Seat Arrangement Problems under B-utility and W-utility

TL;DR

This work extends the seat arrangement problem by introducing two new utilities, $B$-utility and $W$-utility, alongside the traditional $S$-utility, and by considering $1$-dimensional preferences. It provides a comprehensive set of algorithmic and hardness results across four objectives: $MWA$, $MUA$, $EFA$, and $STA$, under these utilities. Key findings include NP-hardness of $MWA$, $MUA$, and $EFA$ for $B$- and $W$-utilities (even with symmetry or strictness) and polynomial-time exchange-stability results for symmetric $B$-utility, with linear-time $1$-D results for paths and cycles. The paper also refines existing $S$-utility results and demonstrates that certain decision problems become tractable on special graph classes (e.g., matching graphs), while establishing hard boundaries via reductions from Partition Into Triangles and Bin Packing. Overall, it delineates the computational landscape of seat arrangement variants and lays groundwork for future IP/CP approaches and explorations of $1$-dimensional preferences.

Abstract

In the Seat Arrangement problem the goal is to allocate agents to vertices in a graph such that the resulting arrangement is optimal or fair in some way. Examples include an arrangement that maximises utility or one where no agent envies another. We introduce two new ways of calculating the utility that each agent derives from a given arrangement, one in which agents care only about their most preferred neighbour under a given arrangement, and another in which they only care about their least preferred neighbour. We also present a new restriction on agent's preferences, namely 1-dimensional preferences. We give algorithms, hardness results, and impossibility results for these new types of utilities and agents' preferences. Additionally, we refine previous complexity results, by showing that they hold in more restricted settings.

Figures (7)

• Figure 1: On the left, subgraph of an example PIT instance, with one triangle from the partition highlighted. On the right, part of an envy-free arrangement, focused on the agents derived from vertex $v_1$.
• Figure 2: Location of the agents transformed from $x_1$ and $x_2$ in the $1$-dimensional space, with $s_1 = 4$ and $s_2 = 2$
• Figure 3: On the left, subgraph of an example PIT instance, with one triangle from the partition highlighted. On the right, part of an envy-free arrangement.
• Figure 4: On the left, subgraph of an example PIT instance, with one triangle from the partition highlighted. On the right, part of an envy-free arrangement. We can see that in the $17$-cycle every agent from $P_i \backslash \{p_{1,1}^{2,3}\}$ is adjacent to their immediate predecessor and successor in $P_i \backslash \{p_{1,1}^{2,3}\}$.
• Figure 5: Positions of the agents in part of the seat graph $G$. Case $1$.

Theorems & Definitions (34)

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