Parameterized and approximation algorithms for coverings points with segments in the plane

TL;DR

It is proved that Weighted Segment Set Cover without any relaxation is $\mathsf{W}[1]$-hard and, assuming ETH, there does not exist an algorithm running in time $f(k)\cdot n^{o(k / \log k)}$.

Abstract

We study parameterized and approximation algorithms for a variant of Set Cover, where the universe of elements to be covered consists of points in the plane and the sets with which the points should be covered are segments. We call this problem Segment Set Cover. We also consider a relaxation of the problem called $δ$-extension, where we need to cover the points by segments that are extended by a tiny fraction, but we compare the solution's quality to the optimum without extension. For the unparameterized variant, we prove that Segment Set Cover does not admit a PTAS unless $\mathsf{P}=\mathsf{NP}$, even if we restrict segments to be axis-parallel and allow $\frac{1}{2}$-extension. On the other hand, we show that parameterization helps for the tractability of Segment Set Cover: we give an FPT algorithm for unweighted Segment Set Cover parameterized by the solution size $k$, a parameterized approximation scheme for Weighted Segment Set Cover with $k$ being the parameter, and an FPT algorithm for Weighted Segment Set Cover with $δ$-extension parameterized by $k$ and $δ$. In the last two results, relaxing the problem is probably necessary: we prove that Weighted Segment Set Cover without any relaxation is $\mathsf{W}[1]$-hard and, assuming ETH, there does not exist an algorithm running in time $f(k)\cdot n^{o(k / \log k)}$. This holds even if one restricts attention to axis-parallel segments.

Figures (7)

• Figure 1: Example of the construction in the proof of Lemma \ref{['dense_set_exists']} for $M = 7$ and some set of points $C$ (marked with black circles). The top panel shows segments $v_i$. The middle panel shows segments $t_i$. Note that $t_5$ is an empty segment, because there are no points in $C$ that belong to $v_5$, while each of the segments $t_3$ and $t_7$ is degenerated to a single point: $c$ and $d$, respectively. Segments $t_1$ and $t_2$ share one point $b$. The bottom panel shows an example of the second case in the correctness proof: a solution $\mathcal{R}$ of size $4$ whose all segments intersect $t_4$. Then one of $y$ and $z$ will cover the whole of $C_4$ after extension.
• Figure 2: Construction of Lemma \ref{['lem:choice-gadget']} for $N=8$. Elements of $I\cup \{0\}$ are depicted with circles and elements of $I^+\cup I^-$ are depicted with squares. Blue segments represent the set $\mathcal{R}_B$ for $B=\{3,7\}$.
• Figure 3: Example solution in the instance $(\mathcal{U},\mathcal{F})$ constructed in the proof of Lemma \ref{['w1_construction']} for $H=K_4$. Blue segments belong to the sets $\mathcal{S}_i$ for $i\in \{1,2,3,4\}$ and orange segments belong to $\mathcal{D}$.
• Figure 4: Variable-gadget. We denote the set of points marked with black circles as $\mathsf{pointsVariable}_{i}$, and they need to be covered (are part of the set $\mathcal{U}$). Note that some of the points are not marked as black dots and exists only to name segments for further reference. We denote the set of red segments as $\mathsf{chooseVariable}^{\texttt{false}}_{i}$ and the set of blue segments as $\mathsf{chooseVariable}^{\texttt{true}}_{i}$.
• Figure 5: OR-gadget. Segments from $\mathsf{chooseOr}^{\texttt{false}}_{i,j}$ are red, segments from $\mathsf{chooseOr}^{\texttt{true}}_{i,j}$ are blue (both light blue and dark blue), segments from $\mathsf{orMoveVariable}_{i,j}$ are green and yellow. Dark blue segment is the $output$ segment. Grey segments$input_x$ and $input_y$ are input segments that are not part of $\mathsf{segmentsOr}_{i,j}$.

Theorems & Definitions (56)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Lemma 7
• proof
• proof : Proof of Theorem \ref{['thm:few_weights']}.
• Theorem 7