Construction of an NP Problem with an Exponential Lower Bound

In this paper we present a Hashed-Path Traveling Salesperson Problem (HPTSP), a new type of problem which has the interesting property of having no polynomial time solutions. Next we show that HPTSP is in the class NP by demonstrating that local information about sub-routes is insufficient to compute the complete value of each route. As a consequence, via Ladner’s theorem, we show that the class NPI is non-empty.

💡 Research Summary

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The paper introduces a novel variant of the classic Traveling Salesperson Problem (TSP) called the Hashed‑Path Traveling Salesperson Problem (HPTSP). In HPTSP each complete tour is encoded as a string by concatenating the vertices and edge weights, and this string is then fed into a cryptographic hash function (the authors use SHA‑1 as a concrete example). The hash output, a fixed‑size bit‑string, serves as the “cost” of the tour; the decision version asks whether there exists a Hamiltonian cycle whose hash value is less than or equal to a given bound m.

The authors first review the properties of cryptographic hash functions—particularly the avalanche effect and the Strict Avalanche Criterion (SAC)—and argue that because a single bit change in the input causes roughly half of the output bits to flip, any partial information about a sub‑tour is essentially useless for predicting the final hash. Consequently, they claim that traditional algorithmic techniques that rely on composability of sub‑solutions (dynamic programming, branch‑and‑bound, approximation schemes) cannot be applied to HPTSP. They illustrate the idea with a tiny 4‑city example, listing every possible permutation, its corresponding “full route” string, and the SHA‑1 hash of that string. By showing that the only way to find the optimal (lowest‑lexicographic‑order) hash is to enumerate all 4! = 24 tours, they extrapolate that for an n‑city instance the search space grows as n! and cannot be pruned without examining an exponential number of candidates.

On the complexity side, the authors argue that HPTSP is in NP because, given a candidate tour, one can compute its hash in polynomial time and compare it to m. However, they do not provide a reduction from any known NP‑hard problem to HPTSP, nor do they prove that HPTSP is NP‑hard. Their main claim is that HPTSP is not NP‑complete, and therefore, assuming P ≠ NP, Ladner’s theorem guarantees the existence of problems in NP that are neither in P nor NP‑complete (the class NPI). By presenting HPTSP as such a problem, they attempt to demonstrate that NPI is non‑empty.

Critically, the paper’s arguments lack formal rigor. The statement that “local information about sub‑routes is insufficient to compute the complete hash value” is intuitive but not quantified; it does not constitute a lower‑bound proof in the sense of complexity theory. The reliance on cryptographic security properties (one‑wayness, collision resistance) does not automatically translate into deterministic time lower bounds. Moreover, the verification that a given tour satisfies the hash bound is indeed polynomial, but the hardness of finding such a tour remains unproven. No polynomial‑time reduction from a known NP‑hard problem (e.g., Hamiltonian Cycle, Subset Sum) to HPTSP is presented, and the authors do not show that HPTSP is at least as hard as any NP‑complete problem.

The use of Ladner’s theorem is also misplaced. Ladner’s result is conditional on P ≠ NP and guarantees the existence of intermediate‑complexity problems, but it does not identify a concrete problem unless one can prove that the problem lies strictly between P and NP‑complete. Since the paper does not establish HPTSP’s NP‑hardness, the application of Ladner’s theorem is unsupported.

In summary, the paper proposes an intriguing idea—using cryptographic hashes to hide the structure of a combinatorial optimization problem—but fails to provide the rigorous reductions, lower‑bound arguments, or formal definitions required to place HPTSP convincingly within the complexity landscape. The claim that HPTSP demonstrates a non‑empty NPI class remains speculative. Future work would need to (1) give a precise, polynomial‑time many‑one reduction from a known NP‑hard problem to HPTSP, (2) develop a formal proof that any algorithm solving HPTSP must examine an exponential number of tours (perhaps via information‑theoretic arguments), and (3) clarify whether the problem is truly outside P while not being NP‑complete. Only then could HPTSP serve as a concrete example of an NP‑intermediate problem.