 
Summary: COMPLEXITY: Exercise No. 3
due in two weeks
1. For each of the following statements, prove, disprove or show that it is an open problem:
(a) If L 1 ; L 2 2 coNP then L 1 `` L 2 2 coNP. (Test 99)
(b) If L 2 NP;L 1 ( L and L 1 2 NP then L \Gamma L 1 2 coNP.
(c) If L 2 NPC;L 1 ( L and L 1 2 SPACE(log n) then L \Gamma L 1 2 NPC.
(d) If L 2 NPC then fxx : x 2 Lg 2 NPC. (Test 94)
2. Are the following problems NPcomplete or polynomial? (prove)
EXACTLY ONE SAT:
Instance: A CNF Boolean formula \Phi.
Question: Is there an assignments to the variables of \Phi such that in each clause of \Phi there
is exactly one TRUE literal ?
1=2 SAT:(Test 96)
Instance: A CNF formula \Phi.
Question: Is there an assignment that satisfies \Phi in which exactly half of the variables are
TRUE?
VERTEX COVER (VC):
Instance: A graph G = (V; E) and an integer k.
Question: Does G have a vertex cover of size Ÿ k? (a vertex cover is a set U ` V , such that
for every edge (u; v) 2 E either u 2 U or v 2 U (or both))
