08 - NP-complete coloring problems- 3-coloring

# 08 - NP-complete coloring problems- 3-coloring - Packing?...

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3/28/08 - NP-complete coloring proble. .. NP-complete problems 3SAT, k-SAT (k > 3) IND. SET CLIQUE VERTEX COVER HAMILTONIAN PATH/CYCLE TRAV. SALES. PROB. Reduce FROM . .. TO (e.g. set cover) if you could use set cover to solve trav salesman in poly time, then you know it is hard Set Cover Set Packing Set cover/packing: Input consists of a set U. (“universal set”) subsets S 1 , S 2 , . .., S n U a number k > 0 Cover: Output “yes” if U can be covered by k of these Packing: Output “yes” if U contains k disjoint sets among S 1 , . .., S n Packing: A = {1,2,4} B = {1,6} Set Packing/Ind. Set (start from a graph and create an instance of 29-1 29

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set-packing) Set cover/Vertex cover wrong way: start from set packing and reduce to ind. set. Ind. Set in degree-3 graphs thought process of choosing which problem to reduce from. one with same type of target value some are cover problems, some are packing problems (set of bad events - need to make none of them happen) Optimization? k? k? Covering? (Good events - make them all happen)
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Unformatted text preview: Packing? (Bad events - make none of them happen) Sequencing? Ind. Set in Max. Degree-3 Graphs From 3SAT to I-Set. Variable Gadgets Clause gadgets Variable gadgets labeled x1, x ̅ 1, x2, x ̅ 2, x3, x ̅ 3 Create a graph where every vertex in every clause gadget has degree 3. a vertex in a variable gadget can have high degree if connected to many clauses. Improve reduction so that we can create an ind. set instance with degree 3 Can we modify the variable gadgets so when everything hooked up we are in a graph of max. degree-3 k = half of each var. gadget and one extra vtx from each clause gadget 3-DIM MATCHING (Sec. 8.6??) Given 3 sets X, Y, Z each with n elements 29-2 Given m sets S 1 , . .., S m each containing exactly one X, one Y, one Z. Output “yes” if there is a subcollection S i1 , S i2 , . .., S in covering each element of the X ∪ Y ∪ Z exactly once. 29-3...
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## This note was uploaded on 10/02/2008 for the course CS 482 taught by Professor Kleinberg during the Spring '08 term at Cornell.

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08 - NP-complete coloring problems- 3-coloring - Packing?...

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