NP-Hardness Proofs With What Problems We . . . What Problems We . . - PowerPoint PPT Presentation

NP-Hardness Proofs . . . Usual Proofs of NP- . . . Proofs of NP- . . . Pedagogical Problem . . . NP-Hardness Proofs With What Problems We . . . What Problems We . . . Realistic Computers Instead What Is NP-Hard: . . . Proof that . . . of Turing Machines: Towards Proof that . . . Making Theory of Home Page Title Page Computation Course More ◭◭ ◮◮ Understandable and ◭ ◮ Relevant Page 1 of 12 Go Back Olga Kosheleva 1 and Vladik Kreinovich 2 Full Screen Departments of 1 Teacher Education and 2 Computer Science Close University of Texas at El Paso, El Paso, TX 79968, USA olgak@utep.edu, vladik@utep.edu Quit

NP-Hardness Proofs . . . Usual Proofs of NP- . . . 1. NP-Hardness Proofs Are Important Proofs of NP- . . . • In many applications areas, certain problems are known Pedagogical Problem . . . to be NP-hard (= provably computationally intractable). What Problems We . . . What Problems We . . . • Knowing that a general problem is NP-hard helps the What Is NP-Hard: . . . researchers to concentrate on easier-to-solve problems: Proof that . . . – to find a practically useful easier-to-solve subclass Proof that . . . of problems, or Home Page – to replace the original problem with a relaxed easier- Title Page to-solve problem. ◭◭ ◮◮ • For example, we may only want an approximate solu- ◭ ◮ tion, or an answer which is correct w/high probability. Page 2 of 12 • It is important to make sure that the new problem is indeed easier-to-solve. Go Back • Thus, it is desirable that the students learn how to Full Screen prove NP-hardness or different problems. Close Quit

NP-Hardness Proofs . . . Usual Proofs of NP- . . . 2. Usual Proofs of NP-Hardness Proofs of NP- . . . • A historically first problem proven to be NP-hard is Pedagogical Problem . . . propositional satisfiability . What Problems We . . . What Problems We . . . • This problem is about propositional formulas , i.e., ex- What Is NP-Hard: . . . pressions F like ( x 1 & x 2 ) ∨ ( x 2 & ¬ x 3 ) obtained: Proof that . . . – from propositional (“yes”-“no”) variables x 1 , . . . , x n , Proof that . . . – by using “and” (&), “or” ( ∨ ), and “not” ( ¬ ). Home Page • We are given a propositional formula F , we must find Title Page values x 1 , . . . , x n that make it true. ◭◭ ◮◮ • The usual NP-hardness proof uses Turing machines , a ◭ ◮ simple theoretical computer designed in 1936. Page 3 of 12 • A Turing machine is, in effect, a tape recorder with a Go Back simple controller and a potentially extendable tape. Full Screen • For example, in the Turing machine, there is no imme- diate access to a memory cell at a given location. Close Quit

NP-Hardness Proofs . . . Usual Proofs of NP- . . . 3. Proofs of NP-Hardness (cont-d) Proofs of NP- . . . • The only way to get to a cell #1,000,000 is to go from Pedagogical Problem . . . cell #0 to cell #1, to cell #2, . . . , to cell #1,000,000. What Problems We . . . What Problems We . . . • It is amazing to learn that complex computations can What Is NP-Hard: . . . be performed on such a primitive computer. Proof that . . . • However, when it comes to proving that no efficient Proof that . . . algorithm exists: Home Page – the fact that, for some problem, no efficient solu- Title Page tions are possible on a Turing machine ◭◭ ◮◮ – is not a very convincing argument that this is im- ◭ ◮ possible on (more complex) real computers. Page 4 of 12 • Yes, there are proofs that Turing machines are suffi- Go Back cient for proving NP-hardness. Full Screen • However, these proofs are beyond the scope of most textbooks. Close Quit

NP-Hardness Proofs . . . Usual Proofs of NP- . . . 4. Pedagogical Problem and What We Do About Proofs of NP- . . . It Pedagogical Problem . . . • As we mentioned, for students, Turing-machine-based What Problems We . . . NP-hardness proofs are not convincing at all. What Problems We . . . What Is NP-Hard: . . . • We propose a new version of the proof of NP-hardness Proof that . . . of propositional satisfiability. Proof that . . . • This proof that uses a much more realistic (and gen- Home Page eral) model of a computer than Turing machine. Title Page • This proof is somewhat more complex than the Turing- ◭◭ ◮◮ machine-based proofs. ◭ ◮ • However, our model (and hence this proof) is closer to Page 5 of 12 the actual computers and is, thus, easier to understand. Go Back Full Screen Close Quit

NP-Hardness Proofs . . . Usual Proofs of NP- . . . 5. What Problems We Are Solving: Examples Proofs of NP- . . . • In mathematics, we are given a statement x and we Pedagogical Problem . . . want to find the proof y of either x or ¬ x . What Problems We . . . What Problems We . . . • Once we have a detailed proof y , it is easy to check its What Is NP-Hard: . . . correctness, but inventing a proof is hard. Proof that . . . • A proof cannot be too long: it must be checkable. Proof that . . . Home Page • In physics, we have observations x , and we want to find a law y that describes them. Title Page • Once we have y we can easily check whether it fits x , ◭◭ ◮◮ but coming up with y is often difficult. ◭ ◮ • A law cannot be too long: otherwise, we can take the Page 6 of 12 data sa the law. Go Back • In engineering, we have a specification x , and we need Full Screen to find a design y that satisfies x . Close Quit

NP-Hardness Proofs . . . Usual Proofs of NP- . . . 6. What Problems We Are Solving: General De- Proofs of NP- . . . scription Pedagogical Problem . . . • In general : What Problems We . . . What Problems We . . . – we have a string x , and What Is NP-Hard: . . . – we need to find y s.t. C ( x, y ) and len( y ) ≤ P ℓ (len( x )). Proof that . . . • Here, C ( x, y ) is a feasible property, i.e., a property that Proof that . . . can be checked feasibly (in polynomial time). Home Page • In such problems: Title Page ◭◭ ◮◮ – once we have a guess y , – we can check its correctness in polynomial time. ◭ ◮ Page 7 of 12 • “Computations” allowing guesses are known as non- deterministic . Go Back • Thus, such problems are called Non-deterministic Poly- Full Screen nomial (NP). Close Quit

NP-Hardness Proofs . . . Usual Proofs of NP- . . . 7. What Is NP-Hard: Reminder Proofs of NP- . . . • Ideally, we would like to call a problem hard if it cannot Pedagogical Problem . . . be solved by a feasible (polynomial-time) algorithm. What Problems We . . . What Problems We . . . • Alas, for neither of the problems from NP, we can prove What Is NP-Hard: . . . that this problem is hard in this sense. Proof that . . . • What we do know is that some problems are harder Proof that . . . than others in the following sense: Home Page – every instance of a problem A Title Page – can be reduced to an appropriate instance of the ◭◭ ◮◮ problem B . ◭ ◮ • A problem is called NP-hard if every problem from NP Page 8 of 12 can be reduced to it. Go Back • In other words, a problem is NP-hard if it is harder Full Screen than all other problems from the class NP. Close Quit

NP-Hardness Proofs . . . Usual Proofs of NP- . . . 8. Proof that Satisfiability Is NP-Hard: Idea Proofs of NP- . . . • We have an instance of an NP problem: given x find y Pedagogical Problem . . . for which C ( x, y ) is true and len( y ) ≤ P ℓ (len( x )). What Problems We . . . What Problems We . . . • We want to reduce it to propositional satisfiability. What Is NP-Hard: . . . • We start with a computational device that, given a Proof that . . . string x of length len( x ) = n and y , checks C ( x, y ). Proof that . . . Home Page • Computing C requires polynomial time T ≤ P ( n ). Title Page • During this time, only cells at distance ≤ R = c · T from the origin can influence the result. ◭◭ ◮◮ • Let ∆ V be the smallest cell volume. ◭ ◮ • Within the sphere of volume V = 4 3 · π · R 3 ∼ T 3 , there Page 9 of 12 are ≤ V Go Back ∆ V ∼ T 3 cells, fewer than ≤ const · ( P ( n )) 3 . Full Screen • So, we have no more than polynomially many cells. Close Quit

NP-Hardness Proofs . . . Usual Proofs of NP- . . . 9. Proof that Satisfiability Is NP-Hard (cont-d) Proofs of NP- . . . • Let ∆ t be a time quantum. Pedagogical Problem . . . What Problems We . . . • The state S i,t +1 cell i at moment ( t +1) · ∆ t can only be What Problems We . . . influenced by states S j,t of cells at distance ≤ r = c · ∆ t . What Is NP-Hard: . . . 3 · π · r 3 • In this vicinity, there are ≤ N neighb = 4 ∆ V cells; Proof that . . . this number does not depend on the inputs size n : Proof that . . . Home Page S i,t +1 = f i,t ( S i,t , S j,t , . . . ( ≤ N neighb terms)) . Title Page • Let S be the largest number of states of each cell. ◭◭ ◮◮ • We can describe each state as 0, 1, 2, . . . ◭ ◮ def • Then we need B = ⌈ log 2 ( S ) ⌉ bits s i,b,t , 1 ≤ b ≤ B , to Page 10 of 12 describe each state S i,t , so: Go Back s i,b,t +1 = f i,t ( s i, 1 ,t , . . . , s i,B,t , s j, 1 ,t , . . . , s j,B,t , . . . ) . Full Screen • We can then use a truth table to transform each such Close equation to a propositional formula F i,b,t . Quit