Evacuation of rectilinear polygons

We investigate the problem of creating fast evacuation plans for buildings that are modeled as grid polygons, possibly containing exponentially many cells. We study this problem in two contexts: the confluent'' context in which the routes to exits remain fixed over time, and the non-confluent’’ context in which routes may change. Confluent evacuation plans are simpler to carry out, as they allocate contiguous regions to exits; non-confluent allocation can possibly create faster evacuation plans. We give results on the hardness of creating the evacuation plans and strongly polynomial algorithms for finding confluent evacuation plans when the building has two exits. We also give a pseudo-polynomial time algorithm for non-confluent evacuation plans. Finally, we show that the worst-case bound between confluent and non-confluent plans is 2-2/(k+1).

💡 Research Summary

The paper studies evacuation planning for buildings modeled as orthogonal (rectilinear) grid polygons, where each unit grid cell initially contains a person. Two evacuation flow models are considered. In a confluent flow, every cell has a unique successor direction, meaning all people passing through that cell head toward the same exit; this corresponds to static signage or guidance that directs everyone uniformly. In a non‑confluent flow, cells may have multiple possible successors, allowing individuals to choose different routes; this can yield faster evacuations but requires more complex, dynamic instructions.

The authors first establish hardness results. When the polygon contains holes (enclosed rooms), the problem of finding an optimal confluent flow is shown to be NP‑complete via a reduction from the classic PARTITION problem. The reduction constructs a polygon whose area can be exponential in the number of vertices, yielding only weak NP‑hardness. They then strengthen the result: even for simple (hole‑free) polygons, if the number of exits is allowed to grow linearly with the input size, the problem becomes strongly NP‑complete. This is proved by adapting the right‑tromino tiling reduction of Moore and Robson, which itself is a reduction from Cubic Planar Monotone 1‑in‑3 SAT. Gadgets representing variables, clauses, wires, and turns are embedded in the polygon, and exits are added so that a feasible evacuation with bounded exit‑class sizes corresponds exactly to a valid tiling, and thus to a satisfying assignment.

Despite these hardness results, the paper presents efficient algorithms for important special cases. When the polygon has exactly two exits, an optimal confluent flow can be computed in linear time. The method relies on a rectangular decomposition of the polygon into vertical and horizontal strips, forming an overlay graph ω and its dual ω*. The problem reduces to finding a minimum cut separating the two exit vertices in ω*, which yields two contiguous regions each assigned to one exit. An initial O(n³) algorithm is described and then refined to O(n) by sweeping the polygon and maintaining cumulative area counts, where n is the number of vertices of the polygon (the description size, not the number of cells).

For non‑confluent flows, the authors give a pseudo‑polynomial time algorithm. Let A be the total number of cells (the total number of people) and k the number of exits. Using dynamic programming, they compute, for each time step, the maximal number of people that can have evacuated given the capacity constraints of each exit. The runtime is O(k·A·poly(n)), which is polynomial in the numeric value A but exponential in the input size only if A itself is exponential. Thus the algorithm is practical when the actual number of occupants is given explicitly.

Finally, the paper quantifies the advantage of non‑confluent over confluent flows. It proves that for any polygon with k exits, the worst‑case ratio between the evacuation time of an optimal confluent flow and that of an optimal non‑confluent flow is 2 − 2/(k+1). This bound is tight and mirrors the classic approximation ratio of list‑scheduling for parallel machines. As k grows, the ratio approaches 2, indicating that non‑confluent flows can be at most twice as fast as confluent ones, while for small k (especially k = 2) the improvement can be substantial.

In summary, the paper contributes both complexity-theoretic insights and concrete algorithms for evacuation planning in grid‑based building models. It shows that while optimal confluent evacuation is computationally intractable in general (especially with holes or many exits), it becomes tractable for the practically important case of two exits. Moreover, it provides a pseudo‑polynomial method for the more flexible non‑confluent model and establishes a tight worst‑case performance gap between the two models. These results have direct implications for architectural design, emergency signage planning, and the development of evacuation‑support software.