Dynamic programming algorithms, efficient solution of the LP-relaxation and approximation schemes for the Penalized Knapsack Problem

TL;DR

This paper introduces an exact dynamic programming approach for the Penalized Knapsack Problem, demonstrating its effectiveness on hard instances and exploring approximation schemes for special cases.

Contribution

It presents a novel dynamic programming-based exact solution method and analyzes approximation schemes for specific cases of the Penalized Knapsack Problem.

Findings

The proposed approach outperforms commercial solvers and existing algorithms.

The method effectively solves hard instances of PKP.

Certain special cases admit fully polynomial time approximation schemes.

Abstract

We consider the 0-1 Penalized Knapsack Problem (PKP). Each item has a profit, a weight and a penalty and the goal is to maximize the sum of the profits minus the greatest penalty value of the items included in a solution. We propose an exact approach relying on a procedure which narrows the relevant range of penalties, on the identification of a core problem and on dynamic programming. The proposed approach turns out to be very effective in solving hard instances of PKP and compares favorably both to commercial solver CPLEX 12.5 applied to the ILP formulation of the problem and to the best available exact algorithm in the literature. Then we present a general inapproximability result and investigate several relevant special cases which permit fully polynomial time approximation schemes (FPTASs).