CMPSCI 611: Advanced Algorithms Micah Adler Summary: CMPSCI 611: Advanced Algorithms Micah Adler Problem Set 4 Out: November 14, 2002 Due: November 21, 2002 1. (a) Recall the Maximum Weight Forest (MWF) and Minimum Spanning Tree (MST) problems. De ne appropriate decision problems for MWF and MST, and demonstrate that MWF P MST. (b) A natural attempt to show that Max-Cut P Min-Cut would be a reduction of the following sort: Given G, k as input to the Max-Cut problem, produce an input to the Min-Cut problem G 0 , k 0 , where G 0 is the complement of G (i.e., there is an edge between u and v in G 0 if and only if there is no edge between u and v in G.) Describe why this approach does not work. (c) Show that Max-Cut P Min-Cut if and only if P = NP. 2. Show that if SAT 2 P, then, given a Boolean formula, we can nd a satisfying assignment to that formula (if one exists) in polynomial time. Keep in mind that SAT is a decision problem, but you are being asked to nd an actual assignment. 3. [CLRS] 34-3: Graph coloring. 4. Consider the 1 3 -Clique problem: Input: A graph G. Question: Does G have a Clique of size exactly dn=3e, where n is the number of nodes in G? Collections: Computer Technologies and Information Sciences