# On Reachable Assignments in Cycles

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Part of the Lecture Notes in Computer Science book series (LNAI,volume 13023)

## Abstract

The efficient and fair distribution of indivisible resources among agents is a common problem in the field of Multi-Agent-Systems. We consider a graph-based version of this problem called Reachable Assignment, introduced by Gourvès, Lesca, and Wilczynski [IJCAI, 2017]. The input for this problem consists of a set of agents, a set of objects, the agent’s preferences over the objects, a graph with the agents as vertices and edges encoding which agents can trade resources with each other, and an initial and a target distribution of the objects, where each agent owns exactly one object in each distribution. The question is then whether the target distribution is reachable via a sequence of rational trades. A trade is rational when the two participating agents are neighbors in the graph and both obtain an object they prefer over the object they previously held. We show that Reachable Assignment is solvable in $$\mathcal {O}(n^3)$$ time when the input graph is a cycle with n vertices.

### Keywords

• Multi-Agent Systems
• Resource allocation
• Polynomial-time algorithm
• Reduction to 2-SAT

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## Notes

1. 1.

If $$\{\sigma _0^{-1}(p),\sigma ^{-1}(p)\} \in E \cap P_{\gamma }(q)$$, then $$\xi _\gamma (p,q)$$ contains two disjoint paths, one with $$\sigma _0^{-1}(p)$$ as endpoint and one with $$\sigma ^{-1}(p)$$.

2. 2.

Note that p is one of the m objects.

3. 3.

In this paper, we represent 2-SAT clauses by implications, equalities, and inequalities. Note that $$p \rightarrow q \equiv (\lnot p \vee q)$$, $$p = q \equiv (p \vee q) \wedge (\lnot p \vee \lnot q)$$, and $$p \ne q \equiv (p \vee \lnot q) \wedge (\lnot p \vee q)$$.

4. 4.

We use the notation $$q \ne d(p,e)$$ for some object q to avoid case distinctions. Since d(pe) is precomputed, this clause is equivalent to $$\lnot q$$ if $$d(p,e)=1$$ and q otherwise.

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Correspondence to Matthias Bentert .

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### Cite this paper

Müller, L., Bentert, M. (2021). On Reachable Assignments in Cycles. In: Fotakis, D., Ríos Insua, D. (eds) Algorithmic Decision Theory. ADT 2021. Lecture Notes in Computer Science(), vol 13023. Springer, Cham. https://doi.org/10.1007/978-3-030-87756-9_18

• DOI: https://doi.org/10.1007/978-3-030-87756-9_18

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