Solution independence and self-referential instances

Abstract: In this paper, we investigate the hitting set problem and demonstrate that solution independence is the crucial property underlying the construction of self-referential instances. As a special case of the hitting set problem, the vertex cover problem lacks the solution independence property. This distinction accounts for its ability to evade exhaustive search, as correlations among candidate solutions can be leveraged to compress the overall search space. In contrast, the dominating set problem on hypergraphs, which is also a special case of the hitting set problem, satisfies the solution independence property, thereby enabling the construction of self-referential instances. Moreover, we prove that these self-referential instances possess an irreducible property, implying that any algorithm for solving such instances must process nearly the entire graph to yield a correct solution.

• The paper identifies solution independence as the structural property enabling self-referential instances and shows that near-independent candidate solutions can prevent algorithms from deciding solvability without examining nearly the entire input.
• The paper finds that Vertex Cover has exponentially strong positive correlations among candidate solutions, while size-ln n dominating sets in random d-uniform hypergraphs are asymptotically independent for d ≥ 3.
• The paper proves that random-hypergraph Dominating Set is irreducible with high probability: no induced subgraph of size n^c for any constant c < 1 can reliably determine whether a valid dominating set exists.

This paper investigates the structural property that governs whether a combinatorial problem admits self-referential instances—instances whose solvability cannot be distinguished by any algorithm inspecting less than the entire input. The authors, Zhou, Wang, Wang, and Xu, work within the hitting set framework and isolate solution independence (the near-independence of candidate solutions under joint probability) as the property that enables such constructions. They contrast two special cases of hitting set: the vertex cover problem, whose candidate solutions exhibit strong positive correlations and which consequently admits subexhaustive algorithms, and the dominating set problem on random $d$-uniform hypergraphs, whose candidate solutions are asymptotically independent and for which they prove an irreducibility result.

Background: self-reference and hardness

Self-reference has long served as a vehicle for impossibility results, from the Liar Paradox through Gödel's incompleteness theorems and the undecidability of the halting problem, to Hartmanis and Stearns' time hierarchy. A recent line of work extends this methodology to computational hardness: Xu and Zhou for SAT [(2605.02174) references], Li et al. for Clique, and Zhou for Dominating Set constructed infinite families of instances that are closed under symmetric mappings, in exact analogy to a Gödel sentence whose unprovability is equivalent to itself. The solvability of such instances is provably indistinguishable to any non-exhaustive algorithm, reflecting the gap between the part (what an algorithm inspects) and the whole (the instance).

The unifying property behind these constructions is solution independence: for two randomly drawn candidate solutions $S_1, S_2$ of size $k$, the probability that both are valid is nearly the product of the individual probabilities. The authors argue, via a "Coin and Box Problem" analogy, that independence forces exhaustive inspection: after examining $n-1$ boxes, the state of the last remains entirely undetermined. Correlations, by contrast, allow search-space compression—Vertex Cover, for instance, admits substantially faster algorithms than naive enumeration. The paper positions this as a complement to parameterized complexity (FPT versus W[1]-hard/W[2]-hard classifications) and to SETH-based lower bounds, offering a direct, self-contained mechanism for proving the necessity of exhaustive search.

Vertex cover: correlated solutions

For the vertex cover problem on $G(n,p)$, the authors compute the dependence ratio for two candidate covers $S_1, S_2$ of size $k$ with overlap $i$:

$$\frac{\Pr(S_1, S_2 \text{ are covers})}{\Pr(S_1 \text{ is a cover})\,\Pr(S_2 \text{ is a cover})} = (1-p)^{-\binom{n-2k+i}{2}}.$$

In both sparse ($p = c/n$, with $S_1, S_2$0) and dense ($S_1, S_2$1, with $S_1, S_2$2) regimes, this ratio is $S_1, S_2$3 for overlaps with $S_1, S_2$4—exponentially large. The same phenomenon extends to $S_1, S_2$5-uniform random hypergraphs with ratio $S_1, S_2$6. These positive correlations among overlapping candidate covers violate the near-independence condition required by the second moment method, and the authors identify them as a fundamental obstruction both to second-moment analysis and, more importantly, to the construction of self-referential instances: correlations can be exploited to compress the search space, so no argument forcing exhaustive inspection can go through.

Dominating set on hypergraphs: independence and irreducibility

The positive result concerns weak dominating sets in the random $S_1, S_2$7-uniform hypergraph $S_1, S_2$8 for $S_1, S_2$9, reformulated as a hitting set instance over the family $k$0 where $k$1 contains $k$2 and all vertices sharing a hyperedge with $k$3. The hyperedge probability is calibrated so that the expected number $k$4 of dominating sets of size $k$5 satisfies $k$6 for a constant $k$7, which forces $k$8. A key structural contrast with the $k$9 case of prior work is precisely this scaling of $n-1$0 as $n-1$1; the proof strategy, however, is largely analogous.

Solution independence. The dependence ratio for two candidate dominating sets simplifies to

$n-1$2

so two random $n-1$3-dominating sets are asymptotically independent. Via a careful second-moment computation showing $n-1$4, the authors establish $n-1$5, and hence

$n-1$6

a probability bounded strictly away from zero. The complementary regime is handled by introducing quasi-dominating sets: $n-1$7-subsets dominating all but exactly one vertex. A second-moment argument over a case analysis of the locations of the two undominated vertices shows $n-1$8 while $n-1$9, yielding quasi-dominating sets with high probability whenever no dominating set of size $G(n,p)$0 exists.

Irreducibility. The main theorem states that for $G(n,p)$1, the dominating set problem on $G(n,p)$2 is irreducible with high probability: no induced subgraph on $G(n,p)$3—indeed, no subgraph on at most $G(n,p)$4 vertices for any constant $G(n,p)$5—contains sufficient information to decide whether a dominating set of size $G(n,p)$6 exists.

The proof constructs self-referential instance pairs via a local symmetry mapping. In the unique-dominating-set case, since $G(n,p)$7 for any subgraph $G(n,p)$8 of order $G(n,p)$9, the solution lies almost surely outside the inspected region. With high probability there exists a vertex $S_1, S_2$0 dominated by exactly one vertex $S_1, S_2$1 through exactly one hyperedge (the authors compute $S_1, S_2$2 conditional on domination). Swapping the hyperedges $S_1, S_2$3 and $S_1, S_2$4 for $S_1, S_2$5 and $S_1, S_2$6—a transformation touching only vertices outside $S_1, S_2$7 and preserving all degrees and the hyperedge count—destroys $S_1, S_2$8 as a dominating set while creating no new one (the probability of any extension of $S_1, S_2$9 to a dominating set is $k$0). The quasi-dominating case applies the mapping in reverse, flipping a no-instance into a yes-instance. In both directions, the induced subgraph $k$1 is unchanged, so any algorithm restricted to $k$2 cannot distinguish the two cases. This establishes the theorem and, with it, the claim that any correct algorithm must process nearly the entire hypergraph in the worst case.

Limitations and open questions

Several caveats bear directly on the strength of the results. The analysis is confined to the random hypergraph ensemble $k$3 at a specific edge probability $k$4; irreducibility is established with high probability over this distribution, not for arbitrary or worst-case hypergraphs, and the paper does not address whether analogous self-referential constructions extend to deterministic instance families. The uniqueness bound $k$5 is a positive-probability rather than a high-probability guarantee for the yes-case, though the quasi-dominating existence is high probability. The proof of Lemma on the non-existence of dominating sets is deferred to an appendix and explicitly relies on techniques imported from the $k$6 treatment in prior work, so the novelty lies in adapting rather than re-deriving that machinery. Finally, the paper concedes that while solution independence cleanly separates Vertex Cover from Dominating Set here, a quantitative theory relating the degree of correlation among candidate solutions to achievable search-space compression—beyond the qualitative observation that 2-SAT is more strongly correlated than 3-SAT—remains undeveloped, and the authors note this gap is one reason a correlation-based proof of P $k$7 NP remains out of reach.

Conclusion

This paper identifies solution independence as the structural property underlying the construction of self-referential instances, demonstrating its absence in vertex cover—where exponentially large positive correlations among candidate solutions enable subexhaustive algorithms—and its presence in the dominating set problem on $k$8-uniform random hypergraphs for $k$9. The resulting irreducibility theorem shows that no sublinear-sized induced subgraph captures the global combinatorial structure of the problem, so any correct algorithm must in the worst case inspect nearly the entire input. The authors connect this mechanism to Cantorian diagonalization, arguing that independence of coordinates is likewise what makes the diagonal argument work, and thereby propose self-reference plus diagonalization as a general framework for proving the necessity of exhaustive search and characterizing the distinguishability boundary of non-exhaustive algorithms.