Main Idea: Graph Coloring and DAG Construction

How do we eliminate cycles and guarantee $$$r_1 \ne r_N$$$?

We can translate this problem into Graph Coloring.

1. Guaranteeing a DAG

Assign a positive integer color $$$C[u]$$$ to each resource vertex $$$u$$$.

For every directed edge $$$u \to v$$$, enforce:

Along any directed path

the colors must satisfy:

Therefore, a directed path can never return to a previously visited vertex. Hence the directed graph is a DAG.

This eliminates deadlocks caused by cyclic wait chains.

2. Minimizing the Chain Length

By the Gallai-Roy-Vitaver theorem, for an undirected graph $$$G$$$:

where $$$\chi(G)$$$ is the chromatic number of $$$G$$$.

Therefore, if the graph can be colored using $$$k$$$ colors, we can orient every edge from the vertex with the smaller color to the vertex with the larger color.

The resulting DAG has longest directed path of at most $$$k-1$$$ edges.

Thus, the problem reduces to finding the chromatic number of a graph with at most $$$15$$$ vertices.

problem: https://vjudge.net/problem/UVA-1439

Share better approaches!!