Fast Matching-based Approximations for Maximum Duo-Preservation String Mapping and its Weighted Variant

Abstract

We present a new approach to approximating the Maximum Duo-Preservation String Mapping Problem (MPSM) based on massaging the constraints into a tractable matching problem. MPSM was introduced in Chen, Chen, Samatova, Peng, Wang, and Tang [Chen et al., 2014] as the complement to the well-studied Minimum Common String Partition problem (MCSP). Prior work also considers the k-MPSM and k-MCSP variants in which each letter occurs at most k times in each string. The authors of [Chen et al., 2014] showed a k^2-appoximation for k >= 3 and 2-approximation for k = 2. Boria, Kurpisz, Leppänen, and Mastrolilli [Boria et al., 2014] gave a 4-approximation independent of k and showed that even 2-MPSM is APX-Hard. A series of improvements led to the current best bounds of a (2 + epsilon)-approximation for any epsilon > 0 in n^{O(1/epsilon)} time for strings of length n and a 2.67-approximation running in O(n^2) time, both by Dudek, Gawrychowski, and Ostropolski-Nalewaja [Dudek et al., 2017]. Here, we show that a 2.67-approximation can surprisingly be achieved in O(n) time for alphabets of constant size and O(n + alpha^7) for alphabets of size alpha.
Recently, Mehrabi [Mehrabi, 2017] introduced the more general weighted variant, Maximum Weight Duo-Preservation String Mapping (MWPSM) and provided a 6-approximation. Our approach gives a 2.67-approximation to this problem running in O(n^3) time. This approach can also find an 8/(3(1-epsilon))-approximation to MWPSM for any epsilon > 0 in O(n^2 epsilon^{-1} lg{epsilon^{-1}}) time using the approximate weighted matching algorithm of Duan and Pettie [Duan and Pettie, 2014].
Finally, we introduce the first streaming algorithm for MPSM. We show that a single pass suffices to find a 4-approximation on the size of an optimal solution using only O(alpha^2 lg{n}) space.