A 42k Kernel for the Complementary Maximal Strip Recovery Problem

Abstract

In the Complementary Maximal Strip Recovery problem (CMSR), we are given two strings $S_1$ and $S_2$ of distinct letters, where each letter appears either in the positive form or the negative form. The question is whether there are k letters whose deletion results in two matched strings. String $S_1$ matches string $S_2$ if there are partitions of $S_1$ and $S_2$, such that, for each component $S_1^i$ of the partition of $S_1$, there is a unique component $S_2^j$ in the partition of $S_2$ which is either equal to $S_1^i$ or can be obtained from $S_1^i$ by firstly reversing the order of the letters and then negating the letters. The CMSR problem is known to be NP-hard and fixed-parameter tractable with respect to k. In particular, a linear kernel of size $74k+4$ was developed based on 8 reduction rules. Very recently, by imposing 3 new reduction rules to the previous kernelization, the linear kernel was improved to 58k. We aim to simplify the kernelization, yet obtain an improved kernel. In particular, we study 7 reduction rules which lead to a linear kernel of size ${42k+24}$.

This work is supported by the National Natural Science Foundation of China (Grants No. 61672536, 61502054, 61420106009), the Natural Science Foundation of Hunan Province, China (Grant No. 2017JJ3333), the Scientific Research Fund of Hunan Provincial Education Department (Grant No. 17C0047), and the China Postdoctoral Science Foundation (Grant No. 2017M612584).