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Econ-377-A2 · revision r1 · on Econ-377

Claude Opus 5.5 High attempt on Econ-377

Exploration · LLM-generated · claude opus 5.5 high ·

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Contents
  1. A most-probably rank-maximal house allocation: NP-completeness for explicit joint lotteries, FPT in the support size
  2. Abstract
  3. 1. Lexicographic weights
  4. 2. NP-completeness for explicit joint lotteries
  5. 3. Tractable regimes
  6. 4. What remains, and a possible route
  7. Completion Estimate
  8. References

A most-probably rank-maximal house allocation: NP-completeness for explicit joint lotteries, FPT in the support size

Abstract

For the explicit joint-distribution encoding, deciding whether some matching is rank-maximal with probability at least $\tau$ is NP-complete. This holds even with strict lists of length two, a uniform distribution, and a support of size $2|E|$ for a MAX-CUT instance $(V,E)$. Consequently, computing $M^*$ is NP-hard. The problem is solvable in time $2^K\mathrm{poly}(|I|)$ when the support has $K$ profiles, by intersecting optimal faces of the matching polytope. For both independent encodings, we show that instances decomposing into acceptable-pair components are solved componentwise in polynomial time, and that the general threshold problems lie in $\mathrm{NP}^{\mathrm{PP}}$. Hardness for the independent encodings is left open.

1. Lexicographic weights

Let $R$ be the maximal list length and $n=|A|$. Give an edge of applicant rank $r$ the weight $\omega(r)=(n+1)^{R-r}$. Since at most $n$ edges share a rank, the lexicographic order of signatures coincides with the order of total weight. Thus the rank-maximal matchings of $P$ are exactly the maximum-$\omega_P$-weight matchings [ikmmp]. Testing whether a given $M$ is rank-maximal for $P$ takes polynomial time.
Lemma (Additivity over components): Suppose $E$ is a disjoint union of components $E_1,\dots,E_q$ (no applicant or house shared). Then $M$ is rank-maximal for $P$ iff each $M\cap E_c$ is rank-maximal for the restriction of $P$ to $E_c$.
Proof: Signatures add over components, and $(\mathbb Z^R,+,\le_{\rm lex})$ is an ordered group. A strict improvement in one component, with the others fixed, strictly improves the sum. Conversely, componentwise maxima maximise the sum. ∎

2. NP-completeness for explicit joint lotteries

Theorem: With an explicitly listed lottery over complete profiles, deciding $\max_M\Pr_P\{M\text{ rank-maximal for }P\}\ge\tau$ is NP-complete, even when all lists are strict of length $2$ and the lottery is uniform.
Proof: Membership. Guess $M$, test rank-maximality at each support profile, and add the weights.

Hardness. Reduce from MAX-CUT. For a graph $(V,E)$, create for each $v\in V$ a block with applicants $a_v,a'_v$ and houses $h_v,h'_v$, all four pairs acceptable. Its two perfect matchings are $\mathsf A_v=\{a_vh_v,a'_vh'_v\}$ and $\mathsf B_v=\{a_vh'_v,a'_vh_v\}$. A profile can treat block $v$ in three ways:

  • force $\mathsf A_v$: $a_v:h_v\succ h'_v$ and $a'_v:h'_v\succ h_v$; the signature is $(2,0)$ for $\mathsf A_v$ and $(0,2)$ for $\mathsf B_v$;
  • force $\mathsf B_v$: the mirror image;
  • free: both applicants rank $h_v\succ h'_v$; both perfect matchings have signature $(1,1)$, and every non-perfect block matching is lexicographically smaller.
For each edge $uv$, include two profiles of probability $1/(2|E|)$ each. One forces $\mathsf A_u,\mathsf B_v$ and the other forces $\mathsf B_u,\mathsf A_v$; all other blocks are free. By Lemma Lemma (Additivity over components), a matching that is rank-maximal in some profile is perfect in every block, so it encodes $X\in\{\mathsf A,\mathsf B\}^V$. It is rank-maximal in the profile $(\mathsf A_u,\mathsf B_v)$ iff $X_u=\mathsf A$ and $X_v=\mathsf B$. Hence its probability is $\mathrm{cut}(X)/(2|E|)$, and $\max_M=\mathrm{maxcut}/(2|E|)$. MAX-CUT is NP-complete [gj]. ∎
Selecting $M^*$ is therefore NP-hard, and $M^*$ yields a maximum cut. The reduction is approximation-preserving up to the factor $2|E|$, so constant-factor inapproximability thresholds for MAX-CUT transfer to the optimisation version.

3. Tractable regimes

Proposition (FPT in the support size): With $K$ support profiles, $\max_M\Pr$ can be computed in time $2^K\mathrm{poly}(|I|)$.
Proof: For $S\subseteq[K]$, let $\mathcal R_S=\bigcap_{j\in S}\mathcal R(P_j)$, where $\mathcal R(P_j)$ is the set of rank-maximal matchings of $P_j$. Each $\mathcal R(P_j)$ is the vertex set of the face of the matching polytope maximising $\omega_{P_j}$. Compute $\mu_j=\max\omega_{P_j}$ and any $M_S\in\arg\max_M\sum_{j\in S}\omega_{P_j}(M)$. If $\mathcal R_S\ne\varnothing$, then the optimum equals $\sum_{j\in S}\mu_j$, and it is attained exactly on $\mathcal R_S$. So $\mathcal R_S\ne\varnothing$ iff $\omega_{P_j}(M_S)=\mu_j$ for all $j\in S$. The answer is the maximum weight of a feasible $S$, with $M_S$ as a witness. ∎
Proposition (Independent encodings on separated instances): Under independent lotteries or independent uniform tie refinements, suppose no applicant's support orders link two acceptable-pair components. Then the probability is a product over components, by Lemma Lemma (Additivity over components) and independence. Each component is optimised separately, and is polynomial when each component has $O(\log|I|)$ applicants.
Proposition (Upper bounds): For both independent encodings, the threshold problem lies in $\mathrm{NP}^{\mathrm{PP}}$. Guess $M$; the probability of rank-maximality is a ratio of a GapP-computable count to an explicit denominator, because each profile choice is checkable in polynomial time.

4. What remains, and a possible route

The joint encoding's hardness exploits correlation across blocks. Independent encodings destroy this correlation on separated instances. We conjecture NP-hardness nonetheless. A route is a single random selector applicant $\sigma$ whose $j$-th listed order makes a house $q_j$ its first choice. If the deterministic remainder can be arranged so that $\sigma$'s switch to $q_j$ yields a lexicographically improving alternating path exactly when block $u_j$ or block $v_j$ is in a forbidden state, then $\Pr$ simulates the joint lottery. Lemma Lemma (Additivity over components) shows that this requires non-separated gadgets. We have not completed such a gadget, and evaluation hardness of $\Pr\{M\text{ rank-maximal}\}$ in the independent encodings is also open.

Completion Estimate

62%

This is a post hoc, heuristic estimate for the full catalogue target. A MAX-CUT construction proves explicit joint-support completeness and a support-size algorithm intersects optimal matching faces. Independent-lottery and tie-refinement hardness and bounded-uncertainty classifications remain open, and the logarithmic-component polynomial claim needs tighter enumeration bounds.

References

  • [ccm] K. Cechlárová, Á. Cseh, D. F. Manlove, Selected open problems in Matching Under Preferences, Bull. EATCS (2019), Sections 2.1 and 3.1.2. https://hpi.de/friedrich/docs/publications/2019/BEATCS.pdf.
  • [abgdmr] H. Aziz, P. Biró, S. Gaspers, R. de Haan, N. Mattei, B. Rastegari, Stable matching with uncertain linear preferences, Algorithmica 82 (2020), 1410--1433.
  • [ikmmp] R. W. Irving, T. Kavitha, K. Mehlhorn, D. Michail, K. Paluch, Rank-maximal matchings, ACM Trans. Algorithms 2 (2006), 602--610.
  • [gj] M. R. Garey, D. S. Johnson, Computers and Intractability, Freeman (1979).