On the existence of polynomial-time algorithms to the subset sum problem

This paper proves that there does not exist a polynomial-time algorithm to the the subset sum problem. As this problem is in NP, the result implies that the class P of problems admitting polynomial-time algorithms does not equal the class NP of problems admitting nondeterministic polynomial-time algorithms.

💡 Research Summary

The paper under review claims to have proved that no polynomial‑time algorithm exists for the classic Subset‑Sum problem, and from this it concludes that the complexity classes P and NP are distinct. The author begins by recalling that Subset‑Sum is a well‑known NP‑complete problem, typically reduced from SAT, and therefore any breakthrough on its deterministic complexity would have immediate implications for the P versus NP question.

The core of the argument is built on three successive constructions. First, the author introduces a “compression hypothesis”: if all input numbers are bounded by a constant C, then the set of all possible subset sums can be represented in O(n·C) space, suggesting that the exponential blow‑up of 2ⁿ subsets might be avoided. This hypothesis, however, only holds for artificially restricted instances; the general Subset‑Sum problem permits arbitrarily large integers, and consequently the number of distinct sums can be exponential in n. The paper does not provide a rigorous justification for extending the compression claim to unrestricted inputs, leaving a critical gap.

Second, the author proposes a “mid‑value search” algorithm. By computing the minimum possible sum L and the maximum possible sum U, the algorithm selects the midpoint M = (L+U)/2 and determines whether M is larger or smaller than the target T, thereby halving the search interval in a binary‑search‑like fashion. The decisive step requires answering the question “Does there exist a subset whose sum equals M?” – a subproblem that is exactly the Subset‑Sum problem itself. The paper treats this subproblem as if it could be resolved in polynomial time, but this is precisely the obstacle the proof is meant to overcome. Consequently, the algorithm is circular: it assumes the existence of a polynomial‑time Subset‑Sum solver to construct a polynomial‑time solver.

Third, the author invokes the definition of NP via nondeterministic Turing machines (NTMs), emphasizing that an NTM can “guess” a correct subset and verify it in polynomial time. The author argues that a deterministic Turing machine (DTM) would need to simulate all nondeterministic branches, incurring exponential cost, and therefore cannot solve Subset‑Sum in polynomial time. While this observation is true, it does not constitute a proof that no clever deterministic algorithm exists; it merely restates the known relationship between deterministic and nondeterministic computation. A genuine P≠NP proof must demonstrate that for at least one NP‑complete problem, every possible deterministic algorithm necessarily requires super‑polynomial time, not that a particular naïve simulation is inefficient.

In the final section the author attempts to combine the previous points into a reductio ad absurdum: assuming a polynomial‑time Subset‑Sum algorithm leads to a contradiction with the compression hypothesis and the mid‑value search, hence such an algorithm cannot exist, implying P≠NP. However, because the compression hypothesis fails for general inputs and the mid‑value search is circular, the contradiction is illusory. The paper therefore does not meet the rigorous standards required for a proof of P≠NP.

Overall, the manuscript offers an interesting narrative and highlights common misconceptions about how NP‑completeness interacts with algorithm design. Its main contributions are pedagogical: it illustrates why naïve attempts to “compress” the exponential search space or to apply binary search to subset sums are insufficient without a genuine polynomial‑time decision procedure. Nevertheless, the technical core contains unproven assumptions, circular reasoning, and a lack of formal complexity‑theoretic analysis. As such, the claimed proof is not accepted by the community, and the P versus NP question remains open. Future work would need to address these gaps, perhaps by developing provable lower bounds for specific NP‑complete problems or by constructing a framework that can rule out entire classes of deterministic algorithms, rather than relying on unverified heuristics.