Why does the greedy matching algorithm guarantee a 2-approximation for the Vertex Cover problem?
Conditions
- The algorithm selects arbitrary edges and adds both endpoints until no edges remain
Reasoning, step by step
- Identify the set of disjoint edges M selected by the algorithm.
- Recognize that any valid vertex cover must include at least one endpoint for each edge in M.
- Conclude that the optimal cover size C* is at least the number of edges in M.
- Note that the greedy algorithm adds both endpoints of every edge in M, resulting in exactly 2|M| vertices.
- Combine these facts to prove that the algorithm's cost is at most twice the optimal cost.
Example
The narrator details the four steps: 'Step one... selects a set of completely disconnected, disjoint edges... Step two... optimal perfect solution C star must select at least one vertex for every single edge... Step three... added both endpoints... Step four... combine them.'
Common misconceptions
- Thinking the algorithm always finds the exact minimum vertex cover.
- Assuming the 2-approximation bound is arbitrary and not mathematically proven.
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