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The geometric intersection number of a curve on a surface is the minimalnumber of self-intersections of any homotopic curve, i.e. of any curve obtainedby continuous deformation. Given a curve $c$ represented by a closed walk oflength at most $\ell$ on a combinatorial surface of complexity $n$ we describesimple algorithms to (1) compute the geometric intersection number of $c$ in$O(n+ \ell^2)$ time, (2) construct a curve homotopic to $c$ that realizes thisgeometric intersection number in $O(n+\ell^4)$ time, (3) decide if thegeometric intersection number of $c$ is zero, i.e. if $c$ is homotopic to asimple curve, in $O(n+\ell\log\ell)$ time. The algorithms for (2) and (3) arerestricted to orientable surfaces, but the algorithm for (1) is also valid onnon-orientable surfaces.  To our knowledge, no exact complexity analysis had yet appeared on thoseproblems. An optimistic analysis of the complexity of the published algorithmsfor problems (1) and (3) gives at best a $O(n+g^2\ell^2)$ time complexity on agenus $g$ surface without boundary. No polynomial time algorithm was known forproblem (2) for surfaces without boundary. Interestingly, our solution toproblem (3) provides a quasi-linear algorithm to a problem raised by Poincar\'emore than a century ago. Finally, we note that our algorithm for problem (1)extends to computing the geometric intersection number of two curves of lengthat most $\ell$ in $O(n+ \ell^2)$ time.

Computing the Geometric Intersection Number of Curves