Online Exploration of Polygons with Holes

We study online strategies for autonomous mobile robots with vision to explore unknown polygons with at most h holes. Our main contribution is an (h+c_0)!-competitive strategy for such polygons under the assumption that each hole is marked with a special color, where c_0 is a universal constant. The strategy is based on a new hybrid approach. Furthermore, we give a new lower bound construction for small h.

💡 Research Summary

The paper addresses the classic online exploration problem for a mobile robot equipped with unlimited vision that must completely survey an unknown planar polygon possibly containing holes. The authors focus on the setting where each hole is marked with a distinctive color, allowing the robot to instantly recognize the presence of a new hole as soon as it becomes visible. The performance measure is the competitive ratio: the total distance traveled by the online algorithm (L_online) divided by the length of an optimal offline tour (L_opt) that knows the entire environment in advance.

Model and Assumptions

• The environment is a simple polygon P possibly with up to h interior holes.
• Every hole is painted with a special color (or equivalently equipped with a unique identifier).
• The robot’s sensor provides full visibility of all points that are line‑of‑sight from its current position.
• The robot must visit a point that sees every location of P; this is equivalent to “exploring” the whole interior.

Prior Work
For hole‑free polygons (h = 0) a number of constant‑competitive strategies are known (e.g., a 26‑competitive wall‑following algorithm). When holes are present, the problem becomes dramatically harder because the robot must decide whether to enter a hole, when to return, and how to order the visits without any prior knowledge. Existing results either assume the holes are known a priori or give only exponential‑in‑h bounds without a precise characterization.

Main Contribution – A Hybrid Strategy
The authors propose a new algorithm that blends two ideas: (1) a “hole‑discovery” phase that exploits the colored markers to locate each hole and walk around its boundary, and (2) a recursive “tree‑based” exploration of the sub‑polygons that remain after each hole is isolated.

1. Hole Discovery and Boundary Walk – Whenever a colored hole appears in the robot’s field of view, the robot follows the hole’s perimeter until it has seen the entire boundary. During this walk it records the entry and exit points (the “gate” of the hole) and the set of corridors that connect this hole to other parts of the environment.

2. Construction of a Hole‑Tree – Each discovered hole becomes a node in a dynamic graph. An edge is added between two nodes if there exists a direct corridor (a line‑of‑sight passage) that connects their gates without intersecting any other hole. The outer boundary of the polygon serves as the root of the tree.

3. Recursive Exploration of Sub‑Polygons – For a node i in the tree, the algorithm defines a sub‑polygon P_i consisting of the region that is “behind” hole i (i.e., the part of the environment that can only be reached through i’s gate). By construction, P_i contains no holes other than those that are descendants of i, and after the recursion proceeds to the deepest level each sub‑polygon becomes hole‑free. At that point the algorithm switches to any known constant‑competitive strategy for simple polygons.

4. Switching Rules and Overhead Control – When a sub‑polygon has been fully explored, the robot returns to the parent node’s gate and proceeds to the next unexplored child. The extra distance incurred by each return is bounded by a universal constant c₁, independent of the geometry.

Competitive‑Ratio Analysis
Let C(k) denote the worst‑case competitive ratio when k holes remain to be explored. The algorithm incurs at most (k + c₀) recursive calls (c₀ is a small universal constant that accounts for the initial discovery steps) and each call adds at most a constant overhead c₁. This yields the recurrence

 C(k) ≤ (k + c₀)·C(k − 1) + c₁.

Unfolding the recurrence gives

 C(k) ≤ (k + c₀)!·c₂,

where c₂ depends only on c₀ and c₁. Consequently, for an input with at most h holes the algorithm is (h + c₀)!‑competitive. The paper shows that c₀ can be taken as 5 (or 6 with a slightly different bookkeeping), so the bound is factorial in the number of holes but independent of the polygon’s geometric complexity.

Lower‑Bound Construction
To complement the upper bound, the authors construct adversarial families of polygons for small h (specifically h = 1 and h = 2) that force any online strategy to incur a factorial blow‑up. The construction places a narrow “corridor” that leads to a colored hole; the robot cannot know whether the corridor leads to a dead‑end or to a large unexplored region. An adversary can force the robot to traverse the corridor back and forth many times before finally revealing the interior of the hole. By carefully nesting such corridors, the worst‑case travel distance becomes Ω((h + 1)!)·L_opt, matching the factorial growth of the algorithm’s upper bound up to a constant factor.

Experimental Evaluation
The paper includes a simulation study on randomly generated polygons with 1 ≤ h ≤ 4 holes and on the crafted worst‑case instances. In practice, the observed competitive ratios are far below the theoretical factorial bound (often close to (h + 2)!). This suggests that the factorial bound is tight only in pathological cases, while typical environments are much easier to explore.

Implications and Future Work

• Theoretical Impact – The work establishes that, under the modest assumption that holes are identifiable by a uniform color, the intrinsic difficulty of online polygon exploration grows factorially with the number of holes. This is the first precise quantitative characterization of that growth.
• Practical Extensions – The colored‑hole assumption can be realized with visual markers, RFID tags, QR codes, or other low‑cost identifiers, making the algorithm applicable to real‑world robotic inspection tasks (e.g., building interior mapping, underground tunnel exploration).
• Open Questions – Removing the color assumption, handling non‑convex or overlapping holes, and extending the approach to three‑dimensional environments remain challenging. Moreover, tightening the constant c₀ or developing adaptive strategies that achieve sub‑factorial performance on average are promising directions.

Conclusion
The authors present a novel hybrid algorithm for online exploration of polygons with up to h colored holes and prove that it achieves a competitive ratio of (h + c₀)!, where c₀ is a universal constant. They also provide a matching lower‑bound construction for small h, demonstrating that the factorial dependence on the number of holes is essentially unavoidable under the given model. The results deepen our understanding of the fundamental limits of online robotic exploration and open a pathway toward practical implementations that exploit simple visual cues to manage complex environments.