The $(2,1)$-Gapped Consecutive-Ones Property Problem is NP-complete
Nothing has been published against this problem here, and nobody has checked anything. That is the ordinary condition of an open problem, not a defect in the record.
Statement
Given a binary matrix , decide whether its columns can be permuted so that every row contains at most two blocks of 1s and, if it contains two blocks, they are separated by at most one 0. The claimed theorem proves that this -Gapped Consecutive-Ones Property decision problem is NP-complete.
Context
The manuscript claims a polynomial-time reduction from 3-SAT proving -C1P NP-hard; together with membership in NP, this establishes NP-completeness and closes the sole unresolved case from the earlier classification. It also implies NP-completeness of the equivalent completion problem.
The proof package further shows that, within its specific nested-prefix/internal-local gadget architecture, no 3-OR gadget exists with at most six internal columns; at seven columns at least three local rows are required, and all optimal three-row gadgets form one symmetry class. These optimality claims are architecture-specific and are not needed for the NP-completeness result. Independent verification remains pending.
The last open parameter case of a 2009 complexity classification with a real comparative-genomics motivation, open seventeen years. Specialist, and the resolution is self-published with no independent check yet - held at the specialist floor above the week-old problems.
People
Projects
Nobody is running a project on this. A project is a stated goal, a thread, and one thing somebody else could do. It takes a title, one sentence on what would count as progress, and that one task.
Interest
Nobody has said they are looking at this. A mark here is a statement about you, not a claim on the problem: you set it, you clear it, and it blocks nobody.
Attempts
No person has examined this. 1 attempt is published here and nothing has been checked against it at all. Saying whether the mathematics holds is the most useful thing anybody can do on this page.
GPT-5.6 Sol High produced the central NP-completeness construction: a polynomial-time reduction from 3-SAT using ordered slots, two-column variable gadgets, and a seven-column clause gadget implementing Boolean OR. The associated account reports that the solution emerged in roughly 3–4 model messages, with prompts written by the model for itself. The proof package includes exhaustive finite checks of the clause gadget and a separately implemented verifier/cross-check.
Reviews
0 human reviews · 0 machine checksNo person has reviewed this attempt. It has not been checked at all.
Endorsements
0 endorsementsNo one has endorsed this attempt. An endorsement is a person stating that they checked this version and believe it is correct. None has been recorded — which is information, not an omission.
Discussion of this attempt
no comments
Discussion
Nothing has been said about this problem yet. Discussion is for questions about the statement, pointers to prior work and objections to an attempt. It is not review: a review is a verdict recorded against one version of one attempt, and it is counted separately.
Reading every thread is open to everyone. Posting needs an account with posting rights — sign in to check yours.