COS 423 Theory of Algorithms Spring 2005
Assignment 7
Answer each problem below. This assignment is due Wednesday, April 20 at the beginning of lecture.
Collaboration is allowed (according to the rules specified in the handout). If you work with a group, be sure
to clearly acknowledge the other members of your study group on the first page.
Read Chapters 10 and 11 in Kleinberg-Tardos.
1. Problem 10.5 (a). (Min cost dominating set on trees.)
2. Problem 11.4. (Load balancing within 20%.) Note: this is Problem 11.5 in the published hardcopy
textbook.
3. Let G=(V, E) be an undirected graph (not necessarily bipartite) with edge weights we≥0. The max-
imum weighted matching problem is to find a matching Mthat maximizes Pe∈Mwe. This problem can
be solved in polynomial-time, but for many practical problems (including partitioning in VLSI design,
refinement of FEM nets, the gossiping problem in telecommunications and input-queued switching in
networks) the best known algorithm may be either too slow or too cumbersome to implement.
Consider the following greedy algorithm. Start with the empty matching M=∅. Add an edge of
maximum weight among all edges which have no endpoints in common with any of the edges in M.
Repeat until no more edges can be added. Prove that this greedy algorithm is a 2-approximation and
can be implemented in O(mlog n) time, where m=|E|and n=|V|.
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