Few Cuts Meet Many Point Sets

Abstract

We study the problem of how to split many point sets in $\mathbb {R}^d$ into smaller parts using a few (shared) splitting hyperplanes. This problem is related to the classical Ham-Sandwich Theorem. We provide a logarithmic approximation to the optimal solution using the greedy algorithm for submodular optimization.

Appendix A Proof of Theorem 4

Proof

This result is by now classical, and we include the proof only for the sake of completeness. Let be the optimal solution. Consider a current solution $\mathcal {C}_i \subseteq \Pi$ at iteration i, and observe that by monotonicity, we have

$$\begin{aligned} f_{\max }= f(\mathcal {O}) \le f(\mathcal {C}_i \cup \mathcal {O}) \le f_{\max }. \end{aligned}$$

As such, we have $f(\mathcal {C}_i \cup \mathcal {O}) = f_{\max }$. Let $\Delta _i = f(\mathcal {O}) - f(\mathcal {C}_i)$ be the of $\mathcal {C}_i$. For $j=0, \ldots , k$, let Set . We have that

Hence, there is an index j, such that $\delta _j \ge \Delta _i/k$. Now, by submodularity, we have that

However, the greedy algorithm adds an element that maximizes the value of , which is at least $\Delta _i/k$. Put differently, the added element decreases the deficiency of the current solution by a factor $\le 1-1/k$. Therefore the deficiency in the end of the ith iteration is at most $\Delta _i \le (1-1/k)^i \Delta _0 = (1-1/k)^if(\mathcal {O}).$ This quantity is less than one for $i = O( k\log f_{\max })$. $\square$