A simple (2+ε)-approximation for knapsack interdiction
Pith reviewed 2026-05-08 14:19 UTC · model grok-4.3
The pith
A (2+ε)-approximation algorithm for knapsack interdiction is presented that is simpler and faster than the known PTAS.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We present a (2+ε)-approximation running in O(n^3 ε^{-1} log(ε^{-1} log ∑_i p_i)) time. Although a polynomial-time approximation scheme (PTAS) is already known for this problem, our algorithm is considerably simpler and faster.
Load-bearing premise
The analysis establishing the (2+ε) guarantee holds under the standard non-negativity of profits, interdiction costs, and weights together with the existence of an efficient subroutine for the underlying knapsack subproblems (details not visible in the abstract).
read the original abstract
In the knapsack interdiction problem, there are $n$ items, each with a non-negative profit, interdiction cost, and packing weight. There is also an interdiction budget and a capacity. The objective is to select a set of items to interdict (delete) subject to the budget which minimizes the maximum profit attainable by packing the remaining items subject to the capacity. We present a $(2+\epsilon)$-approximation running in $O(n^3\epsilon^{-1}\log(\epsilon^{-1}\log\sum_i p_i))$ time. Although a polynomial-time approximation scheme (PTAS) is already known for this problem, our algorithm is considerably simpler and faster. The approach also generalizes naturally to a $(1+t+\epsilon)$-approximation for $t$-dimensional knapsack interdiction with running time $O(n^{t+2}\epsilon^{-1}\log(\epsilon^{-1}\log\sum_i p_i))$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript presents a (2+ε)-approximation algorithm for the knapsack interdiction problem that runs in O(n³ ε⁻¹ log(ε⁻¹ log ∑_i p_i)) time. It claims this algorithm is considerably simpler and faster than the known PTAS for the problem. The approach is shown to generalize to a (1+t+ε)-approximation for t-dimensional knapsack interdiction with running time O(n^{t+2} ε⁻¹ log(ε⁻¹ log ∑_i p_i)).
Significance. If the claimed approximation ratio and runtime are achieved, the result supplies a practical, easy-to-implement alternative to the existing PTAS for an important interdiction problem with applications in security and robust optimization. The simplicity of the construction and the natural extension to the multidimensional case are notable strengths that could facilitate adoption and further algorithmic development.
minor comments (2)
- [Abstract] In the abstract and introduction, the base of the logarithm in the running-time bound is not specified; adding this detail would improve precision.
- [Section 5] The statement of the generalization theorem for t-dimensional interdiction would benefit from an explicit theorem number and a short proof sketch in the main body to make the extension self-contained.
Simulated Author's Rebuttal
We thank the referee for the positive review, accurate summary of our contributions, and recommendation for minor revision. We are pleased that the significance of the algorithm's simplicity, improved runtime, and natural generalization to the t-dimensional case has been noted.
Circularity Check
No significant circularity detected in derivation chain
full rationale
The paper presents a new algorithmic construction for a (2+ε)-approximation to knapsack interdiction, with the guarantee and runtime following directly from the described approach and standard non-negativity assumptions on inputs. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the PTAS comparison is external and the t-dimensional generalization follows the same explicit pattern. The derivation remains self-contained against the problem definition without internal reduction to inputs.
discussion (0)
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