On the complexity of covering points by disjoint segments and by guillotine cuts

Abstract

We show that two geometric cover problems in the plane, related to covering points with disjoint line segments, are NP-complete. Given $n$ points in the plane and a value $k$, the first problem asks if all points can be covered by $k$ disjoint line segments; the second problem treats the analogous question for $k$ guillotine cuts.

Figures (5)

• Figure 1: Optimal coverings for $n=9$ points with (a) three disjoint segments, (b) four guillotine cuts.
• Figure 2: Example of the drawing of a monotone 3-SAT formula with five variables and five clauses.
• Figure 3: Transformed drawing of example in Figure \ref{['fig:monotone-3SAT-example']} after (a) first phase and (b) second phase.
• Figure 4: Variable gadgets. (a) Original variable point (orange). (b) Variable point replaced by two points (red and blue); all pairs of incident lines of different color cross. (c) Additional segments are added (dashed) to have exactly $m$ segment incident to each red and each blue point; in the final instance, one point is added at each intersection (purple points).
• Figure 5: (a) Thicker segments indicate those present in an optimal solution (ignoring that each red/blue segment is actually composed of $m$ segments each). (b) Sequence of guillotine cuts associated with the solution (not showing the details of all $m$ cuts for each variable).

Theorems & Definitions (8)

• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 1
• proof
• Theorem 2
• proof