CS 523 Spring 2014 Assigned Problems

1. Question 2.6 from Design of Approximation Algorithms:
Prove that there can be no α-approximation algorithm for the minimum-degree spanning tree problem for α < 3/2 unless P = NP.
2. Question 2.7 from Design of Approximation Algorithms:
Suppose that an undirected graph G has a Hamiltonian path. Give a polynomial-time algorithm to find a simple path of length Ω(log n/(log log n)). Can you compute the constant hidden in this expression?
3. Question 10.1 from Design of Approximation Algorithms:
In the Euclidean Steiner tree problem, we are given as input a set T of points in the plane called terminals. We are allowed to choose any other set N of points in the plane; we call these points nonterminals. For any pair of points i, j ∈ T ∪ N , the cost of an edge connecting i and j is the Euclidean distance between the points. The goal is to find a set N of nonterminals such that the cost of the minimum spanning tree on T ∪ N is minimized.
Show that the polynomial-time approximation scheme for the Euclidean TSP can be adapted to give a polynomial-time approximation scheme for the Euclidean Steiner tree problem.
4. Do trees have unbounded path-width?
5. Can the treewidth of a subdivision of a graph G be smaller than tw(G)? Can it be larger?
6. Without searching the internet for the answer, prove that bw(G) <= tw(G)+1 <= floor(1.5 * bw(G)) where bw and tw denote the branchwidth and treewidth, respectively.
7. For a graph G with vertex weights and a branch decomposition of width bw(G), give a time O(n * 4^w) algorithm that finds the minimum-weight vertex cover of G.
8. Prove that if every biconnected component of a graph G has branchwidth at most w, then G has branchwidth at most w + 1.
9. A subset D of edges is dominating if every edge not in D has an endpoint in common with some edge in D. Give a linear-time approximation scheme for minimum-weight edge dominating set in planar graphs. (Note: linear in the size of the graph.)
10. Give an H_n-approximation for set multicover: each element e must be covered a specified number of times, r_e; each set can be picked multiple times; if set S is picked k times, the cost is kc(S).
11. Question 1.5 from Design of Approximation Algorithms
12. Question 5.3 from Design of Approximation Algorithms
13. Question 5.6 from Design of Approximation Algorithms
14. Question 6.6 from Design of Approximation Algorithms