# Why is it hard to beat $O(n^2)$ for Longest Common Weakly Increasing Subsequence?

## TL;DR

This paper establishes that the Longest Common Weakly Increasing Subsequence problem likely cannot be solved faster than quadratic time unless a major complexity hypothesis (SETH) is false, using novel reductions and assumptions.

## Contribution

It proves a conditional lower bound for LCWIS based on SETH and NC-SETH, filling a gap in understanding its computational hardness.

## Key findings

- LCWIS cannot be solved in strongly subquadratic time unless SETH is false.
- The developed techniques can be applied to other problems to establish similar lower bounds.
- Provides a new complexity-theoretic foundation for LCWIS's computational difficulty.

## Abstract

The Longest Common Weakly Increasing Subsequence problem (LCWIS) is a variant of the classic Longest Common Subsequence problem (LCS). Both problems can be solved with simple quadratic time algorithms. A recent line of research led to a number of matching conditional lower bounds for LCS and other related problems. However, the status of LCWIS remained open. In this paper we show that LCWIS cannot be solved in strongly subquadratic time unless the Strong Exponential Time Hypothesis (SETH) is false. The ideas which we developed can also be used to obtain a lower bound based on a safer assumption of NC-SETH, i.e. a version of SETH which talks about NC circuits instead of less expressive CNF formulas.