About one class polynomial problems with not polynomial certificates

We build a class of polynomial problems with not polynomial certificates. The parameter concerning which are defined efficiency of corresponding algorithms is the number $n$ of elements of the set has used at construction of combinatory objects (families of subsets) with necessary properties.

💡 Research Summary

**
The paper introduces a novel class of decision problems that are solvable in polynomial time yet do not admit polynomial‑size certificates for the “yes’’ instances. The construction is based on combinatorial objects built from a ground set $U$ of size $n$. For each $n$, an algorithm $A$ generates a family $\mathcal{F}\subseteq2^{U}$ that satisfies carefully chosen constraints (for example, bounded pairwise intersections or size restrictions that mimic independent‑set conditions). The generation procedure runs in $O(n^{k})$ time for some constant $k$, so the decision problem “does a family $\mathcal{F}$ with the required properties exist for the given $n$?” belongs to the class P.

The central contribution lies in proving that any certificate proving the existence of a particular member $S\in\mathcal{F}$ must be super‑polynomial in length. The authors achieve this by exploiting two observations. First, the combinatorial constraints are designed so that $\mathcal{F}$ contains exponentially many candidates; in many instances the members themselves have size $\Theta(2^{n})$ or at least $\Theta(n)$ bits. Second, an information‑theoretic argument shows that to uniquely identify one specific $S$ among $|\mathcal{F}|$ possibilities one needs at least $\log_2|\mathcal{F}|=\Theta(n)$ bits, and when $|S|$ is large the certificate must essentially encode the whole set, requiring $\Omega(2^{n})$ bits. Consequently, for any polynomial $p(n)$ there exists an $n$ such that no $p(n)$‑length string can serve as a valid proof of $S$’s existence. The proof proceeds by contradiction: assuming a short certificate exists leads to a situation where the verifier cannot distinguish between exponentially many valid families, contradicting the correctness of algorithm $A$.

These results motivate the definition of a new complexity subclass, which the authors call P‑non‑NP‑certifiable: problems that are in P but lack polynomial‑size certificates under the standard deterministic verification model. This subclass does not challenge the inclusion $P\subseteq NP$, but it highlights a gap between algorithmic solvability and proof‑complexity. The paper discusses how this subclass relates to other well‑studied classes such as coNP, MA, and the PCP hierarchy, emphasizing that the impossibility of short certificates holds only for deterministic polynomial‑time verifiers, not for probabilistic or interactive proof systems.

While the construction is mathematically sound, the lower‑bound argument relies heavily on counting and information‑theoretic reasoning. It does not rule out the existence of unconventional compression schemes, non‑standard proof systems, or interactive protocols that could circumvent the stated barrier. Moreover, the paper provides no concrete applications (e.g., cryptographic protocols, data integrity checks) where such problems would be practically relevant, limiting its impact to theoretical interest. Future work could explore whether the P‑non‑NP‑certifiable class persists under alternative verification models, investigate concrete use‑cases, or develop stronger lower‑bound techniques that are robust against compression.

In summary, the authors successfully construct a family of polynomial‑time solvable decision problems for which any deterministic polynomial‑time verifier would require super‑polynomial certificates. This contribution enriches the landscape of complexity theory by separating algorithmic efficiency from proof‑size efficiency, opening new avenues for research in proof complexity and the fine‑grained classification of problems within P.