runs. Since the search space is one-dimensional and friendly to parallelization schemes, our work is potential to study further classes of combinatorial problems using swarm-based optimization algorithms and the representation based on numbers.

Tackling the Subset Sum Problem with Fixed Size using an Integer Representation Scheme

The subset-sum problem is a well-known NP-complete decision problem. Many parallel algorithms have been developed to solve the problem within a reasonable computation time, and some of them have been implemented on a GPU. However, the GPU implementations of these parallel algorithms may fail to fully utilize all the CPU cores and the GPU resources at the same time. When the GPU performs some tasks, only one CPU core is used to control the GPU, all the rest of CPU cores are in idle state, this leads to large amounts of available CPU resources are wasted. This paper proposes a novel CPU-GPU cooperative implementation of a parallel two-list algorithm to efficiently solve the subset-sum problem in a heterogeneous CPU-GPU system, which enables the efficient utilization of all the available computational resources of both CPUs and GPUs. In order to find the most appropriate task distribution ratio between CPUs and GPUs, this paper establishes an optimal task distribution model. A series of experiments are conducted on two different hardware platforms. The experimental results show that the CPU-GPU cooperative implementation produces a speedup factor of 9.2 over the best sequential implementation, achieves up to 96.3% performance improvement over the optimized CPU-only implementation, and yields up to 25.7% performance improvement over the optimized GPU-only implementation.

A Novel CPU-GPU Cooperative Implementation of A Parallel Two-List Algorithm for the Subset-Sum Problem

Efficiently representing and generating combinations can allow the seamless visualization, sampling, and evaluation of combinatorial architectures. In this paper, being relevant to tackle resource allocation problems ubiquitously, we address the subset sum problem by (1) using gradient-free optimization with a number-based representation of the combinatorial search space and by (2) generating combinations with minimal change order through parallel reductions in the GPU. Our computational experiments consisting of a relevant set of problem instances and gradient-free optimization algorithms show that (1) it is possible to generate combinations in the GPU efficiently, with quasi-linear complexity, (2) it is possible to tackle instances of the subset sum problem within a reasonable number of function evaluations, and (3) Particle Swarm Optimization with Fitness Euclidean Ratio converges faster. Since the search space of number-based representations is one-dimensional and amenable to parallelization schemes (e.g., GPU), we believe our work opens the door to tackle further combinatorial problems.

Generating combinations on the GPU and its application to the k-subset sum

A novel cooperative accelerated parallel two-list algorithm for solving the subset-sum problem on a hybrid CPU-GPU cluster

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A novel cooperative accelerated parallel two-list algorithm for solving the subset-sum problem on a hybrid CPU-GPU cluster