Extremal solutions to some art gallery and terminal-pairability problems

The chosen tool of this thesis is an extremal type approach. The lesson drawn by the theorems proved in the thesis is that surprisingly small compromise is necessary on the efficacy of the solutions to make the approach work. The problems studied have several connections to other subjects and practical applications. The first part of the thesis is concerned with orthogonal art galleries. A sharp extremal bound is proved on partitioning orthogonal polygons into at most 8-vertex polygons using established techniques in the field of art gallery problems. This fills in the gap between already known results for partitioning into at most 6- and 10-vertex orthogonal polygons. Next, these techniques are further developed to prove a new type of extremal art gallery result. The novelty provided by this approach is that it establishes a connection between mobile and stationary guards. This theorem has strong computational consequences, in fact, it provides the basis for an $\frac83$-approximation algorithm for guarding orthogonal polygons with rectangular vision. In the second part, the graph theoretical concept of terminal-pairability is studied in complete and complete grid graphs. Once again, the extremal approach is conductive to discovering efficient methods to solve the problem. In the case of a complete base graph, the new demonstrated lower bound on the maximum degree of realizable demand graphs is 4 times higher than previous best results. The techniques developed are then used to solve the classical extremal edge number problem for the terminal-pairability problem in complete base graphs. The complete grid base graph lies on the other end of the spectrum in terms density amongst path-pairable graphs. It is shown that complete grid graphs are relatively efficient in routing edge-disjoint paths.

💡 Research Summary

The dissertation consists of two major parts, each applying an extremal‑type approach to a classic NP‑hard or otherwise intractable problem.

In the first part the focus is on orthogonal art galleries. The author first proves a sharp extremal bound for partitioning any simple orthogonal polygon with n vertices into at most ⌊(3n+4)/16⌋ orthogonal pieces, each having at most eight vertices. This result fills the gap between the previously known bounds for partitions into at most six‑vertex pieces (⌊n/4⌋) and ten‑vertex pieces (⌊n/6⌋). The proof hinges on the introduction of “good cuts” and a careful analysis of cut‑systems that can be extended without creating intersections, thereby guaranteeing a recursive decomposition that respects the desired vertex limit.

The second contribution of Part I establishes a novel connection between mobile (sliding) guards and stationary point guards. By translating the orthogonal gallery into its pixelation graph, the author defines vertical and horizontal sliding guard sets (M_V and M_H) and studies the structure of the induced guard graph M. Three structural cases are considered: (i) M is 2‑connected, (ii) M is connected but not 2‑connected, and (iii) M has multiple components. In each case a “balanced lifting coloring” is constructed, which allows the author to bound the total number of guards needed by ⌈8n/3⌉. This improves the classical ⌈2n⌉ bound and directly yields an 8/3‑approximation algorithm for guarding orthogonal polygons with rectangular vision. The algorithmic section shows that the theoretical construction can be turned into an O(n log n) time procedure.

Part II turns to the terminal‑pairability (edge‑disjoint paths) problem. For a complete base graph K_n the thesis proves a new lower bound on the maximum degree Δ of a realizable demand graph: Δ can be as large as four times the previously best known bound, i.e., Δ ≥ 4·⌊n/2⌋. The proof combines a lifting operation that replaces a demand edge by two edges incident to auxiliary vertices with a balanced edge‑coloring argument, ensuring that the resulting multigraph can be immersed in K_n without edge conflicts. Using this bound the author solves the classical extremal edge‑number problem for terminal‑pairability in complete graphs, determining the exact maximum number of demand edges that can always be realized.

The final chapter studies terminal‑pairability when the base graph is a two‑dimensional grid G_{k×k}. Although grid graphs are sparse, the author shows they are surprisingly efficient for routing edge‑disjoint paths. By constructing a hierarchical routing scheme that respects the grid’s product structure and applying the same balanced coloring technique, it is proved that the minimum possible maximum degree of a path‑pairable grid graph is O(log n), improving the earlier Ω(log n / log log n) lower bound. Consequently, complete grid graphs are shown to be relatively dense in the family of path‑pairable graphs.

Overall, the thesis demonstrates that extremal combinatorial reasoning can simultaneously yield sharp theoretical bounds and practical approximation algorithms for geometric guarding and network routing problems. The results tighten long‑standing gaps, provide constant‑factor approximations where exact optimization is infeasible, and open new avenues for applying extremal methods to other combinatorial optimization domains.