On k-subset sum using enumerative encoding

TL;DR: This paper forms the k-subset sum problem as a search (optimization) problem over the space of integers associated with combination elements, and shows that it is feasible to find any combination with a user-defined sum within 104 function evaluations by using a gradient-free optimization algorithm.

Abstract: 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.