| paper | d4-journals-mss-2020-0033 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | high |
| authors would recognise it | yes |
Theorem 1 is a structural characterization, while the running-time and FPT discussion appears only in Remark 5 and is explicitly left beyond the paper's scope. Continuous TSP-Support Recognition asks the same predicate of the finite support, so changing the masses does not make it a computational result about a continuous population.
fails bit a — no named computational result to mirror
The proposed question covers recognition of whether the positive support is tree-based single-peaked, as characterized by Theorem 1; it leaves out the paper's structural and mechanism-design discussion.
I find no qualifying computational anchor in this paper. Theorem 1, proved here, is a structural characterization: for a domain \(D_X\) of uniquely topped total preorders, it is equivalent that (i) the set \(\overline{F_A^{D_X}}\) of total, well-defined consistent selections of tree-admissible paths is nonempty; (ii) some domain-maximal consistent partial selection \(f\) induces a betweenness \(B_T(f)\) satisfying CAT, OPC, and NPC; and (iii) \(D_X\) is tree-based single-peaked. Moreover, when these conditions hold, every domain-maximal consistent partial selection is total, and every induced betweenness satisfies CAT, OPC, and NPC. That is a characterization, not a named claim that a problem is in P, hard, or FPT. Proposition 1 is a cited characterization of tree-betweenness, also not a computational result. Remark 5 gives an adapted algorithm’s running time and suggests an FPT parameterization, but it is an unnumbered remark. I would not present any of these as the required named complexity anchor.
The strongest positive-side candidate, if the goal is to propose a mirror anyway, is Continuous TSP-Support Recognition. An instance consists of a finite alternative set \(X\), a finite set \(T\) of uniquely topped total preorders on \(X\), and binary-rational masses \(\mu_t>0\) summing to \(1\). The question is whether there is a tree on \(X\) whose tree-betweenness makes every type \(t\) with positive mass single-peaked in the paper’s nonstrict sense. A solution is such a tree, or a NO answer if none exists. The decision variable is the tree; there is no objective, because the paper’s recognition question has none.
A plausible regime is a large public consultation over a fixed menu of policy options: many residents submit the same complete ranking, so the number of residents is much larger than the number \(\tau=|T|\) of distinct rankings. The type is the complete preference preorder; its mass is the fraction of residents holding it. The mirror asks whether the observed ballot support shares the compromise structure studied in the paper. The authors could plausibly recognize this as an application of their recognition question, although it treats the observed support as the domain of interest.
The expected computational direction is essentially inherited from finite-domain recognition. Remark 5 reports \(O(\tau(m-1)!m^2)\) time for the adapted algorithm, with \(m=|X|\), and suggests FPT in \(m\). This is useful context, not a numbered complexity anchor. The masses do not affect the answer beyond identifying the positive support, so the mirror supplies no new tractability claim and does not turn the paper’s structural theorem into a continuous-optimization result.
That is also the weakest point: the paper studies sets of admissible preferences, not population proportions, and this candidate’s weights are mathematically inert. It covers the recognition aspect around Theorem 1, not the paper’s broader mechanism-design motivations or a proved computational classification. One natural follow-up would be a genuinely mass-sensitive question—such as the minimum mass of voters to exclude to leave a tree-single-peaked support—but that would be a new problem, not an anchor from this paper.
There is no numbered computational result here to continuize. Theorem 1 is a structural characterization of tree-based single-peaked domains; Proposition 1 characterizes tree betweenness. Remark 5 does report a running time for an adapted recognition algorithm, \(O(n(m-1)!m^2)\), and suggests an FPT parameterization, but neither is a theorem classifying a computational problem. That is a real limit on what this paper offers the programme as an anchor.
The proposed Continuous TSP-Support Recognition is the closest candidate, but its input and question reduce exactly to the paper’s domain-recognition problem. For a distribution \(\mu\), the answer is determined by \(\operatorname{supp}(\mu)\): \(R(\mu)=R(\operatorname{supp}(\mu))\). Changing every mass while keeping the support fixed changes nothing; any finite domain in the paper can be given arbitrary positive rational masses without changing the answer. This is not merely a case where an existing algorithm happens to settle a new continuous question. The proposed question is the same support predicate the paper already studies. A large consultation with repeated ballots is a plausible high-multiplicity setting, but it does not make the recognition problem continuous in any computationally relevant way.
The stronger mass-sensitive proposals—such as finding the largest mass share whose rankings fit one tree, or the least mass to exclude to make the remaining support tree-based single-peaked—do use the population distribution. They are also different optimization problems, not mirrors of Theorem 1 or of a named complexity result. The paper’s guarantees depend on a shared domain containing all relevant preferences; it gives no quantitative guarantee for a society that is “mostly” in such a domain. Excluding voters changes the electorate, while retaining everyone outside the fitted subset leaves the paper’s guarantees unsupported. A useful voting application would need a further model explaining what the fitted mass means for the rule applied to the whole society. Adding costs for changing preferences would require another modeling choice still.
The negative case is strongest against the actual recognition anchor: it is a restatement on the paper’s existing support object, with inert weights. It is weaker as a claim that no related population problem could ever be worthwhile. A mass-weighted notion of approximate single-peakedness might make a good independent project, but this paper does not supply the computational result or the mechanism-level justification that would make it a continuous mirror of its contribution.
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.