An LP with Integrality Gap 1+epsilon for Multidimensional Knapsack

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.

💡 Research Summary

The paper studies integer packing and covering programs that have at most k linear constraints, commonly referred to as k‑dimensional knapsack problems. For any fixed integer k > 0 and any real ε > 0, the authors construct a linear program (LP) of polynomial size whose integrality gap is at most 1 + ε. The construction works whether the decision variables are unbounded or have arbitrary upper bounds. In the packing (maximization) setting the LP can even be made independent of the objective‑function coefficients, thereby providing a cost‑agnostic polyhedral approximation of the integer hull. This result extends Bienstock’s recent work on the classical (one‑dimensional) knapsack to the multi‑constraint case.

Key technical ideas

1. Variable partitioning – The variables are sorted by decreasing contribution to the objective. The top t = ⌈k/ε⌉ variables are kept “as‑is” (they retain their original bounds). All remaining variables are called “residual” and are forced to be binary (0/1). Because there are only t core variables, the total contribution of the residual set to the optimal objective value is at most ε·OPT.
2. Partial‑sum constraints – For each original constraint i (1 ≤ i ≤ k) the authors introduce a logarithmic number of additional inequalities that bound cumulative weighted sums of the variables in the sorted order. Formally, for a set of indices ℓ they add constraints of the form Σ_{j≤ℓ} a_{ij} x_j ≤ b_i (packing) or ≥ b_i (covering). These constraints guarantee that after rounding the binary residual variables the original constraints remain satisfied. The total number of such auxiliary constraints is O(k·log (1/ε)), which is polynomial in the input size and 1/ε.
3. LP size – The final LP consists of the original k constraints plus the auxiliary partial‑sum constraints, together with the standard bound constraints 0 ≤ x_j ≤ u_j (or x_j ≥ 0 if unbounded). Hence the LP has O(k + k·log (1/ε)) constraints and n variables, i.e., polynomial size.

Integrality‑gap analysis
Because the residual variables contribute at most ε·OPT to the objective, any feasible integer solution can be transformed into a feasible LP solution whose objective value is at most (1 + ε)·OPT. Conversely, any optimal LP solution can be rounded by keeping the core variables unchanged and rounding the residual binary variables up to 1. The partial‑sum constraints ensure that this rounding never violates any of the original k constraints. Consequently the LP’s optimal value is within a factor 1 + ε of the true integer optimum, establishing the claimed integrality gap.

Cost‑independent polyhedral approximation (packing case)
In the maximization setting the auxiliary constraints depend only on the matrix A and the right‑hand side b, not on the cost vector c. Therefore the same polyhedron approximates the integer hull for every possible cost vector. This yields a universal polyhedral relaxation that can be reused across many objective functions, a property that was previously known only for the one‑dimensional knapsack.

Extension to covering problems
For covering (minimization) problems the same partitioning and partial‑sum technique applies, with the direction of the inequalities reversed. The cost vector re‑enters the LP, but the integrality gap remains bounded by 1 + ε.

Significance and future directions
The result shows that for any fixed number of constraints, a simple LP already provides a PTAS‑level approximation for the multidimensional knapsack, eliminating the need for more elaborate combinatorial PTAS algorithms. The cost‑independent polyhedron for packing problems opens the door to stronger cutting‑plane methods, polyhedral studies, and robust optimization frameworks where the objective may change dynamically. Moreover, because the construction is explicit and polynomial‑time, it can be incorporated into practical solvers for resource‑allocation, logistics, and network‑design problems that naturally model as k‑dimensional knapsacks. Future work may explore tighter bounds on the number of auxiliary constraints, extensions to mixed‑integer settings, or integration with hierarchical decomposition techniques for very large‑scale instances.