New algorithm for calculating cycle intersection indices

Background. The objects of study are triangulated compact polyhedron $P$, which are $n$-dimensional manifolds with boundary. The goal is to create new efficient algorithms for calculating modulo $2$ intersection indices. Materials and methods. The construction of a closed $n$-dimensional path along a given absolute one-dimensional cycle is used. Results. An algorithm has been developed to calculate the intersection index of a given absolute one-dimensional cycle with an arbitrary relative cycle of dimension $(n - 1)$. A rigorous mathematical justification of the algorithm is given. Conclusions. For the problem under consideration, the solution algorithm was developed for the first time. Its computational complexity is $O(n^{2}N+m)$, where n is the dimension of the manifold $P$, $N$ is the number of its $n$-dimensional simplices, and $m$ is the number of edges that make up the cycle $x$.