Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Subexponential Algorithms for Clique Cover on Unit Disk and Unit Ball Graphs

Tomohiro Koana, Nidhi Purohit, Kirill Simonov

TL;DR

It is shown that Clique Cover also suffers from a "curse of dimensionality", albeit in a significantly different way compared to Maximum Clique, and a $2^{O(\sqrt{n})}$-time algorithm for unit disk graphs is presented and argued that it is tight under the ETH.

Abstract

In Clique Cover, given a graph $G$ and an integer $k$, the task is to partition the vertices of $G$ into $k$ cliques. Clique Cover on unit ball graphs has a natural interpretation as a clustering problem, where the objective function is the maximum diameter of a cluster. Many classical NP-hard problems are known to admit $2^{O(n^{(1 - 1/d)})}$-time algorithms on unit ball graphs in $\mathbb{R}^d$ [de Berg et al., SIAM J. Comp 2018]. A notable exception is the Maximum Clique problem, which admits a polynomial-time algorithm on unit disk graphs and a subexponential algorithm on unit ball graphs in $\mathbb{R}^3$, but no subexponential algorithm on unit ball graphs in dimensions 4 or larger, assuming the ETH [Bonamy et al., JACM 2021]. In this work, we show that Clique Cover also suffers from a "curse of dimensionality", albeit in a significantly different way compared to Maximum Clique. We present a $2^{O(\sqrt{n})}$-time algorithm for unit disk graphs and argue that it is tight under the ETH. On the other hand, we show that Clique Cover does not admit a $2^{o(n)}$-time algorithm on unit ball graphs in dimension $5$, unless the ETH fails.

Subexponential Algorithms for Clique Cover on Unit Disk and Unit Ball Graphs

TL;DR

It is shown that Clique Cover also suffers from a "curse of dimensionality", albeit in a significantly different way compared to Maximum Clique, and a -time algorithm for unit disk graphs is presented and argued that it is tight under the ETH.

Abstract

In Clique Cover, given a graph and an integer , the task is to partition the vertices of into cliques. Clique Cover on unit ball graphs has a natural interpretation as a clustering problem, where the objective function is the maximum diameter of a cluster. Many classical NP-hard problems are known to admit -time algorithms on unit ball graphs in [de Berg et al., SIAM J. Comp 2018]. A notable exception is the Maximum Clique problem, which admits a polynomial-time algorithm on unit disk graphs and a subexponential algorithm on unit ball graphs in , but no subexponential algorithm on unit ball graphs in dimensions 4 or larger, assuming the ETH [Bonamy et al., JACM 2021]. In this work, we show that Clique Cover also suffers from a "curse of dimensionality", albeit in a significantly different way compared to Maximum Clique. We present a -time algorithm for unit disk graphs and argue that it is tight under the ETH. On the other hand, we show that Clique Cover does not admit a -time algorithm on unit ball graphs in dimension , unless the ETH fails.
Paper Structure (2 sections, 3 theorems, 1 figure)

This paper contains 2 sections, 3 theorems, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries

Key Result

Theorem 1

Clique Cover can be solved in time $2^{O(\sqrt n)}$ on $n$-vertex unit disk graphs, when a geometric representation of the graph is given in the input, with bit-length of the vectors bounded by $\mathop{\mathrm{poly}}\nolimits(n)$.

Figures (1)

  • Figure 1: Edge gadgets encoding the edge between vertices $u, v$: left, $2$-subdivision of the edge, suitable for maximum independent sets; center, triangle-like gadget suitable for $3$-colorings; right, improved gadget preserving $3$-colorings --- here vertices $c_1$ and $c_2$ are connected in the same way to all edge gadgets.

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3