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Econ-179 · economics · computer science · Computational economics

Output-sensitive enumeration of weakly stable matchings with ties

LLM status Open unverified

Statement

MM-17 |  Matching markets and allocation

Published question

Definitions and assumptions

An SMT instance $I$ has two disjoint sets $U,W$ of size $n$, complete acceptable lists, and a weak order on each list. A matching is perfect. It is weakly stable if no pair outside it has both endpoints strictly preferring each other to their assigned partners. Let $\mathcal S(I)$ be all such matchings, counted as distinct sets of pairs. The tied orders are given explicitly, without enumerating all strict refinements. Input length is denoted by $L$. No quotient by tied-agent permutations or by identical preference profiles is taken.

Formal question

Does there exist an algorithm that lists every member of $\mathcal S(I)$ exactly once, terminates after the final output, and has total running time \[ O\left(p(L)+|\mathcal S(I)|q(L)\right) \] for fixed polynomials $p,q$? If this total-output-polynomial guarantee is achievable, can it be strengthened to polynomial delay and polynomial working space? Otherwise, establish an appropriate complexity-theoretic obstruction to the total-time guarantee itself.

Scope and status

The source asks the total-time question for complete lists with ties. Polynomial delay is a stronger supplementary target and must not be silently identified with total-output-polynomial time. Enumerating stable matchings for every strict tie refinement can generate extensive duplication, so deterministic strict-instance enumeration alone does not establish the requested bound. The survey records efficient enumeration for strict lists and a special two-master-list case; these are established baselines. Merely finding one matching different from a given stable matching is also weaker than complete duplicate-free enumeration. Counting can remain hard even where enumeration is output efficient, because the number of outputs may be exponential.

References for the source of the problem

Katarína Cechlárová, Ágnes Cseh, and David F. Manlove (2019), Selected open problems in Matching Under Preferences, Section 3.2.4, especially the displayed total-time requirement: https://hpi.de/friedrich/docs/publications/2019/BEATCS.pdf. Dan Gusfield (1987), Three fast algorithms for four problems in stable marriage, SIAM Journal on Computing 16(1), pp. 111--128, supplies the strict-preference baseline cited as reference [44]; the master-list result is reference [54].

Common conventions: Source mathematical conventions

Well-defined objects. All probability spaces and measurable variables are specified or inherited from a local statistical model. Expectations used as finite targets require the stated integrability. A decision rule or estimator is measurable with respect to its available information; randomized rules may use an auxiliary independent random variable. Norms without a subscript are Euclidean in finite-dimensional spaces. An optimizer is requested only when attainment is assured; otherwise the question concerns the optimal value and $\varepsilon$-optimal admissible rules for every $\varepsilon>0$.

Identification. A structural model $m\in\mathcal M$ implies an observable law $P_m^{\rm obs}$ and target $\theta(m)$. The target is point identified on $\mathcal M$ if \[ P_{m_1}^{\rm obs}=P_{m_2}^{\rm obs} \quad\Longrightarrow\quad\theta(m_1)=\theta(m_2) \qquad(m_1,m_2\in\mathcal M). \] When this fails, the sharp identified set is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M, P_m^{\rm obs}=P\}. \] Specifying an intervention or estimand does not establish identification. Positivity, exclusion, causal sufficiency, measurement restrictions, and transport assumptions are substantive local inputs, never automatic consequences of notation.

Minimax risk and nontrivial inference. For a statistical experiment with data law $P^{(n)}$, target $\theta(P)$, and nonnegative loss $\ell$, define \[ \mathcal R_n(\mathcal P,\ell) =\inf_{\widehat\theta_n}\sup_{P\in\mathcal P} \E_{P^{(n)}}\ell(\widehat\theta_n,\theta(P)). \] The infimum is over measurable estimators. A sharp rate is a sequence $r_n$ for which $0<c\leq C<\infty$, independent of $n$, satisfies $c r_n\leq\mathcal R_n\leq C r_n$ for all sufficiently large $n$. Squared-error parametric risk is of order $n^{-1}$; the corresponding root-mean-squared error is of order $n^{-1/2}$. These different conventions are distinguished locally. A uniform asymptotic confidence region $C_n$ must satisfy \[ \liminf_{n\to\infty}\inf_{P\in\mathcal P} \Pr_{P^{(n)}}\{\theta(P)\in C_n\}\geq 1-\alpha, \qquad 0<\alpha<1, \] together with an informative diameter or expected-width requirement. Coverage alone permits the vacuous whole parameter space. Testing questions similarly impose both size control and power against a declared separated alternative class.

Binary causal baseline. When an entry explicitly invokes this baseline, one observes independent copies of $O=(X,W,Y)$, with $W\in\{0,1\}$ and potential outcomes $Y(0),Y(1)$. Assume consistency $Y=Y(W)$, no interference, conditional exchangeability $(Y(0),Y(1))\perp W\mid X$, and overlap $\epsilon\leq e(X)\leq1-\epsilon$ almost surely for a fixed $0<\epsilon<1/2$, where \[ e(x)=\Pr(W=1\mid X=x),\qquad m_w(x)=\E[Y\mid W=w,X=x],\qquad \tau(x)=m_1(x)-m_0(x). \] These assumptions identify conditional mean effects almost everywhere under the covariate law. Pointwise or uniform effects require appropriate regularity and a specified support. Continuous treatment, longitudinal data, adaptive experiments, and network interference require their own assumptions in place of this baseline. Bounded outcomes, densities, and smoothness are additional assumptions only where stated.

Function classes. On $[0,1]^d$, $\mathcal H_d^s(L)$ is the isotropic Hölder ball of radius $L>0$: put $k=\lceil s\rceil-1$ and $\gamma=s-k\in(0,1]$, bound derivatives through order $k$, and require the order-$k$ derivatives to be $\gamma$-Hölder with constant at most $L$. Thus integer orders use a Lipschitz final derivative convention. The class $\operatorname{BV}_d(L)$ consists of functions $f\in L^1((0,1)^d)$ whose distributional derivative is a finite vector measure and for which $\|f\|_{L^1}+|Df|((0,1)^d)\leq L$. An $L^\infty$ bound or a particular pointwise version is imposed separately when needed. Sparse and neural-network classes require a declared dictionary or architecture and coefficient bounds.

An anisotropic Hölder class with declared block smoothness indices means coordinatewise Hölder regularity in each block, uniformly in the other blocks, with all corresponding derivative and Hölder bounds at most the stated radius. Derivative orders and the integer-order convention in each block are those just given. Mixed-derivative bounds are additional restrictions only when explicitly stated.

Decision and computational questions. Policy regret compares admissible feasible policies under the same law and constraints. In computational entries, finite data and thresholds have rational encodings; running time is measured in their total bit length. Real-valued equilibrium search uses the explicitly declared algebraic or FIXP encoding. An approximation ratio for maximization is written $\operatorname{OPT}/\operatorname{ALG}$ and for minimization as $\operatorname{ALG}/\operatorname{OPT}$, with zero optima and infeasible instances handled locally. Historical approximation bounds are not automatically current bounds.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-179-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-179

Enumerating weakly stable matchings A tie-parameter delay bound with polynomial space

Problem ID: C78-175. Status: Unresolved; restricted results and reductions.

Target and parameter

The target asks for duplicate-free total-output-polynomial enumeration for complete tied preference lists, with polynomial delay and space as stronger goals. There are $n$ agents on each side. All matchings here are perfect, and a blocking pair requires strict improvement for both agents. The verified source [survey, Section 3.2.4] separates this question from strict-list enumeration and from finding one different matching.

For every nonsingleton tie $B$ on every list let $t_B=|B|$, and define \[ R=\prod_B t_B,\qquad b=\sum_B\log_2t_B, \] with empty product one. We prove delay $R\operatorname{poly}(L)$ and polynomial space. Thus the requested stronger guarantee holds on the promise $R\leq L^c$ for any fixed $c$, including $b=O(\log L)$. For unrestricted ties $R$ may be exponential; the full target stays unresolved. We do not claim this parameterization is new.

A small covering family of strict refinements

Fix a base ordering of the members of each tie. Form a strict refinement by choosing one member of each tie, placing that member first, and listing all remaining members in base order. There are exactly $R$ choices (ties of size two included).
Lemma (Partner-first covering): Every matching stable in any strict refinement is weakly stable in the original tied instance. Conversely every weakly stable matching is stable in at least one of the $R$ refinements just constructed. Both assertions remain valid when one requires the matching to contain an arbitrary fixed set of disjoint pairs $P$.
Proof: An original strict blocking pair stays strict in every refinement, which proves the first assertion. Given a weakly stable $M$, put each agent's partner first in the tie containing that partner. For every other tie choose its base-first member. If $(u,w)\notin M$ blocked in this refinement, neither agent could be tied with her assigned partner: that partner was placed first in the relevant tie. Thus both agents would strictly prefer the other in the original instance, contradicting weak stability. The argument never alters $M$, so it also preserves $P$. ∎

Only the first position in a tie is needed for this coverage. Enumerating all $\prod_B t_B!$ strict refinements is unnecessary for the construction. Coverage alone does not yet control duplicate outputs; the search tree below supplies that control.

A strict-list completion oracle, proved explicitly

Fix a strict refinement and a set $P$ of disjoint forced pairs. First reject if two agents in distinct forced pairs form a blocking pair. Remove all endpoints of $P$. For each remaining agent $a$, examine each forced agent $b$ on the other side who strictly prefers $a$ to $P(b)$. Require that the eventual partner of $a$ be strictly preferred by $a$ to every such $b$. Delete all residual edges violating any requirement at either endpoint. The remaining strict instance can have incomplete lists. Run the usual proposing-side deferred acceptance procedure: an unmatched proposer offers to her best untried acceptable receiver; a receiver keeps her preferred proposal and rejects the other, if any.
Lemma (Forced-pair oracle): The strict complete instance has a stable perfect matching containing $P$ if and only if the residual deferred-acceptance matching is perfect and the initial forced-pair blocking test passed. This can be decided in $O(n^2)$ operations after preference ranks are available.
Proof: Any stable completion must satisfy every deletion requirement: otherwise its residual agent and the corresponding forced endpoint would block. Its residual restriction must also be stable among residual agents. Conversely a perfect residual stable matching plus $P$ has no blocking pair inside $P$, none between $P$ and the residual set by the deletion requirements, and none within the residual set. Pairs deleted from the residual graph cannot block this union either. If an edge $(a,c)$ was deleted because $a$ had to beat a forced endpoint $b$, then the retained partner of $a$ is above $b$, whereas $c$ is not above $b$; strict orders make the retained partner strictly better than $c$.

Deferred acceptance terminates after at most $n^2$ proposals and is stable: a proposer who prefers another acceptable receiver proposed to her earlier and was rejected, and that receiver's held partner only improved thereafter. To justify the perfectness test, prove by induction on rejections that a rejected pair belongs to no stable matching of the residual instance. Suppose receiver $w$ rejects $u$ for $v$. If a stable matching $S$ contained $(u,w)$, all receivers preferred by $v$ to $w$ have previously rejected $v$, so the induction hypothesis forbids their being $S(v)$. Since $w$ is paired to $u$ in $S$, proposer $v$ must be unmatched or assigned below $w$. Then $(v,w)$ blocks $S$, a contradiction. At the end any unmatched proposer has been rejected by every acceptable receiver, and hence is unmatched in every stable matching. Therefore a perfect residual stable matching exists exactly when the deferred-acceptance output is perfect.

All forced-pair comparisons and residual edge deletions use $O(n^2)$ rank comparisons, and the proposal bound gives the claimed complexity. ∎

Corollary (Weak completion oracle): Whether a weakly stable perfect matching contains a specified set $P$ of disjoint pairs is decidable in $O(R n^2)$ time and polynomial space.
Proof: Loop through the $R$ refinements using a mixed-radix counter and apply Lemma Lemma (Forced-pair oracle). Lemma Lemma (Partner-first covering) proves equivalence of a successful strict completion and a weak completion. Only one refinement, one rank array, and one deferred-acceptance state need be stored at a time. ∎

Duplicate-free search and its delay

Order the agents on side $U$ as $u_1,\ldots,u_n$, and the agents of $W$ by labels. A node at depth $j$ records the distinct partners of $u_1,\ldots,u_j$. Its children append an unused labeled partner for $u_{j+1}$. Keep a child precisely when the weak completion oracle says it has an extension. Traverse the surviving tree depth first, outputting the matching at each leaf.
Theorem (Parameterized delay and total time): The search outputs every weakly stable perfect matching exactly once, uses polynomial working space, and terminates. Including initial and final delay, each delay is $O(R n^4+\operatorname{poly}(L))$. Consequently its total running time is \[ O\left(\operatorname{poly}(L)+(|\mathcal S(I)|+1)R n^4\right). \] For a fixed polynomial bound on $R$, it meets the target's polynomial delay, polynomial space, and total-output-polynomial requirements.
Proof: A leaf uniquely specifies the partner of every labeled proposer, so different leaves cannot represent the same matching. Every surviving leaf is weakly stable by the oracle, and every weakly stable matching has all its prefixes accepted, so every such matching is output.

Between consecutive outputs, depth-first search ascends at most $n$ levels and descends at most $n$ levels along surviving edges. At any visited level it tests at most $n$ candidate children. A failed child is discarded immediately and has no explored subtree. A successful child has a leaf descendant, so descending through successful children cannot consume a second large subtree before producing the next output. Thus at most $2n^2$ completion tests occur between outputs. The same bound applies before the first output and after the last one. Multiplying by Corollary Corollary (Weak completion oracle) proves the delay bound. The original complete tied instance has a weakly stable matching by applying deferred acceptance to any strict refinement, but the bound also covers an empty surviving tree under additional forced pairs.

Store the depth-first stack of length at most $n$, the used-partner array, and the polynomial-space oracle. Refinements are regenerated as needed, so no list of previously output matchings is stored. Termination follows from finiteness of the depth-$n$ tree. Summing the delay bounds proves total time. ∎

The exact unresolved gap

The parameter counts local tie choices rather than outputs. There is no argument here bounding $R$ by a polynomial of $L$ or $|\mathcal S(I)|$ on unrestricted instances. A single tie of size $n$ contributes only $n$, while many unrelated ties can multiply the cost. The extension oracle is a sufficient route to polynomial delay; an unrestricted total-output algorithm need not implement that oracle. Accordingly neither general hardness nor a general efficient enumerator has been established. No finite computational checks are substituted for these quantified algorithmic proofs.

References

Completion Estimate

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