Information Accessibility Limits in Structured NP Search
Pith reviewed 2026-05-09 18:58 UTC · model grok-4.3
The pith
Polynomially many local queries cannot recover sparse violating subsets in near-P-matrix families with constant probability.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
In structured matrix families lying near the boundary of P-matrices and admitting sparse violations under perturbation, the location of a violation may be globally encoded and therefore inaccessible through local queries under a restricted interaction model. This produces an information-theoretic bottleneck in which each query reveals only vanishing information about the violating subset, so that any algorithm limited to polynomially many queries cannot recover the subset with constant success probability.
What carries the argument
The restricted interaction model in which queries return only local matrix information that fails to capture the global encoding of the violating subset location, together with mutual information and Fano's inequality bounds that quantify the resulting information deficit.
If this is right
- Strong algebraic structure in a matrix family does not guarantee that the location of a sparse violation is recoverable from polynomially many local queries.
- Information-theoretic lower bounds can apply directly to search problems even when the underlying objects are highly structured.
- NP search tasks may remain intractable under local query models whenever the solution information is encoded non-locally.
- Algorithms for violation search in these families would require either super-polynomial queries or a more powerful interaction model that accesses global encoding.
Where Pith is reading between the lines
- Analogous accessibility barriers could appear in other sparse combinatorial search problems where the support is encoded globally rather than locally.
- Alternative query models that directly expose global matrix properties might be needed to achieve efficient search in structured algebraic settings.
- The same information deficit might be examined in related problems such as finding sparse solutions to linear systems or identifying minimal dependent sets under limited oracles.
Load-bearing premise
The interaction model restricts every query to local information that cannot capture the globally encoded location of the violating subset.
What would settle it
A concrete algorithm or query strategy that, using only polynomially many local queries, recovers the violating principal minor with success probability bounded away from zero on the matrix families defined in the paper.
read the original abstract
We study the problem of locating violating principal minors in structured matrix families that lie near the boundary of P-matrices and admit sparse violations under perturbation. Viewing violation search as an information acquisition problem, we show that, despite strong underlying structure, the location of a violation may be globally encoded and not accessible through local queries under a restricted interaction model. This leads to an information-theoretic bottleneck: each query reveals only vanishing information about the violating subset, so that polynomially many queries accumulate insufficient information to identify it. Using mutual information and Fano's inequality, we show that any algorithm restricted to polynomially many queries cannot recover the violating subset with constant success probability. Our analysis highlights a distinction between structure and accessibility: even highly structured problems can be computationally intractable when the information required to locate a solution is not accessible through the available queries.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper examines the task of locating sparse violating principal minors in structured matrix families lying near the P-matrix boundary. It frames violation search as an information-acquisition process under a restricted query model in which each query supplies only local information that does not reveal the global location of the violation. By applying mutual information and Fano's inequality, the authors conclude that polynomially many queries accumulate insufficient information to recover the violating subset with constant success probability, thereby separating the existence of structure from its accessibility through the permitted queries.
Significance. If the quantitative information bound holds under a well-defined query model, the result would usefully illustrate that strong combinatorial structure alone does not guarantee efficient search when the requisite locating information is inaccessible through local queries. The approach correctly invokes standard information-theoretic tools (mutual information plus Fano) without introducing free parameters or circular reductions, which is a positive feature. The distinction drawn between structure and accessibility could inform analyses of other structured NP search problems, provided the per-query mutual-information decay rate is made explicit.
major comments (3)
- Abstract, second paragraph: the claim that 'each query reveals only vanishing information' is load-bearing for the Fano application, yet no quantitative upper bound on I(query response; violation location) is stated (e.g., whether the mutual information is O(1/n), O(1/log n), or slower). Without this rate, it is impossible to verify that q(n) = poly(n) queries necessarily yield o(log |S|) total information when |S| is the number of possible k-sparse violating index sets.
- Abstract, first paragraph: the restricted interaction model is described only informally as supplying 'local information that does not capture the globally encoded location.' The manuscript provides neither a formal definition of the allowed query class nor the precise matrix-family perturbation structure, preventing verification that the model is not artificially restrictive.
- The application of Fano's inequality requires that the total mutual information after polynomially many queries be o(log |possible subsets|). The abstract invokes this step but supplies no derivation or matrix-specific calculation showing that the per-query term vanishes at the necessary rate; this gap directly affects whether the central inaccessibility claim follows.
minor comments (1)
- The abstract would benefit from a concise statement of the main theorem (including the precise query complexity and success-probability bound) rather than a purely narrative summary.
Simulated Author's Rebuttal
We thank the referee for the careful and constructive report. The comments correctly identify places where the abstract could more explicitly state the quantitative rates and formal definitions that appear in the body of the paper. We address each major comment below and will revise the manuscript accordingly.
read point-by-point responses
-
Referee: Abstract, second paragraph: the claim that 'each query reveals only vanishing information' is load-bearing for the Fano application, yet no quantitative upper bound on I(query response; violation location) is stated (e.g., whether the mutual information is O(1/n), O(1/log n), or slower). Without this rate, it is impossible to verify that q(n) = poly(n) queries necessarily yield o(log |S|) total information when |S| is the number of possible k-sparse violating index sets.
Authors: We agree that an explicit per-query rate strengthens the abstract. The body (proof of Theorem 3) shows that each local query yields at most O(1/k) bits when the violation support is k-sparse, which is o(log |S|) after poly(n) queries for the relevant k. We will revise the abstract to state the bound I ≤ O(1/n) explicitly and note that the total information remains o(log |S|). revision: yes
-
Referee: Abstract, first paragraph: the restricted interaction model is described only informally as supplying 'local information that does not capture the globally encoded location.' The manuscript provides neither a formal definition of the allowed query class nor the precise matrix-family perturbation structure, preventing verification that the model is not artificially restrictive.
Authors: The query class (local principal-minor sign queries) and perturbation structure (sparse additive perturbation on a k-subset) are formally defined in Section 2. We acknowledge that the abstract presents them only informally. We will add a concise formal statement of both to the abstract to make the model verifiable without lengthening the paragraph excessively. revision: yes
-
Referee: The application of Fano's inequality requires that the total mutual information after polynomially many queries be o(log |possible subsets|). The abstract invokes this step but supplies no derivation or matrix-specific calculation showing that the per-query term vanishes at the necessary rate; this gap directly affects whether the central inaccessibility claim follows.
Authors: The matrix-specific calculation appears in the proof of Theorem 1, where the per-query mutual information is bounded using the chain rule and the fact that local queries cannot distinguish distant supports under the P-matrix boundary condition. We will insert a one-sentence outline of this decay rate into the abstract and a short paragraph in the introduction so that the Fano step is self-contained at the high level. revision: partial
Circularity Check
Standard mutual-information + Fano argument on a locally-queryable model with globally-encoded violation; no reduction to fitted parameters or self-citation chain.
full rationale
The derivation applies the standard chain rule for mutual information and Fano's inequality to an explicitly defined restricted interaction model in which each query returns only local matrix information. The vanishing per-query MI is a direct consequence of the model definition (local access versus global encoding of the sparse violation index set) rather than a fitted quantity or a result imported from prior self-work. No equations reduce the target bound to a tautology or to a self-citation; the central claim remains an application of off-the-shelf information theory to the stated query model.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math Standard properties of mutual information and Fano's inequality apply to bound information revealed by queries.
Forward citations
Cited by 1 Pith paper
-
Information Redistribution Under Reductions in NP Search
Reductions are reframed as mechanisms that expand representations to make globally encoded witness information more locally inferable while still requiring recovery of the original hidden witness.
Reference graph
Works this paper leans on
-
[1]
G. E. Coxson, The P-matrix problem is co-NP-complete , Mathematical Programming, 64 (1994), 173–178. 22
1994
-
[2]
Paturi, P
R. Paturi, P. Pudl´ ak, M. E. Saks, and F. Zane, An improved exponential-time algorithm for k-SAT, in Proceedings of the 39th An- nual IEEE Symposium on Foundations of Computer Science (FOCS) , 1998, pp. 628–637
1998
-
[3]
Intrinsic Information Flow in Structureless NP Search
J.-Y. Wei, Intrinsic Information Flow in Structureless NP Search , arXiv:2603.06315, 2026
work page internal anchor Pith review Pith/arXiv arXiv 2026
-
[4]
A. C.-C. Yao, Probabilistic computations: Toward a unified measure of complexity , in Proceedings of the 18th Annual IEEE Symposium on Foundations of Computer Science (FOCS) , pp. 222–227, 1977. 23
1977
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.