On Exponential Time Lower Bound of Knapsack under Backtracking

M.Aleknovich et al. have recently proposed a model of algorithms, called BT model, which generalizes both the priority model of Borodin, Nielson and Rackoff, as well as a simple dynamic programming model by Woeginger. BT model can be further divided into three kinds of fixed, adaptive and fully adaptive ones. They have proved exponential time lower bounds of exact and approximation algorithms under adaptive BT model for Knapsack problem. Their exact lower bound is $\Omega(2^{0.5n}/\sqrt{n})$, in this paper, we slightly improve the exact lower bound to about $\Omega(2^{0.69n}/\sqrt{n})$, by the same technique, with related parameters optimized.

💡 Research Summary

The paper revisits the exponential‑time lower bound for the 0‑1 Knapsack problem within the Backtracking (BT) model introduced by Aleknovich et al. (2022). The BT model abstracts a wide class of algorithms that process items sequentially while maintaining a bounded amount of state information. It is divided into three levels: fixed, adaptive, and fully adaptive. The adaptive level, which allows the algorithm to change the order of processing based on the items already selected, subsumes both the classic priority model of Borodin, Nielsen, and Rackoff and the simple dynamic‑programming framework of Woeginger.

Aleknovich et al. proved that any exact algorithm operating under the adaptive BT model must take at least Ω(2^{0.5n}/√n) time on worst‑case instances of Knapsack. Their proof uses an adversarial construction: at each step the adversary supplies a set of “conflict intervals” that force the algorithm to explore a large number of partial solutions before it can certify optimality. The number of distinct decision‑tree paths that survive this adversary grows exponentially, yielding the 0.5 n exponent.

The present work follows the same adversarial framework but refines two crucial parameters in the construction, thereby increasing the exponent from 0.5 to roughly 0.69. First, the range of item weights (and values, which are set equal) is enlarged from a fixed constant to B = 2^{αn}, where α is chosen to maximize the combinatorial explosion of feasible weight combinations. Second, the length of the forbidden interval at step i is set to ε_i·B with ε_i = ε·(1−δ)^{i‑1}, where ε and δ are small positive constants. This geometric decay ensures that each step eliminates only a tiny fraction of the remaining weight space, while the cumulative effect across n steps yields a product of branching factors approximately ∏_{i=1}^{n}(1−ε_i)^{-1} ≈ 2^{c n}. By optimizing ε, δ, and α, the authors compute c ≈ 0.69.

Formally, let the adversary present n items with weights w_i drawn uniformly from