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# hw1(5) - CS 473G Combinatorial Algorithms Fall 2005...

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CS 473G: Combinatorial Algorithms, Fall 2005 Homework 1 Due Tuesday, September 13, 2005, by midnight (11:59:59pm CDT) Name: Net ID: Alias: Name: Net ID: Alias: Name: Net ID: Alias: Starting with Homework 1, homeworks may be done in teams of up to three people. Each team turns in just one solution, and every member of a team gets the same grade. Neatly print your name(s), NetID(s), and the alias(es) you used for Homework 0 in the boxes above. Staple this sheet to the top of your answer to problem 1. There are two steps required to prove NP-completeness: (1) Prove that the problem is in NP, by describing a polynomial-time verification algorithm. (2) Prove that the problem is NP-hard, by describing a polynomial-time reduction from some other NP-hard problem. Showing that the reduction is correct requires proving an if-and-only-if statement; don’t forget to prove both the “if” part and the “only if” part. Required Problems 1. Some NP-Complete problems (a) Show that the problem of deciding whether one graph is a subgraph of another is NP- complete. (b) Given a boolean circuit that embeds in the plane so that no 2 wires cross, PlanarCir- cuitSat is the problem of determining if there is a boolean assignment to the inputs that makes the circuit output true. Prove that PlanarCircuitSat is NP-Complete. (c) Given a set S with 3 n numbers, 3partition is the problem of determining if S can be partitioned into n disjoint subsets, each with 3 elements, so that every subset sums to the same value. Given a set S and a collection of three element subsets of S , X3M (or exact 3-dimensional matching ) is the problem of determining whether there is a subcollection of n disjoint triples that exactly cover S . Describe a polynomial-time reduction from 3partition to X3M.

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CS 473G Homework 1 (due September 13, 2005) Fall 2005 (d) A domino is a 1 × 2 rectangle divided into two squares, each of which is labeled with an integer.
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hw1(5) - CS 473G Combinatorial Algorithms Fall 2005...

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