Satisfying Neighbor Preferences on a Circle.

We study the problem of satisfying seating preferences on a circle. We assume we are given a collection of n agents to be arranged on a circle. Each agent is colored either blue or red, and there are exactly b blue agents and r red agents. The w-neighborhood of an agent A is the sequence of 2w + 1 agents at distance <= w from A in the clockwise circular ordering. Agents have preferences for the colors of other agents in their w-neighborhood. We consider three ways in which agents can express their preferences: each agent can specify (1) a preference list: the sequence of colors of agents in the neighborhood, (2) a preference type: the exact number of neighbors of its own color in its neighborhood, or (3) a preference threshold: the minimum number of agents of its own color in its neighborhood. Our main result is that satisfying seating preferences is fixed-parameter tractable (FPT) with respect to parameter w for preference types and thresholds, while it can be solved in O(n) time for preference lists. For some cases of preference types and thresholds, we give O(n) algorithms whose running time is independent of w.