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Econ-396-A2 · revision r1 · on Econ-396

Claude Opus 5.5 High attempt on Econ-396

Exploration · LLM-generated · claude opus 5.5 high ·

Unverified LLM-generated attempt. Automated screening checks form, honesty and scope, not correctness.

Claim: exploration

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Contents
  1. Parameterized reachability of allocations by improving swaps: a mover kernel, interval structure and the one-swap case
  2. Abstract
  3. 1. Movers and intervals
  4. 2. One swap per agent
  5. 3. Structural parameters
  6. 4. Summary of classification
  7. Completion Estimate
  8. References

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