Approximating the least unpopularity factor in house allocation

MM-01 |  Matching markets and allocation

Published question

Definitions and assumptions

Let $A$ be a finite set of applicants, $H$ a finite set of houses, and $E\subseteq A\times H$ the acceptable pairs. Each applicant strictly orders her acceptable houses, all of which are preferred to being unmatched, denoted by $\bot$. A matching is a subset of $E$ with no repeated endpoint; houses have no preferences. Let $\mathcal M(I)$ be the set of matchings, with $M(a)=\bot$ for an unmatched applicant, and define \[ \phi(N,M)=|\{a\in A:N(a)\succ_a M(a)\}|. \] For nonnegative integers $r,s$, set $d(r,s)=r/s$ if $s>0$, $d(0,0)=1$, and $d(r,0)=+\infty$ if $r>0$. Define \[ u_I(M)=\max_{N\in\mathcal M(I)}d(\phi(N,M),\phi(M,N)),\qquad u^*(I)=\min_{M\in\mathcal M(I)}u_I(M). \] The zero-denominator conventions are essential. A matching is popular precisely when $u_I(M)=1$.

Formal question

Determine the optimal polynomial-time approximation guarantee for minimizing $u_I$. In particular, determine the infimum of constants $\rho\ge1$ for which an algorithm outputs $M$ satisfying $u_I(M)\le\rho u^*(I)$ on every instance. If no constant guarantee is achievable, characterize the dependence on $|A|+|H|$.

Scope and status

The source poses improved approximation algorithms and stronger hardness bounds in the one-sided house-allocation setting. It records NP-hardness and exclusion of a $(3/2-\varepsilon)$ approximation, for each $\varepsilon>0$, unless $\mathrm P=\mathrm{NP}$. These are source-dated results. An optimal matching has finite factor: a Pareto-optimal matching admits no comparison with positive numerator and zero denominator. Thus the objective is well defined even though individual matchings can have infinite factor.

References for the source of the problem

Katarína Cechlárová, Ágnes Cseh, and David F. Manlove (2019), Selected open problems in Matching Under Preferences, Bulletin of EATCS 128, Section 3.1.1, Theorem 3 and its concluding paragraph: https://hpi.de/friedrich/docs/publications/2019/BEATCS.pdf. Foundational reference: Robert M. McCutchen (2008), The least-unpopularity-factor and least-unpopularity-margin criteria for matching problems with one-sided preferences, LATIN 2008, pp. 593--604, cited there as reference [71].

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.