# How Hard is it to Find (Honest) Witnesses?

## TL;DR

This paper introduces a framework based on the 3SUM conjecture for proving conditional lower bounds by efficiently representing and enumerating witnesses (candidate solutions) to problems, impacting various computational problems.

## Contribution

The paper presents a novel framework that leverages hashing and encoding to relate witness enumeration to lower bounds, applicable to multiple problem domains.

## Key findings

- Framework establishes 3SUM-hardness for decision problems
- Enables lower bounds for convolution, matrix multiplication, and string problems
- Uses witness enumeration to connect problem complexity with 3SUM conjecture

## Abstract

In recent years much effort was put into developing polynomial-time conditional lower bounds for algorithms and data structures in both static and dynamic settings. Along these lines we suggest a framework for proving conditional lower bounds based on the well-known 3SUM conjecture. Our framework creates a \emph{compact representation} of an instance of the 3SUM problem using hashing and domain specific encoding. This compact representation admits false solutions to the original 3SUM problem instance which we reveal and eliminate until we find a true solution. In other words, from all \emph{witnesses} (candidate solutions) we figure out if an \emph{honest} one (a true solution) exists. This enumeration of witnesses is used to prove conditional lower bound on \emph{reporting} problems that generate all witnesses. In turn, these reporting problems are reduced to various decision problems. These help to enumerate the witnesses by constructing appropriate search data structures. Hence, 3SUM-hardness of the decision problems is deduced. We utilize this framework to show conditional lower bounds for several variants of convolutions, matrix multiplication and string problems. Our framework uses a strong connection between all of these problems and the ability to find \emph{witnesses}. While these specific applications are used to demonstrate the techniques of our framework, we believe that this novel framework is useful for many other problems as well.