# Match up between any two players now every pair of

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match-up between any two players. Now every pair of players can go into a match-up, and the benefit incurred by this match-up will be indicated by the metric B ( Match ). Coach P would like to set up one-on-one games between the players such that the total amount of benefits from all these match-ups is maximized. Can you help him do that through an ILP?
3) This is a modification of the problem in 2). The first modification is that the resulting graph connecting players is a complete graph. In addition, each edge e ∈ E has an associated weight w e equal to the corresponding value of B ( Match ) for the match-up between the corresponding players. Then the problem can be expressed as the following ILP max b e ∈E w e b e subject to e N ( v ) b e 1 , v ∈ { 1 , · · · , |V|} b e ∈ { 0 , 1 } .
Problem 2 ( 4 points ): Consider a graph G = ( V , E ). A matching on G is a collection of M ⊆ E such that, no two edges in M share a vertex. In other words, each vertex in V has at most one connected edge in M . A maximal matching is a matching M such that, if any other edge in E \ M is added to M it no longer becomes a valid matching. Assume that you have an algorithm that takes as input an arbitrary graph G , and outputs a maximal matching M . Propose a heuristic that takes as input G and M , and outputs a valid vertex cover C (a cover is a subset of vertices C ⊆ V such that each edge in E is incident to at least one vertex in C ). The size of the output vertex cover should be within an approximation factor of 2.