Let $P_1,\dots ,P_m\subset \mathbb{R}^n$ be $m<\infty$ convex polytopes in $\mathbb{R}^n$, and $U:=\bigcup_{j} P_j$ their set-theoretic union. What algorithms are known for computing the boundary $\partial U$ of $U$?

Note that $U$ would not always be convex, might have a nontrivial fundamental group, etc.

A naive algorithm I can think about would start from $\partial P_1$, intersect each facet of $P_2$ with it, obtaining $\partial (P_1\cup P_2)$, etc. A less naive point of view might involve working with measures supported on $P_j$'s and inclusion/exclusion in some way.

Any references on this sort of questions?


Not a direct answer, but may help with: "Any references on this sort of questions?"

Computing the convex hull of the union of two polytopes $P_1 \cup P_2$ is NP-hard:

Tiwary, Hans Raj. "On the hardness of computing intersection, union and Minkowski sum of polytopes." Discrete & Computational Geometry 40, no. 3 (2008): 469-479. Author preliminary PDF download.

But the computation is polynomial for special polytopes:

Balas, Egon. "On the convex hull of the union of certain polyhedra." Operations Research Letters 7, no. 6 (1988): 279-283. Journal link.

Conforti, Michele, Marco Di Summa, and Yuri Faenza. "Balas formulation for the union of polytopes is optimal." arXiv:1711.00891 (2017).

Added. Specializing to $\mathbb{R}^3$, there is an algorithm that constructs the boundary of the union, which runs in $O(k^3 + kn \log k \log n)$ expected time, where $k$ is the number of polyhedra and $n$ is the total number of faces:

Aronov, Boris, Micha Sharir, and Boaz Tagansky. "The union of convex polyhedra in three dimensions." SIAM Journal on Computing 26, no. 6 (1997): 1670-1688. Journal link
          Aronov, Sharir, Tagansky.

Subsequently there was a specialized extension to $\mathbb{R}^4$:

Aronov, Boris, Alon Efrat, Vladlen Koltun, and Micha Sharir. "On the union of κ-round objects in three and four dimensions." Discrete & Computational Geometry 36, no. 4 (2006): 511-526. Journal link.

  • 2
    $\begingroup$ The question seems to be about the boundary of set theoretical union, rather than convex hull. $\endgroup$ – Alexandre Eremenko Nov 3 '18 at 2:09
  • $\begingroup$ Indeed, it is about the boundary of a not always convex object. $\endgroup$ – Dima Pasechnik Nov 3 '18 at 9:26
  • $\begingroup$ A related to the complexity question - how about the case of fixed dimension (e.g. $n=3$). $\endgroup$ – Dima Pasechnik Nov 3 '18 at 9:27
  • $\begingroup$ @DimaPasechnik: Yes, I realize that my original post doesn't directly answer your question. Incidentally, it is also NP-hard to compute $P_1 \cap P_2$. $\endgroup$ – Joseph O'Rourke Nov 3 '18 at 12:42
  • $\begingroup$ Thanks, these are very helpful references. $\endgroup$ – Dima Pasechnik Nov 3 '18 at 22:16

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