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1 Design and Analysis of Algorithms Yoram Moses Lecture 11 June 3, 2010 http://www.ee.technion.ac.il/courses/046002
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2 Nondeterministic Polynomial Time (NP)
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3 Shortest Path: Search, Existence, Verification Search problem: Input: (G,w,s,t): a directed graph G with weight function w, a source s, and a sink t. Goal: find a shortest path from s to t. (or reject if none exists) Complexity: our solution runs in O(VE) = O(n 2 ) (Notice n = size of input = O(V+E)) Existence problem: Input: (G,w,s,t,k): G,w,s,t are as before + a number k. Goal: decide whether there is a path from s to t of length ≤ k. Complexity: our solution runs in O(VE) = O(n 2 ) Verification problem: Input: (G,w,s,t,k,p): G,w,s,t,k as before. p is a path in G. Goal: decide whether p is a simple path from s to t of length ≤ k. Complexity: O(V) = O(n).
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4 Max Flow: Search, Existence, Verification Search problem: Input: (G,c,s,t): a directed graph G with capacity function c, a source s, and a sink t. Goal: find a maximum flow in G. (or reject if none exists) Complexity: O(VE 2 ) = O(n 3 ) Existence problem: Input: (G,c,s,t,k): G,c,s,t as before + a number k. Goal: decide whether there is a flow in G with value ≥ k. Complexity: O(VE 2 ) = O(n 3 ) Verification problem: Input: (G,c,s,t,k,f): G,c,s,t,k as before. f is a function from edges of G to real numbers. Goal: decide whether f is a legal flow with value ≥ k. Complexity: O(E) = O(n).
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5 Hamiltonian Cycle: Search, Existence, Verification Search problem: Input: an undirected graph G. Goal: find a Hamiltonian cycle in G (or reject if none exists). Complexity: O(VxV!) = O(n2 n log n ) Existence problem: Input: an undirected graph G. Goal: decide whether G has a Hamiltonian cycle. Complexity: O(VxV!) = O(n2 n log n ) Verification problem: Input: (G,p): an undirected graph G and a sequence of nodes p. Goal: decide whether p is a Hamiltonian cycle in G. Complexity: O(V) = O(n).
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6 3-Coloring: Search, Existence, Verification Search problem: Input: G: an undirected graph Goal: find a 3-Coloring of G. (or reject if none exists) Complexity: O(E 3 V ) = O(n2 n log 3 ) Existence problem: Input: G: as before. Goal: decide whether G has a 3-Coloring. Complexity: O(E 3 V ) = O(n2 n log 3 ) Verification problem: Input: (G, ): G as before and : V {1,2,3}. Goal: decide whether is a 3-Coloring of G. Complexity: O(E) = O(n).
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7 Search and Existence vs. Verification Conclusion: in many natural examples: Search and existence are computationally equivalent Verification is easier Sometimes it’s just a little easier (Shortest Path, Max flow) Sometimes it’s a lot easier (Hamiltonian cycle, 3-Coloring)
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8 Verification Relations Language: L {0,1} * Definition: A verification relation for L is a relation R {0,1} * {0,1} * s.t. for all x {0,1} * : x L there is at least one y {0,1} * s.t. (x,y) R. x L there is no y {0,1} * s.t. (x,y) R. y is called the “certificate” for x A.k.a. its “witness” or “proof” Remarks: Every input x L has at least one certificate y. If (x,y) R, then y is a certificate for x. An input x L may have several certificates. A language L has many verification relations.
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9 Verification Relations: Examples Shortest path: x = (G,w,s,t,k), y = a path p Language: {(G,w,s,t,k): G has an s-t path of length ≤ k} Certificate: s-t path of length ≤ k Verification relation: {((G,w,s,t,k),p): p is an s-t path of length ≤ k in G} Hamiltonian cycle: x = undirected graph G, y = a path p Language: G that has a Hamiltonian cycle Certificate: a Hamiltonian cycle in G Verification relation: {(G,p): p is a Hamiltonian cycle in G}
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10 Nondeterministic Polynomial Time Definition: A binary relation R is polynomially bounded, if there exists some c > 0 s.t. for every (x,y) R, |y| ≤ |x| c. Definition: L is polynomial-time verifiable, if it has a verification relation R, which satisfies both: R is polynomially bounded, and R is polynomial-time decidable. Definition: The class NP (Nondeterministic Polynomial Time) is the set of all polynomial-time verifiable languages.
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11 NP: Examples Examples of languages in NP: Decision Shortest Path, Decision Max Flow, Decision LP Hamiltonian Cycle, TSP, 3-Coloring, SAT, k-SAT, Clique Examples of languages not known to be in NP: HC-complement: given a graph G, decide whether G has no Hamiltonian cycles.
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12 Definition: A nondeterministic algorithm is an algorithm N that, on input x, First, N “nondeterministically” guesses a “witness” y. Then, N runs a deterministic “verification” algorithm on (x,y). Note: N may make different nondeterministic guesses in different runs on the same input x. Nondeterministic Algorithms Nondeterministic guess Nondeterministic Algorithm N Verification x y yes/no (x,y)(x,y)
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13 Decision by Nondeterministic Algorithms Definition: A nondeterministic algorithm N is said to decide language L if: For every input x L, there is at least one guess y s.t. N accepts (x,y). For every input x L, the verification algorithm N rejects (x,y), for all guesses y. A polynomial-time nondeterministic algorithm is one in which The guesses (y’s) are of polynomial size (in |x|), and The verification algorithm runs in polynomial time. Lemma: L NP iff L is decidable by a polynomial-time nondeterministic algorithm.
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14 An NP Algorithm for Clique Nondeterministic guess (input: x = (G,k)) 1.for i = 1,…,k 2. v i nondeterministic guess of a node in V=V(G) 3.output y = (v 1,…,v k ) Verification algorithm (input: (x,y)) 1.If x is not a valid encoding of a graph G and an integer k, reject. 2.If y is not a valid encoding of k nodes v 1,…,v k in G, reject. 3.If v 1,…,v k are not distinct, reject. 4.for i 1,…,k-1 do 5. for j i+1,…,k do 6. if {v i,v j } E reject. 7.accept
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15 Lemma: P NP Biggest open problem of computer science: is P = NP? Two possibilities: Current belief: P NP Search & Existence strictly harder than Verification. P = NP? P vs. NP P = NP P NP
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16 f: N N: a complexity measure. Time(f(n)) = all languages decidable in time O(f(n)). Lemma: Let f(n),g(n) be two complexity measures. If there exists a constant c, s.t. for all n > c, f(n) ≤ g(n), then Time(f(n)) Time(g(n)). Theorem (Time Hierarchy) Let f(n),g(n) be two complexity measures. If there exists a constant c, s.t. for all n > c, f(n) ≤ g(n) 1/2, then Time(f(n)) Time(g(n)). Time Hierarchy
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17 P Definition: Lemma: P EXP but P EXP Lemma: NP EXP (exercise) Open problem: is NP = EXP? 3 Possibilities: P, NP, and EXP P EXP NP P EXP = NP NP = P EXP
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18 NP-Completeness (NPC) Problems in NP not known to be in P: Hamiltonian Cycle, Clique, SAT, k-SAT (k ≥ 3), k-Coloring (k ≥ 3), TSP, …. (many others) All of these are “NP-Complete” NP-Complete Problems: Belong to NP If any of them belongs to P, then NP = P. Two possibilities: NPC P NP NP = NPC = P
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19 NP-Hardness (NPH) Definition: A language L is NP-hard if L ’ ≤ p L holds for all L ’ NP. NPH = class of all NP-hard problems. Lemma: If any NP-hard problem belongs to P, then NP = P. If one NPH problem is easy, then all of NP is easy. Lemma: If L NPH and L ≤ p L ’, then L ’ NPH.
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20 NP-Completeness Definition: A language L is NP-complete if both L NP and L is NP-hard NPC = class of NP-complete problems NPC = NP NPH Theorem: If some NPC language is in P, then P = NP. (P NPC NP = P = NPC). If some NPC language is not in P, then no NPC language is in P. (NPC P P NPC = NP P).
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21 NP-Completeness NPC: “hardest” problems in NP Behave as a “single block”: either all in P or all outside P Lemma: If L 1,L 2 NPC, then both L 1 ≤ p L 2 and L 2 ≤ p L 1.
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22 Proving NP-Completeness How to prove that a given language L is NPC? Show that L NP, and Show that L ’ ≤ p L holds for every L ’ NP. Easier alternative: Show that L NP, and Find some NPC problem L ’ and show L ’ ≤ p L. How do we obtain the first NPC problem? Using the first alternative Cook-Levin theorem: Circuit-SAT is NP-complete.
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23 NP-Completeness: the Full Recipe To show that L is NPC: Prove L is in NP Show a polynomial time nondeterministic algorithm for L Select an NPC problem L ’ Show a polynomial-time reduction f from L ’ to L Prove that x L ’ iff f(x) L Show a polynomial-time algorithm to compute f
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24 Example: Clique is NPC Clique is in NP (seen today) 3-SAT is NPC (will show this later on) 3-SAT ≤ p Clique (seen in previous lecture) Therefore: Clique is also NP-Complete!
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25 End of Lecture 11
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