CS614. Advanced Algorithms. L04 Quiz.

CS614. Advanced Algorithms.

L04 Quiz

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Problem 1. Matroid Intersection Example

Consider a directed graph \(D=(V, E \subseteq V \times V)\). A set \(T \subseteq E\) is an arborescence (oriented forest) if:

  1. \(T\) does not contain a cycle (ignoring directions of edges).

  2. Every vertex in \(V\) has at most one incoming edge.

An arborescence \(T\) with \(|T|=n-1\) will have one incoming edge incident on each node except one. If we denote this special node as root, this is an oriented spanning tree as shown in the figure.

An example arborescence.

Consider the underlying undirected graph \(G_D = (V,E)\) associated with \(D\) (this is the graph obtained by “erasing the arrows” in \(D\)). Consider the universe given by \(E\). Suggest two matroids \({\mathcal M}_1\) and \({\mathcal M}_2\) for which set of arborescences is given by the sets independent in both \({\mathcal M}_1\) and \({\mathcal M}_2\).

Hint: these are both matroids seen in class. Further, you might find it useful to partition \(E\) into \(|V|\) many parts as follows — the part \(P_v\) contains all edges that are incoming arcs for the vertex \(v\) in \(D\). Can you define a matroid based on this partition?

Describe \({\mathcal M}_1\) and \({\mathcal M}_2\).

Problem 2. Maker-Breaker Game

Two players take turns removing edges from an undirected graph until there are no edges left.

Player 2 wins if the edges they remove contains a spanning tree, player 1 wins if the set of edges they remove would disconnect the original graph.

  1. Is it true that exactly one player wins this game? In other words, is the following statement true?

“It is NOT the case that after the game has been played, both players can claim a win.”

  • Yes
  • No
  1. Which player wins on a path?
  • Player 1
  • Player 2
  1. Which player wins on a complete graph?
  • Player 1
  • Player 2
  1. Complete this sentence: player 2 has the winning strategy if and only if the graph contains BLANK.

(No marks for answering this question, take your best guess :) )