hw6(2) - CS 373 Combinatorial Algorithms Fall 2002...

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CS 373: Combinatorial Algorithms, Fall 2002 http://www-courses.cs.uiuc.edu/˜cs373 Homework 6 (Do not hand in!) Name: Net ID: Alias: U 3 / 4 1 Name: Net ID: Alias: U 3 / 4 1 Name: Net ID: Alias: U 3 / 4 1 Neatly print your name(s), NetID(s), and the alias(es) you used for Homework 0 in the boxes above. Please also tell us whether you are an undergraduate, 3 / 4-unit grad student, or 1-unit grad student by circling U, 3 / 4 , or 1, respectively. Staple this sheet to the top of your homework. Required Problems 1. (10 points) Prove that SAT is still a NP-complete problem even under the following constraints: each variable must show up once as a positive literal and once or twice as a negative literal in the whole expression. For instance, ( A ¯ B ) ( ¯ A C D ) ( ¯ A B ¯ C ¯ D ) satisfies the constraints, while ( A ¯ B ) ( ¯ A C D ) ( A B ¯ C ¯ D ) does not, because positive literal A appears twice. 2. (10 points) A domino is 2 × 1 rectanble divided into two squares, with a certain number of pips(dots) in each square. In most domino games, the players lay down dominos at either end of a single chain. Adjacent dominos in the chain must have matching numbers. (See the figure below.) Describe and analyze an e cient algorithm, or prove that it is NP-complete, to determine wheter a given set of n dominos can be lined up in a single chain. For example, for the sets of dominos shown below, the correct output is TRUE. Top: A set of nine dominos Bottom:The entire set lined up in a single chain
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3. (10 points) Prove that the following 2 problems are NP-complete. Given an undirected Graph G = ( V , E ), a subset of vertices V 0 V , and a positive integer k : (a) determine whether there is a spanning tree
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This note was uploaded on 12/15/2009 for the course 942 cs taught by Professor A during the Spring '09 term at University of Illinois at Urbana–Champaign.

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hw6(2) - CS 373 Combinatorial Algorithms Fall 2002...

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