Your Problem on CS 412

Minimum Vertex Cover

Mon, 01 Jan 0001 00:00:00 +0000

Given a simple undirected graph $G=(V,E)$ , a vertex cover is a set $C\subseteq V$ such that every edge has at least one endpoint in $C$ :

\forall\{u,v\}\in E,\qquad u\in C \;\text{or}\; v\in C.

The optimization problem asks for a vertex cover of minimum size. Throughout this project, Minimum Vertex Cover is abbreviated MVC and uses the problem identifier minimum_vertex_cover (short command-line alias mvc).

Minimum Graph Coloring

Mon, 01 Jan 0001 00:00:00 +0000

Given an undirected graph $G=(V,E)$ , a proper vertex coloring assigns a color to every vertex such that no two adjacent vertices receive the same color.

More formally, if $c(v)$ denotes the color assigned to vertex $v$ , then for every edge $(u,v) \in E$ ,

c(u) \neq c(v).

The goal of Minimum Graph Coloring is to find a proper coloring that uses the smallest possible number of colors. The minimum number of colors required to properly color a graph is called its chromatic number.

Maximum Clique

Mon, 01 Jan 0001 00:00:00 +0000

Given an undirected graph $G=(V,E)$ , a clique is a subset of vertices $C \subseteq V$ in which every pair of distinct vertices is connected by an edge.

More formally, for every pair of distinct vertices $u,v \in C$ ,

(u,v) \in E.

The goal of Maximum Clique is to find a clique containing the largest possible number of vertices.

Traveling Salesperson

Mon, 01 Jan 0001 00:00:00 +0000

Given a weighted graph $G=(V,E)$ , a tour is a cycle that visits every vertex exactly once before returning to its starting vertex.

If a tour visits the vertices in the order

v_0,v_1,\ldots,v_{n-1},v_0,

then its total cost is the sum of the weights of the edges used by the tour:

Longest Path

Mon, 01 Jan 0001 00:00:00 +0000

Given an undirected graph $G=(V,E)$ , a simple path is a sequence of vertices

P=(v_0,v_1,\ldots,v_k)

such that every consecutive pair of vertices is connected by an edge and no vertex appears more than once.

More formally,

(v_i,v_{i+1}) \in E

for every $0 \le i < k$ , and the vertices $v_0,v_1,\ldots,v_k$ are all distinct.