[Paper Review] An LP with Integrality Gap 1+epsilon for Multidimensional Knapsack

David Pritchard|arXiv (Cornell University)|May 18, 2010

Optimization and Packing Problems21 references3 citations

TL;DR

This paper presents a polynomial-sized linear programming (LP) relaxation for the k-dimensional knapsack problem with an integrality gap of at most 1+ϵ for any fixed k and ϵ>0. It achieves this by combining filtering (guessing high-value or high-constraint items) with disjunctive programming, enabling near-optimal rounding of fractional solutions, and extends to a cost-independent LP formulation in the packing case.

ABSTRACT

In this note we study packing or covering integer programs with at most k constraints, which are also known as k-dimensional knapsack problems. For any integer k > 0 and real epsilon > 0, we observe there is a polynomial-sized LP for the k-dimensional knapsack problem with integrality gap at most 1+epsilon. The variables may be unbounded or have arbitrary upper bounds. In the packing case, we can also remove the dependence of the LP on the cost-function, yielding a polyhedral approximation of the integer hull. This generalizes a recent result of Bienstock on the classical knapsack problem.

Motivation & Objective

• To develop a polynomial-sized LP relaxation for the k-dimensional knapsack problem with a small, bounded integrality gap.
• To generalize Bienstock's 1D knapsack result to higher dimensions, achieving an integrality gap of 1+ϵ.
• To construct an LP relaxation independent of the objective function c in the packing case, enabling polyhedral approximation of the integer hull.
• To explore whether a similar cost-independent LP with small integrality gap can be constructed for the covering version.
• To provide a theoretical framework using filtering and disjunctive programming that enables near-optimal rounding of fractional solutions.

Proposed method

• Use a filtering technique to guess the γ most profitable items (or those with highest coefficients per constraint) in the optimal solution, where γ = k/ϵ.
• For each such guess g, define a subproblem with modified bounds d^g and right-hand side b^g to enforce that the guessed items are included.
• Construct a disjunctive program by combining all such subproblem relaxations into a single extended LP with additional variables and constraints.
• Ensure the extended LP has polynomial size: n^{O(k/ϵ)} variables and kn^{O(k/ϵ)} constraints for the cost-dependent case.
• For the cost-independent case, replace profit-based filtering with constraint-based filtering—guessing the γ items with largest coefficients per constraint across k dimensions.
• Use rounding: round down (for packing) or up (for covering) the fractional solution, and correct infeasibility by removing a small-cost set of items per constraint.

Experimental results

Research questions

• RQ1 Can a polynomial-sized LP relaxation be constructed for k-dimensional knapsack with integrality gap 1+ϵ for any fixed k and ϵ>0?
• RQ2 Can such an LP be made independent of the objective function c in the packing case, yielding a polyhedral approximation of the integer hull?
• RQ3 Is it possible to extend the cost-independent LP construction to the covering version of k-dimensional knapsack?
• RQ4 What is the trade-off between LP size and integrality gap when removing dependence on c?
• RQ5 Can the filtering and disjunctive programming approach be used to achieve a (1+ϵ)-approximation with a compact LP formulation in higher dimensions?

Key findings

• For any fixed k and ϵ>0, there exists a polynomial-sized extended LP relaxation for k-dimensional knapsack with integrality gap at most 1+ϵ.
• The size of the LP is n^{O(k/ϵ)} variables and kn^{O(k/ϵ)} constraints when the objective function c is used in the construction.
• In the packing case, a cost-independent LP relaxation with integrality gap 1+ϵ can be constructed, at the cost of increasing the size to kn^{O(k²/ϵ)} variables and constraints.
• The construction relies on guessing γ = k²/ϵ items with the largest coefficients per constraint, enabling rounding with bounded cost loss.
• Rounding down (for packing) or up (for covering) the fractional solution from the LP yields an integral solution within a (1−ϵ) or (1+ϵ) factor of the optimal.
• The result generalizes Bienstock’s 1D knapsack result to higher dimensions and provides a polyhedral approximation of the integer hull in the packing case.