The Knapsack Problem with Neighbour Constraints

TL;DR

This paper investigates two variants of the knapsack problem with dependency constraints defined by graph adjacencies, providing algorithms and hardness results for different graph types and weight-profit configurations.

Contribution

It introduces approximation algorithms and hardness results for the 1-neighbour and all-neighbours knapsack problems under various graph and weight-profit conditions.

Findings

Approximation algorithms for specific graph classes.

Hardness results establishing computational limits.

Analysis for uniform and arbitrary weight-profit functions.

Abstract

We study a constrained version of the knapsack problem in which dependencies between items are given by the adjacencies of a graph. In the 1-neighbour knapsack problem, an item can be selected only if at least one of its neighbours is also selected. In the all-neighbours knapsack problem, an item can be selected only if all its neighbours are also selected. We give approximation algorithms and hardness results when the nodes have both uniform and arbitrary weight and profit functions, and when the dependency graph is directed and undirected.