[Paper Review] An Approximate Dynamic Programming Approach to The Incremental Knapsack Problem

Ali Aouad, Danny Segev|arXiv (Cornell University)|Oct 15, 2020

Optimization and Search Problems21 references4 citations

TL;DR

This paper presents the first polynomial-time approximation scheme (PTAS) for the incremental knapsack problem, a strongly NP-hard generalization of the binary knapsack problem with time-varying capacities. By leveraging an approximate dynamic programming formulation with state pruning, rounding, and problem decomposition, the authors achieve a (1−ε)-approximate solution in polynomial time, resolving a long-standing open question in approximation algorithms.

ABSTRACT

We study the incremental knapsack problem, where one wishes to sequentially pack items into a knapsack whose capacity expands over a finite planning horizon, with the objective of maximizing time-averaged profits. While various approximation algorithms were developed under mitigating structural assumptions, obtaining non-trivial performance guarantees for this problem in its utmost generality has remained an open question thus far. In this paper, we devise a polynomial-time approximation scheme for general instances of the incremental knapsack problem, which is the strongest guarantee possible given existing hardness results. In contrast to earlier work, our algorithmic approach exploits an approximate dynamic programming formulation. Starting with a simple exponentially sized dynamic program, we prove that an appropriate composition of state pruning ideas yields a polynomially sized state space with negligible loss of optimality. The analysis of this formulation synthesizes various techniques, including new problem decompositions, parsimonious counting arguments, and efficient rounding methods, that may be of broader interest.

Motivation & Objective

• To close the gap in approximation guarantees for the incremental knapsack problem under general settings, where prior work only achieved constant-factor approximations under structural assumptions.
• To develop a polynomial-time approximation scheme (PTAS) for the incremental knapsack problem despite its strong NP-hardness, which rules out an FPTAS unless P=NP.
• To demonstrate that an approximate dynamic programming formulation—based on state pruning and rounding—can yield near-optimal solutions with provable performance guarantees.
• To synthesize novel techniques, including parsimonious counting and efficient rounding, that may extend beyond this specific problem.

Proposed method

• Formulate the incremental knapsack problem as an exponentially sized dynamic program with states tracking time, item index, and profit.
• Introduce a discretized state space via rounding of profit and weight parameters, reducing the state space to polynomial size while preserving near-optimality.
• Apply state pruning techniques that eliminate suboptimal or redundant states based on profit and weight thresholds, ensuring the remaining states yield a (1−ε)-approximate solution.
• Use a problem decomposition into clusters of items to manage complexity and enable efficient computation of intermediate solutions.
• Leverage a parsimonious counting argument to bound the number of relevant states and ensure polynomial runtime.
• Prove that the approximate value function serves as a lower bound on the true value function, enabling performance guarantees.

Experimental results

Research questions

• RQ1 Can a PTAS be developed for the general incremental knapsack problem, given its strong NP-hardness and the absence of an FPTAS?
• RQ2 Can approximate dynamic programming be effectively applied to a problem with time-varying constraints and nested packing requirements?
• RQ3 What state pruning and rounding strategies preserve near-optimality while reducing the state space to polynomial size?
• RQ4 How can problem decomposition and counting arguments be combined to control the number of relevant states in the dynamic program?
• RQ5 Can the resulting algorithm achieve a (1−ε)-approximation in polynomial time for any ε>0, despite the problem's computational hardness?

Key findings

• The paper presents the first polynomial-time approximation scheme (PTAS) for the incremental knapsack problem, achieving a (1−ε)-approximate solution for any ε>0.
• The algorithm runs in time polynomial in the input size and 1/ε, making it the strongest possible approximation guarantee under known hardness results.
• The approximate dynamic programming formulation, combined with state pruning and rounding, yields a state space of polynomial size with negligible loss in optimality.
• The value function of the approximate dynamic program provides a lower bound on the true value function, enabling the (1−ε)-approximation guarantee.
• The analysis introduces novel techniques—such as parsimonious counting and cluster-based decomposition—that may be applicable to other dynamic programming problems.
• The approach achieves a (1−7ε)-approximation in the final solution, with the final error bound tightened to (1−7ε) through careful handling of rounding and discretization errors.