On the relationship between classes P and NP

In this paper we discusses the relationship between the known classes P and NP. We show that the difficulties in solving problem “P versus NP” have methodological in nature. An algorithm for solving any problem is sensitive to even small changes in its formulation. As we will shown in the paper, these difficulties are exactly in the formulation of some problems of the class NP.

💡 Research Summary

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The paper attempts to shed new light on the long‑standing P versus NP problem by arguing that the core difficulty is methodological rather than purely computational. The author, Anatoly Plotnikov, proposes that the formulation of many NP problems is overly sensitive to small changes, and that this sensitivity is the root cause of the apparent intractability.

1. Formal Model
The author introduces a set‑theoretic framework called a “hereditary system.” A hereditary system is a pair (R, Q) where R is a finite ground set and Q is a family of non‑empty subsets of R that is closed under inclusion: if a set belongs to Q, all its subsets also belong to Q. An NP problem Z is then modeled as a quadruple (R, Q, P, f):

• R – the universe of elementary objects that may appear in a solution,
• Q – the collection of admissible (partial) solutions,
• P – a predicate that decides whether a given subset of R belongs to Q, and
• f – a weight function assigning a numeric value to each admissible solution.

The paper maps three classic NP‑complete problems onto this framework:

• SAT: literals form R; a set of literals is admissible if it contains no complementary pair and together they cover all clauses.
• Hamiltonian Cycle: edges form R; a subset of edges is admissible if it can be extended to a Hamiltonian cycle.
• Maximum Independent Set (MMIS): vertices form R; a subset is admissible if it is an independent set.

2. Sequential Construction
Plotnikov argues that any deterministic Turing machine solving an NP problem must construct a solution step‑by‑step, adding one element of R at a time. He formalizes this as a “sequential method” and proves two elementary theorems: (i) if R₁⊂R₂ then the time to construct R₁ is strictly less than the time to construct R₂, and (ii) every NP problem can be solved by such a sequential method because a deterministic Turing machine can simulate the nondeterministic one. These statements are essentially restatements of the obvious fact that a deterministic simulation proceeds in a linear sequence of configurations.

3. Problems Without Foresight (UF)
The central contribution is the definition of a subclass UF ⊆ NP, called “problems without foresight.” For a given admissible partial solution R₁∈Q, the algorithm must decide whether adding a particular element r∈R\R₁ yields another admissible solution (i.e., whether R₁∪{r}∈Q). If this predicate can be evaluated in polynomial time for every R₁ and every candidate r, the problem is placed in UF; otherwise it is labeled “exponential in nature.”

Three theorems are presented:

• Theorem 3 claims an “if and only if” relationship between being in UF and being able to construct a support solution (a maximal admissible set) in polynomial time. The proof sketches that UF guarantees polynomial‑time extension at each step, and that a support solution contains at most a polynomial number of elements, thus the whole construction is polynomial. Conversely, if a problem is solvable in polynomial time but not in UF, there must exist at least one step requiring exponential time, leading to a contradiction.
• Theorem 4 shows UF is a proper subset of NP (UF⊂NP and UF≠NP) by pointing to the Hamiltonian cycle problem, which requires exhaustive search to verify whether a partial edge set can be extended to a Hamiltonian cycle.
• Corollary 1 concludes P⊂UF and P≠NP, essentially restating the well‑known inclusion P⊆NP while positioning UF as an intermediate class that contains P.

4. Discussion and Conclusions
The author critiques the standard definition of NP for being “redundant” because it allows inclusion of problems whose verification step may itself be exponential. By stripping away those “exponential‑nature” problems, the remaining class UF supposedly captures the truly tractable portion of NP. Plotnikov then suggests that the original P versus NP question should be reformulated as “P versus UF.” He argues that focusing on UF may lead to more realistic expectations about algorithmic breakthroughs, since UF problems admit polynomial‑time construction of a support solution.

However, several issues undermine the paper’s impact:

• The definition of UF hinges on the ability to evaluate the predicate “R₁∪{r}∈Q?” in polynomial time, but this predicate is essentially the same as the verification step in the classic NP definition. The paper does not provide a rigorous separation between the two notions.
• The claim that every NP problem can be solved by a sequential method is trivial; any deterministic Turing machine operates sequentially. The novelty lies not in the observation but in how the sequential steps are chosen, which the paper does not address.
• The “if and only if” proof of Theorem 3 is incomplete. It assumes that a support solution contains at most a polynomial number of elements, which is not guaranteed for all NP problems (e.g., the size of a Hamiltonian cycle is linear, but the number of intermediate admissible subsets can be exponential).
• The paper does not establish whether UF is strictly larger than P; it merely restates P⊂UF without demonstrating any concrete problem known to be in UF but not in P.
• The discussion of reductions is superficial. While the author mentions that any NP problem can be reduced to a UF problem (e.g., SAT), no formal reduction is provided, and the claim that such reductions “reformulate” exponential problems into UF problems is misleading.

In summary, the paper introduces a set‑theoretic formalism for NP problems and defines a subclass UF based on the polynomial‑time evaluability of a one‑step extension predicate. It claims that UF captures exactly those NP problems for which a maximal admissible solution can be built in polynomial time, and that P is a proper subset of UF, thereby offering a “new” perspective on the P versus NP question. Nevertheless, the definitions overlap heavily with existing notions, the proofs contain gaps, and the paper does not deliver a substantive separation between P, UF, and NP. Consequently, while the attempt to reinterpret the P versus NP problem through the lens of problem formulation is interesting, the work falls short of providing rigorous theoretical contributions or practical algorithmic insights.