Analysis of the Deterministic Polynomial Time Solvability of the 0-1-Knapsack Problem

Previously the author has demonstrated that a representative polynomial search partition is required to solve a NP-complete problem in deterministic polynomial time. It has also been demonstrated that finding such a partition can only be done in deterministic polynomial time if the form of the problem provides a simple method for producing the partition. It is the purpose of this article to demonstrate that no deterministic polynomial time method exists to produce a representative polynomial search partition for the Knapsack problem.

💡 Research Summary

The manuscript under review attempts to prove that the 0‑1 Knapsack problem cannot be solved in deterministic polynomial time because a “representative polynomial search partition” (RPSP) does not exist for this problem. The author builds on a previous, unpublished work in which it was claimed that any NP‑complete problem can be solved in polynomial time only if one can partition the exponential input space into a polynomial‑size subset that still contains at least one satisfying assignment. This partition is called a representative polynomial search partition.

The paper first restates the classic 0‑1 Knapsack formulation (as in Horowitz and Sahni, 1974): given a set S of r real numbers and a target M, choose a binary vector δ∈{0,1}^r such that Σ_i δ_i S_i = M. The total number of possible δ vectors is 2^r, which grows exponentially with the size of the instance.

The central claim is that any algorithm that wishes to run in polynomial time must first locate an RPSP – a polynomial‑size collection of input vectors that is guaranteed to contain at least one solution if a solution exists. The author argues that finding such a partition by exhaustive elimination (e.g., discarding all vectors that contain a particular element, or all vectors that contain a particular pair of elements) still leaves an exponential‑size remainder, and therefore the partition cannot be constructed in polynomial time.

Two “necessary conditions” for a polynomial‑time algorithm are enumerated: (1) the problem’s structure must allow a direct method for generating the RPSP in polynomial time, and (2) a deterministic polynomial‑time solution to SAT must exist, because SAT can be reduced to any NP‑complete problem. The author treats these as mutually exclusive, concluding that no polynomial‑time algorithm for Knapsack can exist.

To formalize the impossibility, the author introduces a self‑named “P = NP Search Partition Theorem,” which essentially states: if an RPSP can be found in polynomial time, then P = NP. The paper then argues that this theorem creates a logical trap – any algorithm that hopes to avoid the trap must already possess the very partition it is trying to discover, which is impossible. Consequently, the only way out would be a polynomial‑time algorithm for SAT, which the author assumes does not exist.

The manuscript proceeds to examine potential loopholes. It distinguishes between instances where the elements of S have a “predictable relationship” (e.g., powers of two) and those where the relationship is random. In the predictable case, the author shows a simple greedy‑like construction that reduces the problem to binary conversion, claiming that this yields a polynomial‑time solution for that special class. However, the author concedes that this does not generalize to arbitrary instances.

For the random case, the author argues that because the relationship between S and M cannot be predicted, no deterministic polynomial‑time algorithm can exist. This is encapsulated in the “Knapsack Random Set Theorem,” which asserts that deterministic Turing machines cannot exploit randomness in the input to achieve polynomial time. No formal probabilistic model or rigorous reduction is provided; the statement remains an intuitive claim.

The paper also analyses the role of the target value M. It observes that the number of integer compositions of M grows as 2^M − 1, which is exponential in the magnitude of M. Therefore, enumerating all compositions (which is equivalent to enumerating all satisfying δ vectors) is infeasible in polynomial time. The author notes that if both S and M were restricted to small whole numbers, the problem could be solved by exhaustive search, but this restriction defeats the purpose of studying the general NP‑complete case.

In the conclusion, the author reiterates that because an RPSP cannot be constructed in polynomial time for the general 0‑1 Knapsack problem, no deterministic polynomial‑time algorithm exists for this problem, and consequently P is a proper subset of NP.

Critical assessment: The manuscript suffers from several fundamental issues. First, the notion of a “representative polynomial search partition” is never rigorously defined; the paper treats it as a black‑box object without specifying how it should be constructed, verified, or how it differs from simply selecting a polynomial‑size subset of inputs. Second, the core argument reduces to a circular statement: “If you could find the partition in polynomial time, you would have solved an NP‑complete problem; therefore you cannot find the partition.” This does not constitute a proof that such a partition cannot exist; it merely restates the difficulty of the original problem. Third, the “P = NP Search Partition Theorem” is introduced without proof and is not linked to any established complexity‑theoretic results. Fourth, the analysis of “predictable” versus “random” instances conflates average‑case considerations with worst‑case complexity, ignoring the fact that NP‑completeness is defined with respect to worst‑case instances. Fifth, the paper contains numerous typographical errors, malformed formulas, and ambiguous notation, which further obscures the logical flow.

Overall, while the manuscript attempts to provide a novel perspective on why the 0‑1 Knapsack problem resists polynomial‑time solutions, it fails to deliver a rigorous, verifiable proof. The central concepts are ill‑defined, the arguments are circular, and the work does not engage with the extensive existing literature on reductions, approximation algorithms, or known hardness results for Knapsack. Consequently, the claim that “P is a proper subset of NP” remains unsubstantiated within this paper.