Homework 8 Due Wednesday, April 12

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Homework 7Due Wednesday, April 12
1. Show that 2-SAT is in P where 2-SAT ? ?
is a formula and is in conjunctive normal
form with 2 literals per clause and ? is in
SAT
2. Prove that if P NP then ? i?0, Sip Si1p
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3. True or False or Unknown (Prove your answer
if T or F) ? A,B, If B ? P and Altm B, then A ? P
Problem 4. Sipser 7.26 Sipser describes a game
PUZZLE in which K cards are placed in a box in
one of two orientations flipped around a vertical
axis. There are R rows per card with a hole
either on the left side of the card, right side
of the card, both, or neither. PUZZLE is those
sets of cards that have an set of orientations
that completely covers the bottom of the
box. Show that PUZZLE is NP-complete.
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5 Show that co-NP P1,p where
co-NP L L is in NP P1,p L ? k, a
language L in P s.t. L x ? w,
w O(xk), s.t.
(w,x) ? L
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6. Prove or disprove ATM is NP-hard 7.
Prove that L is NP-complete where L (G,k)
undirected graph G has a clique-cover of size
k A clique-cover of a graph is a set of cliques
s.t. every vertex of the graph is a member of at
least one of the cliques. Note that the size of
the clique-cover is the number of cliques. Use
VERTEX-COVERpairs and eliminate triangles
first. 8. Prove that Undirected_HP is
NP-complete using Direct_HP 9. Define TSP
(Travelling Salesperson Problem) and prove that
TSP is NP_complete using Undirected_HP