UVa 10982 - Troublemakers

10982 - Troublemakers

 * http://acm.uva.es/p/v109/10982.html

Summary
This is a graph theory problem with a greedy solution.

We are given a graph $$G$$ with $$n$$ nodes, numbered from 1 to $$n$$, and $$m$$ edges, and we want to partition the graph into two partitions such that the number of edges which are contained by two nodes in the same partition is at most $$\frac{m}{2}$$.

Explanation
In graph theory terms, a partition of graph's vertices into two disjoint sets is called a cut. Size of a cut is defined as the number of edges, which have endpoints in both partitions. So, what we are really looking for in this problem, is a cut of size at least m/2. This can be done greedily. A related problem of finding a cut of largest possible size (MAX-CUT problem) is NP-hard.

The simplest solution is to randomly split vertices into two partitions (repeating this several times, if needed.) Expected size of a cut is m/2 edges, so with high probability this algorithm will quickly find desirable cut. There are also several deterministic algorithms, which show that such cut always exists.

Consider the following greedy algorithm: A = {}, B = {} for i = 1 to n:	a = 0, b = 0 for j = 1 to i-1: if there is an edge (i, j): if j is in A:				a++ else: b++

if a <= b:		add x to set A	else: add x to set B Output cut {A, B}

Claim: The algorithm outputs a cut of size at least m/2.

Proof: Let $$E_i$$ denote all edges (i, j) in which j b$$, so exactly $$\max(a,b)$$ edges of $$E_i$$ will be in the cut. Since $$a+b=|E_i|$$, it follows that $$\max(a,b) \ge \frac{|E_i|}{2}$$. Hence, the final size of the cut is at least $$\sum_{i=1}^n \frac{|E_i|}{2} = \frac{m}{2}$$, as required.

Gotchas

 * The same edge may be listed multiple times.

Input
10 4 3 1 2 2 3 3 4

4 6 1 2 1 3 1 4 2 3 2 4 3 4

3 0

3 1 1 2

3 1 1 3

3 2 1 2 1 3

3 1 2 3

3 2 1 2 2 3

3 2 1 3 2 3

3 3 1 2 1 3 2 3

Output
Note the output is not unique. Case #1: 2 2 4 Case #2: 2 3 4 Case #3: 3 1 2 3 Case #4: 2 2 3 Case #5: 2 2 3 Case #6: 2 2 3 Case #7: 2 1 3 Case #8: 1 2 Case #9: 1 3 Case #10: 2 2 3