Counting and Enumerating Independent Sets with Applications to Knapsack Problems

TL;DR

This paper develops methods to count and enumerate independent sets in specific graph classes, enabling polynomial-time solutions for certain knapsack and bin packing problems with fixed dimensions.

Contribution

It introduces novel graph-based characterizations for knapsack problems, allowing efficient algorithms for special cases and providing bounds on bin requirements.

Findings

Polynomial-time algorithms for fixed-dimension knapsack instances

Characterizations linking independent sets to knapsack feasibility

Lower bounds on bin numbers in bin packing problems

Abstract

We introduce methods to count and enumerate all maximal independent, all maximum independent sets, and all independent sets in threshold graphs and k-threshold graphs. Within threshold graphs and k-threshold graphs independent sets correspond to feasible solutions in related knapsack instances. We give several characterizations for knapsack instances and multidimensional knapsack instances which allow an equivalent graph. This allows us to solve special knapsack instances as well as special multidimensional knapsack instances for fixed number of dimensions in polynomial time. We also conclude lower bounds on the number of necessary bins within several bin packing problems.