1. Given an algorithm for the following: Given an undirected tree T = (V, E) and a subset M ⊆ V of even cardinality, find a pairing (perfect matching) of the nodes in M such that the paths in T between the matched pairs are edge-disjoint. Prove that your algorithm is correct.
  2. An instance of the Minimum Cost Steiner Tree problem consists of a graph G = (V, E) with positive costs c(e) for each edge e ∈ E, and a subset R ⊆ V of required vertices or terminals. The goal is to find a minimum cost set of edges T ⊆ E that spans the set R; i.e., any pair of terminals in R has a connecting path in T . Let |R| = k. Since the edge set E has minimum cost, it will have no cycles and hence be a tree: this is called a minimum cost Steiner tree on R.
    • Let G' be a complete graph on the same set of nodes V: set the cost c'(e) of edge e = uv to be the shortest-path distance between u and v in G, interpreting the edge-costs in G as lengths. For any set R of terminals, show that T, the optimal Steiner tree on R in G has the same cost as T', the optimal Steiner tree on R in the metric completion G'.
    • Let G'' be the graph induced by the set of terminals R in the graph G', and let T'' be the minimum spanning tree in G'' according to the edge costs c'. Show that c'(T'') is at most 2(1-1/k)c'(T').
    • Use these observations to give a 2(1-1/k)-approximation for the minimum-cost Steiner tree problem.
  3. Question 2 at this link with the additional part:
    • Give a polynomial-time algorithm for the following problem: Given a graph G with nonegative vertex weights and a valid, though not necessarily optimal, coloring of G using k colors, find a vertex cover of weight at most (2-2/k)OPT.
  4. Generalize the dual-fitting algorithm to give an H(n)-approximation algorithm for set multicover. This is the set cover problem with the additional requirement that each element must be covered multiple times. In particular, require that element e must be in r(e) sets of the solution.
  5. Question 3 at this link.
  6. Question 6.6 (p 157) of Design of Approximation Algorithms (Williamson, Shmoys).
  7. In Brenda Baker's original paper, the details are light on using the technique for the minimum dominating set problem. Give a complete proof of how to give a PTAS for the minimum dominating set problem in planar graphs. You may assume you are given a poly-time algorithm for treewidth-k graphs for the following problem: given a subset S of vertices, find the smallest-weight subset of vertices that dominates S. Hint: The challenge is in ensuring feasibility at the boundaries between pieces in the decomposition.
  8. Consider a type of Turing machine that has a tape with n+1 positions. The starting position, x, is unknown. The only allowed operation is to move one position forward or one position backward and the only positions that are recognizable are positions 0 and n. Your goal is to compute x. Give a deterministic strategy such that the number of operations you require is at most a constant times the optimal number of steps. Prove that the constant factor you get is the best possible.