Approximate #Knapsack Computations to Count Semi-Fair Allocations

TL;DR

This paper develops fully polynomial-time approximation schemes (FPTAS) for counting knapsack solutions with specific cardinality and for counting semi-fair allocations of indivisible goods, ensuring at least one player is envy-free.

Contribution

It introduces FPTAS algorithms for counting solutions to cardinality-constrained knapsack problems and semi-fair allocations, extending previous work with new approximation methods.

Findings

FPTAS for counting cardinality-constrained knapsack solutions

FPTAS for counting semi-fair allocations ensuring envy-freeness

Extensions of approximation algorithms to new allocation problems

Abstract

In this paper, we study the problem of counting the number of different knapsack solutions with a prescribed cardinality. We present an FPTAS for this problem, based on dynamic programming. We also introduce two different types of semi-fair allocations of indivisible goods between two players. By semi-fair allocations, we mean allocations that ensure that at least one of the two players will be free of envy. We study the problem of counting such allocations and we provide FPTASs for both types, by employing our FPTAS for the prescribed cardinality knapsack problem.