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Econ-396 · economics · computer science · Computational economics

Parameterized reachability of allocations by improving swaps

LLM status Open unverified

Statement

MM-10 |  Matching markets and allocation

Research agenda formalization

Definitions and assumptions

Let $G=(A,F)$ be an undirected network of $n$ applicants with $n$ houses, strict preferences over acceptable houses, and a bijective acceptable initial allocation $M_0$. A legal swap between adjacent $a,b$ replaces their houses by one another's houses and requires \[ M(b)\succ_a M(a),\qquad M(a)\succ_b M(b). \] No other assignment changes. Write $M_0\leadsto_G M_*$ if there is a finite legal-swap sequence ending at the specified acceptable bijection $M_*$. Every applicant's house strictly improves whenever she participates. Hence every such sequence has at most $n(n-1)/2$ swaps, since each applicant changes house at most $n-1$ times.

Formal question

For specified parameters $\kappa(I)$, classify the decision problem $M_0\leadsto_G M_*$ as fixed-parameter tractable or parameterized-hard. Fixed-parameter tractability means a running time $f(\kappa(I))|I|^c$ with $c$ independent of the parameter. Candidate parameters include the network's treewidth, maximum acceptable-list length, a bound on the total number of swaps, and a bound on each applicant's number of swaps. Classify combinations of parameters when individual ones are insufficient.

Scope and status

The source asks for parameterized study of the NP-hard matching-reachability cases. The parameter list above makes that program concrete and is not claimed to be a catalog of individually published unresolved cases. In particular, the source already records polynomial-time matching reachability on trees and therefore on paths; these are baselines rather than open subproblems. A bounded sequence length adds an explicit budget constraint, while treewidth changes the structural instance class. These distinctions matter when transferring known results concerning reachability of one house, which is a different decision problem.

References for the source of the problem

Katarína Cechlárová, Ágnes Cseh, and David F. Manlove (2019), Selected open problems in Matching Under Preferences, Section 3.1.3, Theorem 5, Table 2 and final paragraph: https://hpi.de/friedrich/docs/publications/2019/BEATCS.pdf. Laurent Gourvès, Julien Lesca, and Anaëlle Wilczynski (2017), Object Allocation via Swaps along a Social Network, IJCAI 2017: https://doi.org/10.24963/ijcai.2017/31.

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 2

Econ-396-A2 · Exploration · LLM-generated · claude opus 5.5 high ·

Claude Opus 5.5 High attempt on Econ-396

Parameterized reachability of allocations by improving swaps: a mover kernel, interval structure and the one-swap case

Abstract

Let $D=\{a:M_0(a)\ne M_*(a)\}$ be the set of movers. We prove that every legal sequence from $M_0$ to $M_*$ uses only swaps between adjacent movers. During the sequence, each mover holds only houses in its interval $[M_0(a),M_*(a)]$ of its preference list. Consequences: (i) a polynomial kernel with at most $2k$ agents for the parameter $k$ = total swap budget; (ii) fixed-parameter tractability in $|D|$, in time $O(|D|!\,|D|^2)$, and in $k$; (iii) a polynomial algorithm when each agent may swap at most once, together with an exact characterisation; (iv) an $O^*(\prod_a|I_a|)$ algorithm, which is FPT in the combined parameter $(|D|,\lambda)$, where $\lambda$ is the maximum interval length, and which also bounds each agent's number of swaps by $|I_a|-1$. The source records polynomial time on trees. We leave treewidth $\ge2$, alone or combined with $\lambda$, open.

1. Movers and intervals

For a mover $a$, let $I_a=\{h:\ M_0(a)\preceq_a h\preceq_a M_*(a)\}$. For a non-mover, $I_a=\{M_0(a)\}$.
Lemma (Mover lemma): Let $M_0=N_0,N_1,\dots,N_s=M_*$ be a legal swap sequence. Then (a) every swap is between two adjacent agents of $D$; (b) $N_t(a)\in I_a$ for all $a,t$; (c) agent $a$ participates in at most $|I_a|-1$ swaps.
Proof: Each participant strictly improves, and nobody else changes house. Hence $a$'s house sequence is non-decreasing in $\succ_a$ and strictly increases at each of $a$'s swaps. If $a$ ever participates, then $N_s(a)\succ_aM_0(a)$, so $a\in D$; this gives (a). The sequence starts at $M_0(a)$, ends at $M_*(a)$ and never decreases; this gives (b). Each swap moves $a$ strictly up within $I_a$; this gives (c). ∎
Corollary (Kernel): For the problem "reach $M_*$ with at most $k$ swaps", answer no if $|D|>2k$. Otherwise delete all non-movers and restrict lists to $\bigcup_{a\in D}I_a$. The result is an equivalent instance with at most $2k$ agents, $O(k^2)$ edges and lists of length at most $2k$.
Proof: Each swap involves two agents, and every mover participates in some swap; so $|D|\le2s\le2k$. By the mover lemma, deleting non-movers loses no legal swap of any sequence reaching $M_*$. Restricting lists preserves the comparisons used by swaps whose houses lie in the intervals. Every house used lies in an interval, by (b). ∎
Theorem (FPT algorithms): $M_0\leadsto_GM_*$ is decidable in time $O(\Lambda\cdot|D|^2)$, where $\Lambda=\#\{\text{bijections }N:D\to M_0(D)\text{ with }N(a)\in I_a\}\le\min(|D|!,\prod_{a\in D}|I_a|)$. A witness sequence of minimum length is produced. In particular, the problem is FPT in $|D|$, in $k$, and in $(|D|,\lambda)$ with $\lambda=\max|I_a|$.
Proof: By the mover lemma, run breadth-first search over the $\Lambda$ admissible allocations of $D$'s houses. From each state, try the at most $|D|^2$ adjacent mover pairs, keeping a swap if it is legal and stays admissible. Every legal path to $M_*$ stays inside this graph, and conversely every path in it is legal. ∎

2. One swap per agent

Proposition: Suppose each agent may participate in at most one swap. Then $M_0\leadsto M_*$ iff the permutation $\pi=M_0^{-1}\circ M_*$ on $D$ is a product of disjoint transpositions $(a\,b)$ with $ab\in F$, $M_0(b)\succ_aM_0(a)$ and $M_0(a)\succ_bM_0(b)$. This can be checked in linear time. If the cap is removed, $M_*$ with this structure is still reachable, by performing the transpositions in any order.
Proof: With one swap each, mover $a$ swaps once and must receive $M_*(a)$ directly from its partner $b$. Then $M_*(a)=M_0(b)$ and $M_*(b)=M_0(a)$, so $\pi$ consists of such transpositions. Disjoint transpositions commute, and each is legal at $M_0$. ∎
Remark (Two swaps per agent): With a cap of two, a mover may pass through one intermediate house, and the order of swaps then matters. For example, on a triangle a 3-cycle of $\pi$ needs two transpositions in a specific order. We do not know whether the two-swap case is polynomial in general graphs. The BFS of Theorem Theorem (FPT algorithms) runs in polynomial time when each agent's interval has at most $3$ houses and the movers split into connected pieces of bounded size. Without the second condition the state space is exponential.

3. Structural parameters

The source records polynomial time on trees, hence on paths and stars [glw,ccm]. Proposition Proposition and Theorem Theorem (FPT algorithms) are independent of graph structure. Two consequences are worth separating. First, the swap-budget parameter has a polynomial kernel (Corollary Corollary (Kernel)), so budget-constrained variants are never harder than their kernelised core. Second, $|D|$ dominates the budget: any yes-instance has $k\ge|D|/2$. For treewidth, we note that the intervals give a local view: on a tree decomposition, a bag's state is the current house of each bag agent, from at most $\lambda^{w+1}$ possibilities. But legality depends on the global order of swaps across bags, so a standard DP over bags would have to synchronise time. We therefore do not obtain an algorithm FPT in $(w,\lambda)$, and we pose it as the central open question. Is $M_0\leadsto M_*$ FPT, or W[1]-hard, parameterized by treewidth plus maximum interval length?

4. Summary of classification

ParameterStatus
total swaps $k$FPT, polynomial kernel ($\le2k$ agents)
movers $|D|$FPT, $O(|D|!\,|D|^2)$
$(|D|,\lambda)$FPT, $O(\lambda^{|D|}|D|^2)$
at most 1 swap per agentpolynomial
treespolynomial [glw]
treewidth $\ge2$; treewidth $+\lambda$; max list length aloneopen here

Completion Estimate

62%

This is a post hoc, heuristic estimate for the full catalogue target. Mover intervals give exact state-space search, a swap-budget reduction and a polynomial one-participation characterization. Treewidth and bounded-list combinations remain unclassified, and the kernel needs houses of deleted nonmovers removed to justify its claimed list-size bound.

References

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Econ-396-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-396

Parameterized reachability of allocations by improving swaps: a research attempt

Target and outcome

Status: unresolved overall; exact parameterized subcases established. The specified target allocation, strict preferences, and bilateral improving swaps together permit a useful reduction: anyone whose initial and target houses agree cannot participate in a successful sequence. This yields fixed-parameter tractability in the number of displaced applicants, hence in a total-swap budget. It also gives polynomial algorithms when each applicant may swap at most once or has at most two acceptable houses. The full classification by treewidth, list length, and participation bounds is not settled here.

The primary IJCAI record [swaps] treats reachability, individual object acquisition and Pareto efficiency as distinct tasks. The present arguments concern a specified complete target, as in the catalogue. They do not transfer automatically to the problem of finding any reachable efficient allocation, whose eventual participants are unknown.

The fixed-agent lemma

Let $M_0,M_*$ be acceptable bijections. Define \[ D=\{a:M_0(a)\ne M_*(a)\},\quad k=|D|,\quad H_D=M_0(D). \] For every $a\in D$, reject immediately unless $M_*(a)\succ_a M_0(a)$.
Lemma: In any legal sequence from $M_0$ to $M_*$, every participating applicant belongs to $D$. The houses held by $D$ remain exactly $H_D$.
Proof: An applicant's house strictly improves each time she participates. By transitivity and strictness, after one participation she can never again hold her original house. Thus $a\notin D$ cannot participate. With every other applicant fixed, all swaps occur inside $D$ and only permute their initially endowed houses. As both endpoint allocations are bijections and agree on the complement, $M_*(D)=H_D$ as well. ∎

Delete applicants outside $D$, delete their houses and incident edges, and restrict each remaining preference list to $H_D$, retaining the original relative order and acceptability. This loses no feasible successful sequence. Conversely every reduced-instance sequence is a legal original sequence by keeping the deleted agents fixed. Houses lying strictly below an applicant's initial house or strictly above her target house can additionally be removed from that applicant's admissible intermediate assignments; neither could occur on an increasing trajectory ending at the target.

A finite-state algorithm and its certificate

Enumerate acceptable bijections between $D$ and $H_D$; there are at most $k!$. Starting at the reduced $M_0$, perform breadth-first search. For every current assignment and every induced network edge $ab$, generate its transposition exactly when both agents prefer the other's current house. Store a predecessor and the generating edge when a state is first reached. Stop when the target is reached or the queue is exhausted.
Proposition: The algorithm decides target reachability in $k!\,k^{O(1)}+|I|^{O(1)}$ time and produces a shortest sequence when one exists. In particular it is fixed-parameter tractable in $k$.
Proof: Every generated edge is a legal swap, so predecessor paths certify acceptance. By the lemma, every successful original path lies in the enumerated state space. Induction on its length shows that breadth-first search reaches each of its states, proving completeness. There are at most $k!$ states and at most $\binom k2$ swap candidates per state; preference comparisons use precomputed ranks. Copying or indexing assignments contributes only a polynomial in $k$. Initial reading and reduction are polynomial in the original binary input size. Breadth-first search gives a shortest path because every edge has unit length. ∎

For $k=0$ the empty sequence succeeds. For $k=1$ a bijection cannot differ at exactly one position, so a correctly formed input never reaches that case. The graph of allocations has no directed cycle: the sum of agents' preference-rank improvements strictly increases at each edge. The algorithm does not require this observation for correctness, but it prevents mistaken claims about beneficial cycles or the need for arbitrarily long certificates.

The total-swap budget has a small kernel

Suppose at most $\ell$ swaps are permitted. Every one of the $k$ displaced agents must participate, and each swap involves exactly two agents; thus $k\leq2\ell$ is necessary. If it fails, reject. Otherwise use the reduced instance and accept exactly when the breadth-first-search distance is at most $\ell$.

The reduced graph and $k$ preference lists can be relabeled using $O(k^2\log k)$ bits, plus a capped budget. Each applicant can change to at most $k-1$ distinct better houses, giving at most $k(k-1)/2$ swaps in any reduced sequence. Replace $\ell$ by the smaller of $\ell$ and this bound. The resulting encoding has size polynomial in $\ell$ after the immediate rejection rule. Its running time is at most $(2\ell)!\,\ell^{O(1)}+|I|^{O(1)}$, with the usual constant-time convention at $\ell=0$. This is an exact, albeit impractical for large budgets, fixed-parameter algorithm. Merely branching over all original network edges for $\ell$ rounds would instead give an exponent depending on $\ell$ and would not establish this conclusion.

One participation per applicant and short acceptable lists

If every applicant may swap at most once, successful swaps have disjoint endpoints. The permutation taking initial houses to target houses must therefore consist of disjoint transpositions and fixed points. This necessary condition is sufficient provided every target transposition is an edge of $G$ and both participating applicants strictly improve. Execute those disjoint swaps in any order; their houses and legal status do not interact. All conditions can be checked in polynomial time.

If every acceptable list has length at most two, each applicant can improve at most once, so the preceding criterion solves unrestricted reachability for that class. This argument assumes unacceptable houses cannot be held as intermediate assignments, precisely as in the catalogue. It does not prove an algorithm for list length three: those agents may take two improving steps, allowing chains of interdependent swaps.

For example, three applicants initially holding $a,b,c$ can reach $(b,c,a)$ through swaps $12$ then $23$ when applicant 1 prefers $b$ to $a$, applicant 2 has $c\succ a\succ b$, and applicant 3 prefers $a$ to $c$. The second applicant must participate twice. If her acceptable list omitted the intermediate house $a$, the first swap would be illegal; final improvements alone would not suffice.

What still needs proof

The displacement parameter depends on both endpoints. Small treewidth or short lists need not bound $k$, so the factorial algorithm says nothing about FPT for those parameters alone. A bound on each applicant's participation likewise permits many distinct participants. Conversely, ordinary token-swapping hardness cannot be imported without checking strict improvements, since the lemma forbids temporary movement of a correctly placed agent. Full classifications require reductions respecting these monotonicity constraints or dynamic programs with sufficient history information at graph separators. No claim of literature priority is made for the elementary parameterized deductions above.

Completion Estimate

66%

This is a post hoc, heuristic estimate for the full catalogue target. The fixed-agent reduction proves witness-producing reachability algorithms for displaced applicants and total swap budget, including a polynomial kernel and length-two-list case. Treewidth, longer lists, higher per-applicant participation caps and parameterized hardness classifications remain unresolved.

References

  • [swaps] L. Gourvès, J. Lesca and A. Wilczynski (2017), Object Allocation via Swaps along a Social Network, IJCAI, 213--219. Primary proceedings record and abstract inspected: https://www.ijcai.org/proceedings/2017/31. The parameterized proofs above are supplied here rather than attributed to the abstract.

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