# Greedy Vertex Cover in C

The Vertex Cover Problem is to find a subset of the vertices of a graph that contains an endpoint of every edge. For example, in the graph shown above, the subset (0, 1) highlighted in red is a vertex cover. The minimisation problem of finding the smallest vertex cover is NP-hard. However a good approximation can be achieved using a greedy algorithm. The algorithm is simply:

• While there are uncovered edges:
1. Find the first uncovered edge
2. Put one of its endpoints in the covering set
3. Mark all of the edges incident upon that endpoint as covered

Below is an implementation in C. The function vertex_cover() takes a graph in edge list format, its size (number of edges) and order (number of vertices), and the address of a pointer through which to return the covering set as an out parameter. The return value is the size of the covering set.

#include <stdlib.h>

typedef struct {
unsigned int first;
unsigned int second;
} edge;

unsigned int vertex_cover(const edge *edges, unsigned int size, unsigned int order,
unsigned int **cover)
{
unsigned int uncovered_size = size;
unsigned int cover_size = 0;
unsigned int *covered = calloc(size, sizeof(unsigned int));
*cover = calloc(order, sizeof(unsigned int));
if (covered == NULL || cover == NULL) {
free(covered);
free(*cover);
*cover = NULL;
return 0;
}
while (uncovered_size > 0) {
unsigned int e, v;
/* Find the first edge that hasn't been covered */
for (e = 0; e < size && covered[e] == 1; e++);
/* Add its first endpoint to the covering set */
v = edges[e].first;
(*cover)[v] = 1;
cover_size++;
/* Mark all uncovered edges containing its first endpoint as covered */
for (e = 0; e < size; e++) {
if (!covered[e] && (edges[e].first == v || edges[e].second == v)) {
covered[e] = 1;
uncovered_size--;
}
}
}
free(covered);
return cover_size;
}


Here is an example program that finds a vertex cover on the graph shown above:

#include <stdio.h>
#include <stdlib.h>

/* Connect two edges */
void edge_connect(edge *edges, unsigned int first, unsigned int second,
unsigned int *pos)
{
edges[*pos].first = first;
edges[*pos].second = second;
(*pos)++;
}

int main(void)
{
const unsigned int size = 4; /* Edges */
const unsigned int order = 5; /* Vertices */
edge *edges = malloc(size * sizeof(edge));
unsigned int i = 0;
edge_connect(edges, 0, 1, &i);
edge_connect(edges, 0, 2, &i);
edge_connect(edges, 0, 3, &i);
edge_connect(edges, 1, 4, &i);
unsigned int *cover;
unsigned int c = vertex_cover(edges, size, order, &cover);
printf("Cover size is %u\n", c);
printf("Vertices in cover:\n");
for (i = 0; i < order; i++) {
if (cover[i]) {
printf("%u ", i);
}
}
putchar('\n');
free(edges);
free(cover);
return 0;
}


The output:

Cover size is 2
Vertices in cover:
0 1