| paper | d4-journals-mss-2018-0025 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | high |
| authors would recognise it | unclear |
The numbered results prove asymptotic probability and moment bounds for finite random instances, but assert no algorithm, complexity bound, or hardness result. The paper attributes its only NP-completeness statement to prior work, so the proposed type-mass decision problem is an adjacent extension, not a continuous analogue of a computational result proved here.
fails bit a — no named computational result to mirror
No result proved in this paper is covered. Its results concern asymptotic probabilities, moments, partner ranks, and existence for finite random preference lists.
I do not find an eligible computational anchor in this paper. The numbered theorems concern asymptotic probabilities and moments for finite random instances: for example, Theorem 4.11 bounds the second moment of the number of two-sided e-stable matchings, and Theorem 5.4 does so for one-sided matchings. Neither asserts an algorithmic or complexity result. The NP-completeness claim in the introduction is attributed to Cechlárová and Manlove; it is unnumbered and not proved here. The paper’s \([0,1]\) variables model random preferences, not a continuous population.
There is a plausible *prospective* mirror of the cited one-sided decision problem, but it cannot serve as the required anchor. Let \(T\) be a finite set of roommate types, each with a strict ranking \(\succ_t\) of partner types, and let \(\mu_t\) be the rational mass of type \(t\), with \(\sum_t\mu_t=1\). A matching is a symmetric nonnegative array \(x_{tu}=x_{ut}\) with \(\sum_u x_{tu}=\mu_t\). It is exchange-stable if, whenever \(x_{ac}>0\) and \(x_{bd}>0\), it is not the case that both \(d\succ_a c\) and \(c\succ_b d\). The decision question is whether such an array exists; a solution is the array itself.
A conceivable regime is a large roommate or placement market where many people share a finite set of preference profiles over partner categories, so the number of people is large relative to the number of types. The paper’s cited NP-completeness result motivates asking this question, but it does not establish whether hardness transfers, dissolves, or becomes continuum-specific. The main weakness is that the paper ranks named individuals, while this mirror ranks partner categories; adding identity-level preferences to the types could erase the small-support regime. So this is a candidate research question, not a continuous analogue anchored in a named result of this paper.
I would reject a continuous *computational mirror of this paper*. Its numbered results are about probabilities and moments for finite random preference lists: Theorems 4.11 and 5.4 bound the second moment of the number of e-stable matchings; Theorems 4.10 and 5.2 bound the probability that a doubly stable matching exists. None states an algorithmic or complexity result. The NP-completeness claim belongs to the earlier Cechlárová–Manlove paper, which Pittel cites as motivation.
The proposed mass model is a coherent high-multiplicity matching problem. Given types with rankings over partner types, rational masses \(\mu_t\), and a symmetric matching array \(x\) with row sums \(\mu_t\), one can ask whether the support of \(x\) is exchange-stable. That may be worth studying on its own. But it does not mirror Pittel’s results: those concern random permutations ranking individually named partners, with the population and partner universe growing together. Keeping those identities makes the type space grow with \(n\); replacing them with partner categories changes the preference model and the random-instance distribution. Either way, the paper’s moment and probability results do not become computational questions about a continuous society.
The same gap applies to a proposed continuous version of the cited NP-complete decision problem. It would mirror that prior problem, not a computational result of this paper. And replacing the paper’s count of finite matchings with a count of mass allocations would need a new statistic: fractional allocations can form a continuum, so that count has no direct counterpart.
The honest limit of this negative case is that it cannot establish that every high-multiplicity e-stability question is unworthy. The type-based decision problem is plausible and could merit a separate project. Under the requested standard—anchoring a mirror on a named computational result of this paper—it does not survive.
The adversarial triple: the proponent anchors on up to three named results; the opponent sees that case and must defeat every anchor; the judge decides which case convinced it. These are the pipeline’s own outputs, generated by tools/triple_run.py — no human edited them. The paper’s own text is not reproduced here beyond the quoted statement above.