We study the problem of simultaneous geometric embedding of two paths without self-intersections on an integer grid. We show that minimizing the length of the longest edge of such an embedding is NP-hard. We also show that we can minimize in $O(n^{3/2})$ time the perimeter of an integer grid containing such an embedding if one path is $x$-monotone and the other is $y$-monotone.

翻译：我们研究了在整数网格上无自相交的两条路径的同步几何嵌入问题。我们证明了最小化此类嵌入中最长边的长度是NP难的。同时我们还证明，若其中一条路径是$x$-单调的而另一条是$y$-单调的，则可以在$O(n^{3/2})$时间内最小化包含此类嵌入的整数网格的周长。