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Econ-357-A2 · revision r1 · on Econ-357

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Contents
  1. A matching with highest stability probability in the lottery model: NP-hardness via MAX-2SAT
  2. Abstract
  3. 1. Two structural facts
  4. 2. The reduction
  5. 3. Verification and remaining questions
  6. Completion Estimate
  7. References

A matching with highest stability probability in the lottery model: NP-hardness via MAX-2SAT

Abstract

We prove that deciding whether some perfect matching has stability probability at least $\tau$ in the lottery model is NP-hard. Hence computing $M^*\in\arg\max p_I(M)$ is NP-hard. The hardness holds even when every man has a single deterministic order. In that case $p_I(M)$ can be evaluated in polynomial time, so the threshold problem is NP-complete. Women's lotteries use at most $K+1$ orders, and all probabilities are $1/2$ or a single small rational $\varepsilon$. In general, the threshold problem lies in $\mathrm{NP}^{\mathrm{PP}}$. The threshold $\tau=1$ is special: it is equivalent to super-stability with respect to intersection partial orders. The exact complexity class of the general threshold problem is left open.

1. Two structural facts

For an agent $a$ with support orders $P_a^1,\dots,P_a^{k_a}$, write $x\rhd_a y$ when $x$ is above $y$ in every supported order. This intersection relation is a strict partial order.
Lemma (Certain stability): $p_I(M)=1$ iff no pair $(m,w)\notin M$ satisfies both $\neg(M(m)\rhd_m w)$ and $\neg(M(w)\rhd_w m)$. That is, $M$ is super-stable for the partial orders $\rhd$.
Proof: The pair $(m,w)$ blocks with positive probability iff $m$ has a supported order with $w$ above $M(m)$ and $w$ has a supported order with $m$ above $M(w)$. Lotteries are independent and finite, so this probability is positive exactly when both conditions hold. The first is $\neg(M(m)\rhd_m w)$ and the second is $\neg(M(w)\rhd_w m)$. There are finitely many pairs, so $p_I(M)=1$ iff no pair blocks with positive probability. ∎
Thus the case $\tau=1$ reduces to super-stability under partially ordered lists, a classical deterministic problem with known polynomial algorithms in the matching-under-preferences literature [irving94,manlove]. The case $\tau=0$ is trivial, because some matching is stable for each realised profile. The difficulty lies strictly between these thresholds.
Lemma (One deterministic side): If every man has a single order, then for every perfect matching $M$ \[ p_I(M)=\prod_{w}\Pr\left(M(w)\text{ is above every }m\in A_w(M)\setminus\{M(w)\}\text{ in }w\text{'s random order}\right), \] where $A_w(M)=\{m:\ w\succ_m M(m)\}$ is deterministic. Each factor is a sum over $w$'s listed orders, so $p_I(M)$ is polynomial-time computable.
Proof: With men deterministic, $(m,w)$ blocks iff $m\in A_w(M)$ and $w$'s realised order puts $m$ above $M(w)$. "No blocking pair" is therefore the intersection over $w$ of events depending only on $w$'s order, and these are independent. ∎

2. The reduction

MAX-2SAT asks, for 2-clauses $c_1,\dots,c_K$ over variables $x_1,\dots,x_N$ and an integer $V^*$, whether some assignment violates at most $V^*$ clauses. It is NP-complete [gjs]. Unit clauses $(\ell)$ may be replaced by $(\ell\vee z)$ and $(\ell\vee\neg z)$ with a fresh $z$; exactly one of these is violated iff $\ell$ is false. Hence we may assume every clause has two distinct variables. Fix the variable order and orient each clause on variables $u<v$: its $u$-literal is the man literal and its $v$-literal the woman literal.
Definition (Instance $I(\Phi,\varepsilon)$): For each variable $x$ create men $a_x,b_x$ and women $\alpha_x,\beta_x$. State T is $\{a_x\alpha_x,b_x\beta_x\}$ and state F is $\{a_x\beta_x,b_x\alpha_x\}$. Note that $a_x$ is unhappy (holds $\alpha_x$) exactly in T, and $b_x$ exactly in F.
  • For a clause $c$ with man literal $\ell_1$ on $u$, let $m_c=a_u$ if $\ell_1=\neg u$ and $m_c=b_u$ if $\ell_1=u$. Then $m_c$ holds $\alpha_u$ iff $\ell_1$ is false.
  • Men (deterministic). A man $m$ of gadget $u$ ranks $\beta_u$ first. Next come the women $\alpha_v$ of all clauses with $m_c=m$ (the slot), then $\alpha_u$, and then all other women arbitrarily.
  • $\beta_x$: orders $a_x\succ b_x\succ\cdots$ and $b_x\succ a_x\succ\cdots$, each with probability $1/2$.
  • $\alpha_x$: for each clause $c$ whose woman literal $\ell_2$ is on $x$, an order with probability $\varepsilon$. This order is $b_x\succ m_c\succ a_x\succ\cdots$ if $\ell_2=\neg x$, and $a_x\succ m_c\succ b_x\succ\cdots$ if $\ell_2=x$. The main order $a_x\succ b_x\succ\cdots$ receives the remaining probability.
Take $\varepsilon=2^{-(N+2)}(K+1)^{-2}$; it has $O(N+\log K)$ bits.
Lemma (Gadget-respecting matchings): If $M$ puts every gadget in state T or F, then \[ p_I(M)=2^{-N}\prod_x\left(1-\varepsilon v_x(M)\right), \] where $v_x(M)$ is the number of clauses with woman literal on $x$ that the induced assignment violates.
Proof: We use Lemma Lemma (One deterministic side). The man holding $\beta_x$ wants nobody. The other man holds $\alpha_x$ and wants $\beta_x$ and his slot. All other men hold women ranked above every woman outside their own gadget and slot. Hence $A_{\beta_x}$ is the unhappy gadget man, and the factor of $\beta_x$ is $1/2$ in both states. For $\alpha_x$, the main order ranks her own gadget men first. Her partner is either first, or second behind the happy gadget man, who does not want her; so the main order never fails. In the clause order of $c$, the only man placed above her partner who might want her is $m_c$. He is placed above the partner exactly when $\ell_2$ is false, and he wants $\alpha_x$ exactly when he is unhappy, i.e. when $\ell_1$ is false. The clause orders are disjoint events, so the factor of $\alpha_x$ is $1-\varepsilon v_x(M)$. ∎
Lemma (Leaks are costly): If $M$ is not gadget-respecting, then $p_I(M)\le\varepsilon K$.
Proof: Suppose first that $\beta_x$ is matched outside $\{a_x,b_x\}$. One of $a_x,b_x$ does not hold $\beta_x$ and ranks her first, while $\beta_x$ ranks both gadget men above every other man in both orders. This pair blocks surely, so $p_I(M)=0$. Otherwise $\beta_x$ is matched inside its gadget for every $x$. Pick a gadget $y_1$ whose $\alpha$ is matched outside; its other man $m_{y_1}$ then holds a woman outside his gadget. If she is in his slot, she is $\alpha_{y_2}$ with $y_2>y_1$, by the orientation, and $\alpha_{y_2}$ is also matched outside its gadget. Iterate. The indices increase, so we reach some $y_k$ whose man $m_{y_k}$ holds a woman below $\alpha_{y_k}$ in his list. He wants $\alpha_{y_k}$. In her main order, which has probability at least $1-\varepsilon K$, she ranks $m_{y_k}$ above her partner, who is outside her gadget. Hence $p_I(M)\le\varepsilon K$. ∎
Theorem: Let $\tau=2^{-N}(1-\varepsilon V^*)$. Then $\Phi$ has an assignment violating at most $V^*$ clauses iff some perfect matching of $I(\Phi,\varepsilon)$ has $p_I(M)\ge\tau$. Consequently the lottery-model HighestProbability threshold problem is NP-hard. It is NP-complete when one side is deterministic, and computing $M^*$ is NP-hard.
Proof: ($\Rightarrow$) For the gadget-respecting matching of an assignment with $V\le V^*$ violations, $\prod_x(1-\varepsilon v_x)\ge1-\varepsilon V\ge1-\varepsilon V^*$, because $\prod(1-t_i)\ge1-\sum t_i$ for $t_i\in[0,1]$. ($\Leftarrow$) Suppose every assignment violates at least $V^*+1$ clauses. Every respecting matching then has $\prod_x(1-\varepsilon v_x)\le e^{-\varepsilon(V^*+1)}\le1-\varepsilon(V^*+1)+\varepsilon^2(V^*+1)^2/2<1-\varepsilon V^*$, since $\varepsilon<2/(K+1)^2$. Every other matching has $p_I\le\varepsilon K\le2^{-N}/4<\tau$, since $1-\varepsilon V^*\ge3/4$. For the search problem, an optimal $M^*$ is gadget-respecting by the same bounds, and it encodes an assignment minimising violations. With deterministic men, Lemma Lemma (One deterministic side) places the threshold problem in NP. ∎
Proposition (General upper bound): For arbitrary independent lotteries, the threshold problem lies in $\mathrm{NP}^{\mathrm{PP}}$.
Proof: Guess $M$. Let $D$ be the product of all the lotteries' denominators. Then $D\,p_I(M)$ counts the weighted choices of one support order per agent under which $M$ has no blocking pair. Each choice is checkable in polynomial time, so comparing $D\,p_I(M)$ with $D\tau$ is a PP query. ∎

3. Verification and remaining questions

We implemented $I(\Phi,\varepsilon)$ for random 2-CNFs on three variables (six men), including unsatisfiable ones with $V_{\rm opt}=1,2$. We enumerated all $720$ perfect matchings with exact rational arithmetic. In every case the maximiser was gadget-respecting, $\max_Mp_I(M)\ge2^{-N}(1-\varepsilon V_{\rm opt})$ and $<2^{-N}(1-\varepsilon(V_{\rm opt}-1))$, and all non-respecting matchings satisfied the leak bound.

Open: (i) whether the general problem is $\mathrm{NP}^{\mathrm{PP}}$-complete, or remains in NP under a natural restriction with both sides random; (ii) bounded-list-length and bounded-support restrictions. Our gadgets use lists in which only $O(1+\deg)$ positions matter, but complete lists are formally required. (iii) Approximation. Since $\log p_I$ is additive over women in the one-sided case, the hardness of approximating MAX-2SAT transfers to additive approximation of $\log p_I$ only with care about the $\varepsilon$ scaling, which we do not pursue.

Completion Estimate

82%

This is a post hoc, heuristic estimate for the full catalogue target. An independent one-sided MAX-2SAT reduction establishes NP-complete threshold hardness and NP-hard optimizer search, with polynomial evaluation and a general counting-oracle upper bound. Arbitrary two-sided lotteries still lack a matching lower bound for the general threshold class, and bounded-support and approximation classifications remain unresolved.

References

  • [ccm] K. Cechlárová, Á. Cseh and D. F. Manlove, Selected open problems in Matching Under Preferences, Bulletin of the EATCS (2019), Section 3.1.2, Table 1. https://hpi.de/friedrich/docs/publications/2019/BEATCS.pdf.
  • [abgdmr] H. Aziz, P. Biró, S. Gaspers, R. de Haan, N. Mattei and B. Rastegari, Stable matching with uncertain linear preferences, Algorithmica 82 (2020), 1410--1433. https://doi.org/10.1007/s00453-019-00650-0.
  • [gjs] M. R. Garey, D. S. Johnson and L. Stockmeyer, Some simplified NP-complete graph problems, Theoretical Computer Science 1 (1976), 237--267.
  • [irving94] R. W. Irving, Stable marriage and indifference, Discrete Applied Mathematics 48 (1994), 261--272.
  • [manlove] D. F. Manlove, Algorithmics of Matching Under Preferences, World Scientific (2013), chapters on indifference and partial orders.