| paper | d4-conf-aaai-2016-0536 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | high |
| authors would recognise it | yes |
Answer: Polynomial-time solvable · discrete side: polynomial · adjudication: proved
full proof & adversarial review →
Corollary 2
statement extracted from the paper’s text layer
Given a finite activity set \(A\), oriented pair set \(E\), and a rational distribution \(\mu\) over \(\tau\) complete report types \(\theta\), each with rational log-reports \(r_\theta^{ab}\) for \((a,b)\in E\), compute normalized qualities \(q\) minimizing \(\sum_\theta \mu_\theta\sum_{(a,b)\in E}|q_a-q_b-r_\theta^{ab}|\), and return the induced consistent tradeoffs \(t^{ab}=e^{q_a-q_b}\).
A society is a rational distribution \(\mu\) over complete response-sheet types with rational log-reports. Choose activity qualities \(q_a\), normalized by fixing one quality to zero, to minimize population-weighted log-disagreement; represent the consistent aggregate by \(t^{ab}=e^{q_a-q_b}\).
The compressed regime depends on repeated complete reports; without a finite response menu or repeated template, exact reports across many activity pairs may leave \(\tau\) near the population size.
fatal: False
The mirror covers Corollary 2’s computation of \(\mathrm{DBR}_{\log}\) outcomes; it leaves the paper’s axiomatizations, likelihood interpretation, invariance results, and other rules outside the mirror.
The strongest mirror is the population DBRlog aggregation problem. A type is a complete response sheet: for every oriented pair of activities \((a,b)\), it gives the voter’s ideal tradeoff. In a large policy consultation using a common response vocabulary, many residents may submit the same sheet; \(\tau\), the number of distinct sheets, can then be much smaller than the population. Let \(\mu_\theta\) be the fraction of society with type \(\theta\), with rational masses summing to \(1\).
Here is a precise version. For each type and edge, encode the reported tradeoff as a rational log-value \(r_\theta^{ab}\), corresponding to tradeoff \(e^{r_\theta^{ab}}\). Choose real activity qualities \(q_a\), normalize one activity’s quality to zero, and return a minimizer of
\[ \sum_{\theta}\mu_\theta\sum_{(a,b)\in E} \left|q_a-q_b-r_\theta^{ab}\right|. \]
The resulting societal tradeoff is \(t^{ab}=e^{q_a-q_b}\), so it is consistent by construction. Ties allow any minimizing vector. This asks for the same underlying societal tradeoff as the paper’s rule; the decision variable is the aggregate, and the objective is the population’s per-capita total log-disagreement. Rational log-values give the problem a clean exact bit model.
My lead anchor is Corollary 2, proved in this paper: the authors state that an outcome under DBRlog can be computed in polynomial time. In full, the result is that, given a finite activity set and a profile of voters’ pairwise tradeoff reports, one can compute a consistent aggregate tradeoff vector minimizing the sum of logarithmic distances to those reports. The paper obtains this by transforming to additive qualities and solving a linear program. For the population problem above, introduce an epigraph variable for each type and edge. The result is a rational LP with \(O(\tau m^2)\) variables and constraints, hence solvable in time polynomial in \(m\), \(\tau\), and the input bit length \(L\).
This is a particularly direct high-multiplicity regime. If a finite profile has \(n_\theta\) voters of each type, set \(\mu_\theta=n_\theta/n\). The paper’s objective is then \(n\) times the population objective, so the set of optimal aggregates is identical. The compressed input describes masses by type rather than listing all \(n\) voters. With a standardized elicitation form and a modest set of activities, many respondents can share complete response sheets while \(n\gg\tau\). The model preserves the paper’s pairwise reports, consistency requirement, and logarithmic disagreement objective. The authors should recognize it as their aggregation problem with population shares in place of voter counts.
I expect this mirror to be tractable, Class A. Its scope is Corollary 2’s computation of DBRlog outcomes; it does not claim a mirror for every axiomatic result in the paper. The continuous society is the object being optimized over, while the societal tradeoff vector remains the same kind of outcome as in the original problem.
The weakest point is that the paper’s named computational anchor is already a positive algorithmic result, not a discrete hardness result whose hardness might disappear under continuization. Also, the weighted LP is a natural extension of the authors’ sum-of-voter-costs formulation, so this is stronger as a clear, useful population model than as a claim of a new algorithmic phenomenon. It raises further questions about how the aggregate changes under small shifts in type masses, and whether sampling a large population gives reliable approximations to the population DBRlog outcome.
Corollary 2 is a genuine computational anchor, and the proposed mirror is well formed. I can make a case that it is a low priority for the programme, but I can’t honestly argue that no worthwhile mirror survives.
The strongest objection is that the tractable population problem is almost forced by the paper’s existing LP. With \(r_\theta^{ab}\) as the log tradeoff reported by type \(\theta\), the objective is
\[
\sum_\theta \mu_\theta\sum_{(a,b)\in E}|q_a-q_b-r_\theta^{ab}|.
\]
Epigraph variables give an LP of size \(O(\tau m^2)\). There is no new pricing problem or structural obstacle: the population weights simply replace repeated voter terms. For rational masses, clearing denominators turns the instance into a finite profile with \(D\mu_\theta\) copies of each type, and the population objective is the original objective divided by \(D\). This makes the high multiplicity question mathematically sound, but its algorithmic result looks like an immediate compression of the paper’s LP rather than a substantial new application of continuous optimization.
There is also a limit to the proposed scenario. A type is a complete response sheet with roughly \(m^2\) real-valued entries; for independently elicited exact tradeoffs, repeated sheets may be rare, leaving \(\tau\) close to the population size. A shared finite response menu or repeated institutional template would make \(\tau\ll n\) plausible, and that is a defensible mirror. But then the computational gain comes from the chosen vocabulary and repeated templates, not from a broad continuous-population phenomenon. The rational-log encoding also gives a clean exact LP only after choosing log-values as the input representation; with rational tradeoff ratios, the LP coefficients are generally logarithms of rationals.
Those points weaken the case that this is a high-value programme target; they do not defeat its best version. A population using common templates is plausible, the types preserve all information the rule uses, and the authors would likely recognize the weighted objective as their aggregation problem. The stated scope also explicitly accepts high-multiplicity compression, even when the continuous answer is straightforward. So the negative case is weak: Corollary 2 survives, and I would not claim that no worthwhile continuous mirror exists here.
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.