# Widely distributed clusters of the constraint satisfaction problem model d-k-CSP

## TL;DR

This paper investigates the structure of solution spaces in the d-k-CSP model, revealing that near-critical constraint densities lead to widely distributed, well-separated small clusters, explaining the problem's computational hardness.

## Contribution

It uncovers the relationship between solution space structure and problem hardness in d-k-CSP, highlighting the presence of widely distributed small cluster-regions near critical constraint densities.

## Key findings

- Hard instances occur when r is near 1.
- Solution space contains many small, well-separated cluster-regions.
- These structures likely cause increased computational difficulty.

## Abstract

Relation between problem hardness and solution space structure is an important research aspect. Model d-k-CSP generates very hard instances when $r=1$ and $r$ is near 1, where $r$ represents normalized constraint density. We find that when $r$ is below and close to 1, the solution space contains many widely distributed well-separated small cluster-regions (a cluster-region is a union of some clusters), which should the reason that the generated instances are hard to solve.