d4-journals-jair-2026-0058

The strongest mirror I see is the paper’s \(t\)-approval hardness result. Theorem 1, proved in this paper, says that for every fixed \(t\ge 3\), finding a minimum-cost constructive \(t\)-approval-TNC strategy is NP-hard. The underlying problem takes an election and a preferred candidate \(p\), and seeks a sequence of demotion operations that makes \(p\) a co-winner; in the unpriced model, each operation costs one. The paper’s Lemma 1 gives the decision form used in the proof: for fixed \(t\ge 3\), deciding whether a strategy with at most \(tq\) demotions exists is NP-hard. The proof reduces from a restricted exact-cover-by-3-sets problem, showing the construction for \(t=3\) and extending it to each fixed \(t\ge4\). That is a substantive hardness anchor, not just an algorithmic lead.

My lead mirror is Continuous \(t\)-Approval-TNC. Fix \(t\ge3\). An instance consists of \(m\) candidates, a target \(p\), and a society supported on \(\tau\) complete rankings \(r_1,\ldots,r_\tau\), with binary-rational masses \(\mu_i\ge0\) summing to \(1\). A mass of agents with the same ranking can be split among different campaign sequences. Demoting a candidate by any valid number of positions on mass \(x\) costs \(x\), the mass-weighted version of one unpriced operation. The question is to find a finite aggregate plan of such demotions minimizing total cost, subject to \(p\) being a co-winner afterward. The final score of candidate \(c\) is the mass whose resulting ranking puts \(c\) in the top \(t\).

This is a direct high-multiplicity version of the paper’s problem: rankings, demotion operations, the scoring rule, the constructive goal, and the per-operation cost are all retained. The population is the continuous object; the campaign actions are applied to portions of it. A plausible regime is a large election with, say, hundreds of thousands or millions of voters but only hundreds or thousands of recurring complete ranking types. Each type is complete for this unpriced problem: its members have the same ranking and face the same operation costs. A campaign that can target a fraction of a large audience makes fractional mass a natural quantity.

There is a compact exact formulation. For each initial ranking \(r_i\) and each size-\(t\) set \(S\), let \(y_{i,S}\) be the mass ending with \(S\) as its approved set. Let \(w_i(S)\) be the least-preferred member of \(S\) in \(r_i\), and let \(\rho_i(c)\) be \(c\)’s rank there. The minimum number of operations per unit mass needed to make \(S\) the top \(t\) is \(\kappa_i(S)=\rho_i(w_i(S))-t\): demote the candidates above \(w_i(S)\) that are not in \(S\). Thus the mirror asks to minimize \(\sum_{i,S}\kappa_i(S)y_{i,S}\), subject to \(\sum_S y_{i,S}=\mu_i\) for every \(i\), \(y_{i,S}\ge0\), and \(\sum_{i,S}y_{i,S}(\mathbf{1}[p\in S]-\mathbf{1}[c\in S])\ge0\) for every \(c\ne p\).

Although this LP has exponentially many set variables, its pricing problem is polynomial. For a given initial ranking and dual weights on candidates, enumerate which candidate \(w\) is the least-preferred member of \(S\). Its cost is then fixed; among candidates ranked above \(w\), choose the \(t-1\) with the best reduced-cost coefficients. Trying each \(w\) takes polynomial time in \(m\). The standard separation-to-optimization machinery therefore gives an exact algorithm polynomial in \(m,\tau,L\), where \(L\) is the bit length of the masses. I expect this mirror to be Class A: tractable. This is a concrete case where the population relaxation changes the computational question: the discrete reduction relies on choosing whole set-voters, while the continuous plan can split each type’s mass.

The scope is deliberately narrow: it mirrors Theorem 1’s constructive, unpriced \(t\)-approval result. It does not continuize outcomes, time, or probabilities, and it makes no claim about the paper’s Borda, candidate-specific price, or destructive results. It raises useful follow-ups: what integrality gap or additive rounding guarantee relates the continuous optimum to an election with \(N\) individual voters, and which pricing extensions preserve the polynomial pricing oracle?

The weakest point is that a fractional campaign can allocate, for example, \(0.3\) of a ranking type to one sequence and \(0.7\) to another. That is not an exact finite-election strategy when individual voters are indivisible. It is the intended high-multiplicity relaxation, but the bridge back to discrete campaigns needs to be established. The mirror remains plausible because it preserves the authors’ underlying intervention and asks what happens when the audience is measured in shares—the same kind of demographic targeting that motivates their model.