CS614. Advanced Algorithms. L02 Quiz.

CS614. Advanced Algorithms.

L02 Solutions

Problem 1. Identify the Circuits

Let G be a simple, undirected, and connected graph. Consider the graphic matroid discussed in class, i.e, where:

the universe U is the set of edges of G, i.e, E(G);
the family \mathcal{F} of independent sets is the collection of all subsets of edges that are acyclic.

A maximal independent set in a matroid is called a basis, and for this example, the maximal independent sets correspond to spanning trees.

A minimal dependent set in a matroid is called a circuit. In this example, what are the circuits?

Solution

The circuits of the graphic matroid are the cycles of the graph G.

Problem 2. Matchings

Let G be a simple, undirected, and connected graph. Consider the following set system:

the universe U is the set of edges of G, i.e, E(G);
the family \mathcal{F} of independent sets is the collection of all subsets of edges that are matchings.

Is this a matroid? Why or why not? Justify your answer.

Solution

Not a matroid: consider the graph on the vertex set \{a,b,c,d\} with the edges \{ab, cd, ad\}.

There are two matchings in this instance:

M_1 := \{ab,cd\}
M_2: \{ad\}

However, although |M_1| > |M_2|, neither of the edges from M_1 can be added to M_2.

Problem 3. Independent Sets

Let G be a simple, undirected, and connected graph. Consider the following set system:

the universe U is the set of vertices of G, i.e, V(G);
the family \mathcal{F} of independent sets is the collection of all subsets S of that are independent in G, i.e, the subgraph G[S] has no edges.

Is this a matroid? Why or why not? Justify your answer.

Solution

Not a matroid: consider the graph on the vertex set \{a,b,c\} with the edges \{ab, ac\}. There are two independent sets: S_1 := \{b,c\} and M_2: \{a\}, but neither of the vertices from S_1 can be added to S_2.

If the independent sets formed a matroid the problem of finding a maximum independent set would not be NP-complete.

— Comment in class