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Math/CS 262 Spring 2011 Homework #10 (Due Monday, May 2nd by 4p) 1. Satisfiability and 3-Sat NOTE: 3-Sat is NP If we assume 𝑛 variables and 𝑚 clauses in a 3-Sat instance. If it takes 𝑘 steps to verify that a clause is satisfied then it just takes 𝑘𝑚 steps to verify whether a given guess satisfies all the clauses. a. A “brute force” approach to 3-Sat would work through all possible truth assignments to the Boolean variables to try to find one that satisfies all the clauses. If each truth assignment takes 𝑘𝑚 steps to verify whether it satisfies, how many steps would this approach take to find there is no satisfying truth assignment? b. Now let 𝑣 represent the number of variables, and 𝑐 represent the number of clauses in a Satisfiability instance. Further, assume that the largest size clause has size 𝑠. What is a bound on the size of the 3-Sat instance our transformation produces? (Can you bound the number of variables and clauses produced?) For the transformation to be valid, each of these must be bounded by a polynomial in 𝑣 and 𝑐. 2. Graph 3-Color: Assume the instance is a graph with 𝑣 vertices and 𝑒 edges. NOTE: Graph 3-Color is NP If it takes 𝑘 steps to verify that the two endpoints of an edge are different colors. (𝑘 would be a fixed number.) Then it just takes 𝑘𝑒 steps to verify whether a given guess is a valid coloring. a. A “brute force” approach would work through all possible color assignments (using three colors) to try to find one that represented a valid coloring. If each color assignment takes 𝑘𝑒 steps to verify whether it’s valid, how many steps would this approach take to find that there is no 3-coloring? b. Given an instance of 3-Sat with 𝑛 variables and 𝑚 clauses, what is the size of the graph our transformation produces? (How many vertices and edges would the graph have?) Our transformation is only valid if these sizes are bounded by a polynomial in 𝑛 and 𝑚. 3. Consider the following three problems: a. Set Packing Instance: A collection 𝐶 = {𝑐𝑖 : 1 ≤ 𝑖 ≤ 𝑛} of finite sets and a positive integer 𝐾 < 𝑛. Question: Does C contain 𝐾 disjoint sets? b. Exact Cover by 3 Sets Instance: A set 𝑆 with |𝑆| = 3𝑛 and a collection 𝐶 = {𝑐𝑖 : 1 ≤ 𝑖 ≤ 𝑘} where 𝑐𝑖 ⊂ 𝑆 and |𝑐𝑖 | = 3 for each 𝑖. Question: Does 𝐶 contain an exact cover? (I.e., is there a subcollection 𝐵 of 𝐶 such that each element of 𝑆 appears in exactly one of the sets in 𝐵?) c. 3-Dim Matching Instance: A complete set of triples S = W x X x Y (with |𝑊| = |𝑋| = |𝑌| = 𝑛) and a subset A ⊆ S of “allowable” triples Question: Does A contain a matching? (I.e., is there a subset B of A with the property that each element of W, X and Y appears in exactly one of the triples in B?) Your task: Use the fact that “3-Dim Matching” is NP Hard to show that “3-Set Exact Cover” is NP Hard. Then show “Set Packing” is NP Hard. Hint: Use the restriction method – i.e. show that an instance of the known NP Hard problem already is an instance of the new problem as a special case. 4. Vertex Cover Show that “Vertex Cover” is NP hard by transforming an arbitrary instance of “3-Sat” into an equivalent instance of “Vertex Cover”. I suggest you use the method of “component design”. The vertex cover examples from the last HW are intended to be a hint. 5. Partition into Triangles Instance: a graph G = (V, E) with |𝑉| = 3𝑛. Question: Is there a partition of V into disjoint subsets 𝑉 = ⋃𝑛𝑖=1 𝑇𝑖 such that each 𝑇𝑖 is consists of three vertices that form a triangle in G. (In other words, all three potential edges between the three vertices are in G.) Show that “Partition into Triangles” is NP hard by transforming an arbitrary instance of “Exact cover by 3 Sets” into an equivalent instance of “Partition into Triangles”. What follows are hints for this last problem. Consider the following gadget: x y z In your constructed graph, the blue vertices will have no edges other than the ones shown. The black vertices may be connected to multiple gadgets. If all three black vertices are available then there are two ways to partition the blue vertices of the gadget into triangles. Triangles Choice 1: This may only be chosen if the three black vertices are available. x y z Triangles Choice 2: If the gadget is a piece of a bigger graph then we would also have the following option, which excludes {x,y,z}. x y z Practice Exercise: Does the following graph have a partition into triangles? a b d e c f