Formulation of the . . . Case of Probabilistic . . . How the Answer . . . Main Result: . . . Coming Up with a Good What Is NP-Hard? . . . Question Is Not Easy: Case of Fuzzy Uncertainty How Degrees Change . . . A Proof Main Result: Fuzzy Case What Happens in the . . . Joe Lorkowski 1 , Luc Longpr´ e 1 , Home Page Olga Kosheleva 2 , and Salem Benferhat 3 Title Page Departments of 1 Computer Science and 2 Teacher Education ◭◭ ◮◮ University of Texas at El Paso, 500 W. University El Paso, Texas 79968, USA ◭ ◮ lorkowski@computer.org, longpre@utep.edu, olgak@utep.edu Page 1 of 17 3 Centre de Recherche en Informatique de Lens CRIL Go Back Universit´ e d’Artois, F62307 Lens Cedex, France benferhat@cril.univ-artois.fr Full Screen Close Quit
Formulation of the . . . Case of Probabilistic . . . 1. Formulation of the Problem How the Answer . . . • Even after a very good lecture, some parts of the ma- Main Result: . . . terial remain not perfectly clear. What Is NP-Hard? . . . Case of Fuzzy Uncertainty • A natural way to clarify these parts is to ask questions How Degrees Change . . . to the lecturer. Main Result: Fuzzy Case • Ideally, we should be able to ask a question that im- What Happens in the . . . mediately clarifies the desired part of the material. Home Page • Coming up with such good questions is a skill that Title Page takes a long time to master. ◭◭ ◮◮ • Even for experienced people, it is not easy to come up ◭ ◮ with a question that maximally decreases uncertainty. Page 2 of 17 • In this talk, we prove that the problem of designing a Go Back good question is computationally difficult (NP-hard). Full Screen Close Quit
Formulation of the . . . Case of Probabilistic . . . 2. Towards Describing the Problem in Precise How the Answer . . . Terms: General Case Main Result: . . . • A complete knowledge about an area means that we What Is NP-Hard? . . . have the full description. Case of Fuzzy Uncertainty How Degrees Change . . . • Uncertainty means that several different variants Main Result: Fuzzy Case v 1 , v 2 , . . . , v n are consistent with our knowledge. What Happens in the . . . • A “yes”-‘’no” question is a question an answer to which Home Page eliminates some possible variants: Title Page – if the answer is “yes”, then we are limited to vari- ◭◭ ◮◮ ants v ∈ Y ⊂ { v 1 , . . . , v n } consistent with “yes”; ◭ ◮ – if the answer is “no”, then we are limited to variants v ∈ N ⊂ { v 1 , . . . , v n } consistent with “no”. Page 3 of 17 • These two sets are complements to each other. Go Back • For the question “is v = v 1 ?”, Y = { v 1 } and N = Full Screen { v 2 , . . . , v n } . Close Quit
Formulation of the . . . Case of Probabilistic . . . 3. Case of Probabilistic Uncertainty How the Answer . . . • In the probabilistic approach, we assign a probability Main Result: . . . n What Is NP-Hard? . . . p i ≥ 0 to each of the possible variants: � p i = 1. i =1 Case of Fuzzy Uncertainty • The probability p i is the frequency with which the i -th How Degrees Change . . . variant was true in similar previous situations. Main Result: Fuzzy Case What Happens in the . . . • In the probabilistic case, Shannon’s entropy S de- Home Page scribes the amount of uncertainty: Title Page n � S = − p i · ln( p i ) . ◭◭ ◮◮ i =1 ◭ ◮ • We want to select a question that maximizes the ex- Page 4 of 17 pected decrease in uncertainty. Go Back Full Screen Close Quit
Formulation of the . . . Case of Probabilistic . . . 4. How the Answer Changes the Entropy How the Answer . . . • If the answer is “yes”, then for i ∈ N , we get p ′ i = 0, Main Result: . . . and for i ∈ Y , we get What Is NP-Hard? . . . p i Case of Fuzzy Uncertainty p ′ � i = p ( i | Y ) = p ( Y ) , where p ( Y ) = p i . How Degrees Change . . . i ∈ Y Main Result: Fuzzy Case • So, entropy changes to S ′ = − � p ′ i · ln( p ′ i ) . What Happens in the . . . i ∈ Y Home Page • If the answer is “no”, then for i ∈ Y , we get p ′′ i = 0, Title Page and for i ∈ N , we get p i ◭◭ ◮◮ � p ′′ i = p ( i | N ) = p ( N ) , where p ( N ) = p i . ◭ ◮ i ∈ N • So, entropy changes to S ′′ = − � Page 5 of 17 p ′′ i · ln( p ′′ i ) . i ∈ N Go Back • We want to maximize the expected decrease in entropy: Full Screen p ( Y ) · ( S − S ′ ) + p ( N ) · ( S − S ′′ ) . Close Quit
Formulation of the . . . Case of Probabilistic . . . 5. Main Result: Probabilistic Case How the Answer . . . • Our main result is that the problem of coming up with Main Result: . . . the best possible question is NP-hard. What Is NP-Hard? . . . Case of Fuzzy Uncertainty • What is NP-hard: a brief reminder. How Degrees Change . . . • In many real-life problems, we are looking for a string Main Result: Fuzzy Case that satisfies a certain property. What Happens in the . . . • For example, in the subset sum problem: Home Page – we are given positive integers s 1 , . . . , s n represent- Title Page ing the weights, and ◭◭ ◮◮ – we need to divide these weights into two groups ◭ ◮ with exactly the same weight. Page 6 of 17 • So, we need to find a set I ⊆ { 1 , . . . , n } s.t. Go Back � n � s i = 1 � � 2 · s i . Full Screen i ∈ I i =1 Close Quit
Formulation of the . . . Case of Probabilistic . . . 6. What Is NP-Hard? (cont-d) How the Answer . . . • The desired set I can be described as a sequence of n Main Result: . . . 0s and 1s: the i -th term is 1 if i ∈ I and 0 if i �∈ I . What Is NP-Hard? . . . Case of Fuzzy Uncertainty • In principle, we can solve each such problem by simply How Degrees Change . . . enumerating all possible strings. Main Result: Fuzzy Case • For example, in the above case, we can try all 2 n pos- What Happens in the . . . sible subsets of the set { 1 , . . . , n } . Home Page • This way, if there is a set I with the desired property, Title Page we will find it. ◭◭ ◮◮ • The problem is that for large n , the number 2 n of com- ◭ ◮ putational steps becomes unreasonably large. Page 7 of 17 • For example, for n = 300, the resulting computation Go Back time exceeds lifetime of the Universe. Full Screen • Can we solve such problems in feasible time, i.e., in time ≤ a polynomial of the size of the input? Close Quit
Formulation of the . . . Case of Probabilistic . . . 7. What Is NP-Hard? (cont-d) How the Answer . . . • It is not known whether all exhaustive-search problems Main Result: . . . ? can be thus solved – this is the famous P =NP problem. What Is NP-Hard? . . . Case of Fuzzy Uncertainty • Most computer science researchers believe that some How Degrees Change . . . exhaustive-search problems cannot be feasibly solved. Main Result: Fuzzy Case • What is known is that some problems are the hardest What Happens in the . . . (NP-hard) in the sense that Home Page – any exhaustive-search problem Title Page – can be feasibly reduced to this problem. ◭◭ ◮◮ • This means that, unless P=NP, this particular problem ◭ ◮ cannot be feasibly solved. Page 8 of 17 • The above subset sum problem has been proven to be Go Back NP-hard, as well as many other similar problems. Full Screen Close Quit
Formulation of the . . . Case of Probabilistic . . . 8. How Can We Prove NP-Hardness How the Answer . . . • A problem is NP-hard if every other exhaustive-search Main Result: . . . problem Q can be reduced to it. What Is NP-Hard? . . . Case of Fuzzy Uncertainty • So, if we know that a problem P 0 is NP-hard, then How Degrees Change . . . every problem Q can be reduced to it; thus, Main Result: Fuzzy Case – if P 0 can be reduced to our problem P , What Happens in the . . . – then, by transitivity, any problem Q can be reduced Home Page to P , Title Page – i.e., P is indeed NP-hard. ◭◭ ◮◮ • Thus, to prove that P is NP-hard, it is sufficient to ◭ ◮ reduce a known NP-hard problem P 0 to P . Page 9 of 17 • We will prove that the subset sum problem P 0 (which Go Back is known to be NP-hard) can be reduced to P . Full Screen Close Quit
Formulation of the . . . Case of Probabilistic . . . 9. Simplifying the Expression for Entropy De- How the Answer . . . crease � p i Main Result: . . . � p i • For “yes”-answer, S ′ = − � What Is NP-Hard? . . . p ( Y ) · ln . p ( Y ) i ∈ Y Case of Fuzzy Uncertainty �� � 1 How Degrees Change . . . • Thus, S ′ = − p ( Y ) · p i · (ln( p i ) − ln( p ( Y )) . Main Result: Fuzzy Case i ∈ Y What Happens in the . . . �� � 1 Home Page • So, S ′ = − p ( Y ) · p i · ln( p i ) + ln( p ( Y )) . Title Page i ∈ Y �� � ◭◭ ◮◮ 1 • Similarly, S ′′ = − p ( N ) · p i · ln( p i ) + ln( p ( N )) . ◭ ◮ i ∈ N • So, S ( Y ) = p ( Y ) · ( S − S ′ ) + p ( N ) · ( S − S ′′ ) = Page 10 of 17 Go Back − p ( Y ) · ln( p ( Y )) − p ( N ) · ln( p ( N )) . Full Screen • This expression is known to be the largest when Close p ( Y ) = p ( N ) = 0 . 5 . Quit
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