# Ppt on polynomials for class 10

##### 6. The Most Difficult NP Problems: The Class NPC

tree parity P 7. Partial Order Dimension NP-complete 8. Precedence constrained 3-processor schedu open 9. Linear Programming P 10. Total Unimodularity P 11. Composite number open (P) 12. Minimum length triangulation open Integer Programming 2011 NP-complete in the/strongly NP-complete) if  is in NP and there exists a polynomial function p for which p is NP-complete. It is the problem class for which there might not exist even pseudo-polynomial time algorithms. If a problem is not a number problem and /

##### NP-Complete Problems Polynomial time vs exponential time

O(n) (polynomial in the input size). Class P and Class NP PNP If we can design a polynomial time algorithm for problem A, then A is in P. However, if we have not been able to design a polynomial time algorithm for A, then two possibilities: No polynomial time algorithm for A exists /: (Directly connect two nodes in R based on the visited order of the traversal.) 7 1 2 3 4 5 6 8 9 10 From triangle inequality, w(ST)≤w(traversal) ≤2OPT(I). ..........(2) Inequality(2) says that the cost of ST is no more than/

##### The Theory of NP-Completeness

is an input for R; x’ is an input for S (R  S) x x’ Algorithm for S T Yes or No answer Algorithm for R P NPC NP NP: Non-deterministic Polynomial P: Polynomial ? NP: Non-deterministic Polynomial P: Polynomial NPC: Non-deterministic Polynomial Complete NP : the class of decision problem /5) x(2)=7 v i≠2 v FAILURE (6) -FAILURE v -SUCCESS (7) SUCCESS (8) x(1)=7 (9) x(2)≠7 (10) The NP searching algorithm reduces to SAT Satisfiable at the following assignment : i=1 satisfying (1) i≠2 satisfying (2), (4) and (6) /

##### NP Completeness Tractability Polynomial time Computation vs. verification Power of non-determinism Encodings Transformation & reducibilities P vs. NP “Completeness”

of other “completeness” theories; 9.Helps explain why verifying proofs seems to be easier than constructing them; 10.Illuminates the fundamental nature of algorithms and computation; NP Completeness Benefits 11.Gave rise to new and novel /  “Karp” reduction (transformation) Polynomial-time Turing reduction: solves A by polynomially-many calls to “oracle” for B.  “Cook” reduction B A Open: do polynomial-time-bounded many-one and Turing reductions yield the same complexity classes? (NP, co-NP, NP-complete/

##### CLOSE Please YOUR LAPTOPS, and get out your note-taking materials.

lecture, during the last 10 minutes of class. CLOSE Please YOUR LAPTOPS, and get out your note-taking materials. and turn off and put away your cell phones, and get out your note-taking materials. Section 6.1 Introduction to Factoring Polynomials: Greatest Common Factor (GCF/ that on quizzes and tests!) How would you check this? Example Factor x3 + 4x + x2 + 4 by grouping. SOLUTION: First, look for a GCF. (Always do this first!) There isn’t one, so now separate the four terms into two groups of two: x3 + 4x /

##### Computability and Complexity 13-1 Computability and Complexity Andrei Bulatov The Class NP.

polynomial p(x) such that, for any w  L, there is a certificate c with |c|  p(|w|) All problems from the previous slide have a polynomial time verifier Computability and Complexity 13-5 The Class NP Definition The class of languages that have polynomial time verifiers is called NP Definition The class of languages that have polynomial/ for whites in a checker game on n  n board? Checkers Computability and Complexity 13-10 Non-deterministic Machines We can get an alternative definition of the class /

##### P and NP Sipser 7.2-7.3 (pages 256-270). CS 311 Fall 2008 2 Polynomial time P = ∪ k TIME(n k ) … P = ∪ k TIME(n k ) … TIME(n 3 ) TIME(n 2 ) TIME(n)

s t CS 311 Fall 2008 9 Hamiltonian paths HAMPATH = { | ∃ Hamiltonian path from s to t} s t CS 311 Fall 2008 10 Checking for Hamiltonian paths Brute force method E = "On input : 1.Generate all orderings, p 1, p 2,..., p n, of the nodes of / for some string c} –A polynomial time verifier runs in polynomial time in the length of w –A language A is polynomially verifiable if it has a polynomial time verifier CS 311 Fall 2008 15 The class NP Definition 7.19: NP is the class of languages that have polynomial /

##### 1 Polynomial Church-Turing thesis A decision problem can be solved in polynomial time by using a reasonable sequential model of computation if and only.

solved in polynomial time by using a reasonable sequential model of computation if and only if it can be solved in polynomial time by a Turing machine. 2 The complexity class P P := the class of decision problems (languages) decided by a Turing machine so that for some polynomial p and all/resulting in languages L 1 and L 2 we have L 1 2 P iff L 2 2 P. 10 Terminology When we say, “Problem X can be solved in polynomial time”, we mean L binary X 2 P, i.e., we assume binary representation of integers of input/

##### 1 Intractable Problems Time-Bounded Turing Machines Classes P and NP Polynomial-Time Reductions.

“an O(T(n)) running-time algorithm.” 3 The class P uIf a DTM M is T(n)-time bounded for some polynomial T(n), then we say M is polynomial-time (“polytime ”) bounded. uAnd L(M) is said to be in the class P. uImportant point: when we talk of P, it doesn/uEach table needs only space O(m) to represent all the positive and negative differences we could achieve. uEach table can be constructed in time O(n). 10 Knapsack – (4) uSince n < m, we can build the final table in O(m 2 ) time. uFrom that table, we can see /

##### The Polynomial Toolbox for MATLAB. Index Introduction The Polynomial Matrix Editor Polynomial matrix fractions Control system design Robust control with.

Kharitonov Polynomials. For continuous-time interval polynomials we have an even simpler method available: An interval polynomial of invariant degree (with real coefficients) is known to be stable if and only if just its four “extreme” polynomials (called the Kharitonov polynomials) are stable. For the interval polynomial of Example 3 the Kharitonov polynomials are computed by [stability,K1,K2,K3,K4] = kharit(pminus,pplus) Polytopes of polynomials A more general class/

##### 1 Polynomial Church-Turing thesis A decision problem can be solved in polynomial time in a reasonable sequential model of computation if and only if it.

of computation if and only if it can be solved in polynomial time by a Turing machine. P is the class of these decision problems 2 Search Problems: NP L is in NP iff there is a language L’ in P and a polynomial p so that: 3 Intuition The y-strings are the /-hard is called NP-complete. NPC := the class of NP-complete problems. Proposition: L 2 NPC ) [L 2 P iff P=NP]. 10 L is in NP means: There is a language L’ in P and a polynomial p so that L 1 · L 2 means: For some polynomial time computable map r : 8 x: x /

##### NP-Complete Problems Reading Material: Chapter 10 Sections 1, 2, 3, and 4 only.

: Chapter 10 Sections 1, 2, 3, and 4 only. NP-Complete Problems Problems in Computer Science are classified into –Tractable: There exists a polynomial time algorithm that solves the problem O(n k ) –Intractable: Unlikely for a polynomial time algorithm/algorithm? The Class NP Definition: The class of decision problems NP consists of those decision problems for which there exists a nondeterministic algorithm that runs in polynomial time Example Show that the coloring problem belongs to the class of NP /

##### NP-Complete Problems (Fun part)

can solve a problem of size k (i.e., compute 2k operations) in a hour/week/month. If the new computer is 10 times faster than ours, then the new computer can solve the problem of size k+3. The improvement is very little. Story All/ the first NP-complete problem. S. A. Cook in 1971 Won Turing prize for this work. Significance: If Satisfiability problem can be solved in polynomial time, then ALL problems in class NP can be solved in polynomial time. If you want to solve PNP, then you should work on NPC/

##### Chapter 10 P and NP.

program... while (!optimal) { a = generator(i); if (is_optimal(a)) optimal = true; } ....and its been running for 10 years? It could still be polynomial (imagine O(n10) for n = 1,000) and halt(p,i) might be just about to finish and return true. But you will never know/exist according to its own rules. For the halting problem, you can also agree that the wtf program can’t exist according to its own rules. But the wtf program does exist, feel free to implement it after class. The real show stopper is that/

##### 1 NP-Complete Problems (Fun part) Polynomial time vs exponential time –Polynomial O(n k ), where n is the input size (e.g., number of nodes in a graph,

is the first NP-complete problem. –S. A. Cook in 1971 –Won Turing prize for this work. Significance: –If Satisfiability problem can be solved in polynomial time, then ALL problems in class NP can be solved in polynomial time. –If you want to solve P  NP, then you should work on NPC/: (Directly connect two nodes in R based on the visited order of the traversal.) 7 1 2 3 4 5 6 8 9 10 From triangle inequality, w(ST)≤w(traversal) ≤2OPT(I)...........(2) 37 Inequality(2) says that the cost of the spanning tree ST/

##### FA to Classroom Practice

a Partner. Use the guiding questions to focus your work. Time: 10 minutes 4 Fraction Considerations from CTS Need experience before alogrithms Use of /when multiplying) of two polynomials will always be. 18 18 I can define “closure” as it applies to polynomials. I can tell what it means for polynomials to be closed under/… Anticipation of success (11G projects) Zeros Flexible due dates Retesting Confidence and in-class behavior (Cody) Routine/mental ease (Tianna & Morgan) Encouraging risk-taking (11G)/

##### The Remainder and Factor Theorems Check for Understanding 2.3 – Factor polynomials using a variety of methods including the factor theorem, synthetic division,

following function. c(x) = 2x 4 – 4x 3 – 7x 2 – 13x – 10 4 2 –4 –7 –13 –10 8 16 36 92 2 4 9 23 82 Time for Class work Time for Class work Evaluate each function at the given value. The Factor Theorem The binomial (x – a) is a factor of the polynomial f(x) if and only if f(a) = 0. The Factor Theorem/

##### Quadratic Equation- Session1. Session Objective 1.Definition of important terms (equation,expression,polynomial, identity,quadratic etc.) 2. Finding roots.

for all values of the variable (x+1) 2 = x 2 +2x+1 Equation holds true for all real x _H001 Polynomial identity If a polynomial equation of degree n satisfies for/Find the roots of the equation x 2 -10x+22=0 Here a=1, b=-10, c=22 Apply the general solution form _H003 Illustrative Problem Solution: Ans: Roots are /quadratic equation. Solution: As irrational roots are not in conjugate form. Co-efficient are not rational. Class Exercise6 If the sum of the roots of is zero, then prove that product of the roots is/

##### Course Notes for CS 1501 Algorithm Implementation By John C. Ramirez Department of Computer Science University of Pittsburgh.

similar in nature to thermodynamic entropy, which you may have discussed in a physics or chemistry class The more entropy, the less we can compress, and the less entropy the more we/way we learn to multiply in school –However, we are using base 2 rather than base 10 –See example on board Run-time of algorithm? How to implement? –We need to /your CS research WWhat should you do? T ry to find an efficient solution (polynomial) for the problem I f unsuccessful (or if you think it likely that it is not possible/

##### 1 The Theory of NP-Completeness 2012/11/6 P: the class of problems which can be solved by a deterministic polynomial algorithm. NP : the class of decision.

for running nondeterministic algorithms do not exist and they would never exist in reality. (They can only be made by allowing unbounded parallelism in computation.) (Quantum computation may be an approximation.) Nondeterministic algorithms are useful only because they will help us define a class of problems: NP problems 5 NP (Nondeterministic Polynomial/ we are addressing the lower bound of a problem and D ∝ O 10 To express Nondeterministic Algorithm by three special commands Choice(S) : arbitrarily chooses/

##### BCH_Code 2004/5/5Yuh-Ming Huang, CSIE, NCNU1 BCH Code a larger class of powerful random error-correcting cyclic codes a remarkable generalization of the.

BCH Code a larger class of powerful random error-correcting cyclic codes a remarkable generalization of the Hamming codes for multiple-error correction Def : Let q and m be given and let α be any element of GF(q m ) of order n. Then for any positive integer / Peterson-Gorenstein-Zierler decoder. BCH_Code 2004/5/5Yuh-Ming Huang, CSIE, NCNU23 Eg: BCH(15,5) triple-error-correcting code. g(X) = X 10 + X 8 + X 5 + X 4 + X 2 + X + 1 received code polynomial  V(X) = X 7 + X 2 In GF(2 4 ) S 1 = α 7 + α 2 = α 12 S 2/

##### MTH 209 Week 1 Third. Due for this week…  Homework 1 (on MyMathLab – via the Materials Link)  The fifth night after class at 11:59pm.  Read Chapter.

-Wesley Section 5.2 Addition and Subtraction of Polynomials Copyright © 2013, 2009, and 2005 Pearson Education, Inc. Objectives Monomials and Polynomials Addition of Polynomials Subtraction of Polynomials Evaluating Polynomial Expressions Monomials and Polynomials A monomial is a number, a variable, or/5 + 2x− 5 2x – 5 −5x 2 + 0 + 10 The quotient is Try Q: 41 pg 349 Due for this week…  Homework 1 (on MyMathLab – via the Materials Link)  The fifth night after class at 11:59pm.  Read Chapter 6.1-6.4  Do /

##### CSCI 3160 Design and Analysis of Algorithms Tutorial 10 Chengyu Lin.

into bit strings The corresponding language contains all the strings of “YES” instances 5 Class P V.S. Class NP P stands for what? Polynomial ! Class P: Problems solvable in deterministic polynomial time What about NP?  No Problem?  Not Polynomial (i.e. polynomial time unsolvable)? Nondeterministic Polynomial ! Class NP: Problems solvable in nondeterministic polynomial time 6 Deterministic/Nondeterministic Polynomial time? Where do these terms come from? They’re based on different computation/

##### Exploiting Vanishing Polynomials for Equivalence Verification of Fixed-Size Arithmetic Datapaths GIEE, NTU ALCom Lab Presenter: 陳炳元.

Vanishing Polynomials for Equivalence Verification of Fixed-Size Arithmetic Datapaths GIEE, NTU ALCom Lab Presenter: 陳炳元 Outline Introduction Modeling Vanishing Polynomials Algorithm Conclusions and Future Work Outline Introduction Modeling Vanishing Polynomials Algorithm / c  a,b,c  R Introduction to Rings Example: 我們在  上定義一個 equivalence relation, 若 (a,b)  R  a  b mod n 對應之 equivalence class set  n ={[0],[1],…,[n-1]}. 在  n 上定義兩個 binary operation +,  by [a]+[b]=[a+b mod n] 與 [a]  [b]=[ab mod n/

##### Cdlh, Barcelona, July 20071 TCS for Machine Learning Scientists Barcelona July 2007 Colin de la Higuera.

the size of an instance is n. With a problem , we associate the co- problem co-  The set of positive instances for  is denoted I+( ,) cdlh 2007 cdlh, Barcelona, July 2007 108 10 Complexity Classes P : deterministic polynomial time NP : non deterministic polynomial time cdlh 2007 cdlh, Barcelona, July 2007 109 Turing machines Only one tape Alphabet of 2 symbols An input of length n/

##### Unit 9: Coping with NP-Completeness ․ Course contents:  Complexity classes  Reducibility and NP-completeness proofs  Coping with NP-complete problems.

TSP  NP. Complexity Classes NP and co-NP ․ Is class NP closed under complement? ․ Class co-NP: class of problems whose complement problems/polynomial time. Significance of Reduction ․ Significance of L 1  P L 2 :   polynomial-time algorithm for L 2   polynomial-time algorithm for L 1 (L 2  P  L 1  P).  polynomial-time algorithm for L 1  polynomial- time algorithm for/with the conjunction (  ) of clauses describing the operation of each gate, e.g.,  = x 10  (x 4   x 3 )  (x 5  (x 1  x 2 )) /

##### MATLAB-Tutorial 2 Class ECES-304 Presented by : Shubham Bhat.

MATLAB-Tutorial 2 Class ECES-304 Presented by : Shubham Bhat % Nise, N.S. % Control /. "tf" needs two MATLAB-vectors, one containing the coefficients of the numerator polynomial - taken in descending orders - and one for the denominator polynomial of the transfer function. As an example we will create the transfer function H0/=zpk(Gtf); [r,p,k]=residue(numg,deng); Homework Problems % 10. Write the transfer function in phase variable form ‘a’ Num=100; Den=[1 20 10 7 100]; G=tf(num,den) [Acc,Bcc,Ccc,Dcc]=tf2ss(/

##### 1. 2 Lecture outline Basic definitions: Basic definitions: P, NP complexity classes P, NP complexity classes the notion of a certificate. the notion of.

classes Def: A Decision problem for a language L  {0,1} * is to decide whether a given string x belongs to the language L. Def: P is the class of languages (decision problems) that can be recognized by a deterministic polynomial time Turing machine. Def: NP is the class/, finding a solution can only be harder than just deciding whether it exists. Note: NP = { L(R) : R is polynomially-verifiable } 10 Problem: 3-coloring graphs Instance: An undirected graph G=(V,E) Corresponding relation: R 3COL = { (G,  ) :/

##### Course Notes for CS 1501 Algorithm Implementation By John C. Ramirez Department of Computer Science University of Pittsburgh.

similar in nature to thermodynamic entropy, which you may have discussed in a physics or chemistry class The more entropy, the less we can compress, and the less entropy the more we/way we learn to multiply in school –However, we are using base 2 rather than base 10 –See example on board Run-time of algorithm? How to implement? –We need to /your CS research WWhat should you do? T ry to find an efficient solution (polynomial) for the problem I f unsuccessful (or if you think it likely that it is not possible/

##### Lecture 12 P and NP Introduction to intractability Class P and NP Class NPC (NP-complete)

has power of todays supercomputers... And each processor works for the life of the universe... Will not help solve 1,000 city TSP problem via brute force. 1000! >> 10^1000 >> 10^{79+13+17} Exponential growth (Source from Wikipedia) Sessa/ in polynomial time. (Tractable) For some problems, we still cannot find polynomial time solutions. (Intractable) Decision problems Turing machine Where are we? Introduction to intractability Class P and NP Class NPC Class P Definition: ClassP A problem π is in class P /

##### 1 Ch 10 - NP-completeness Tractable and intractable problems Decision/Optimization problems Deterministic/NonDeterministic algorithms Classes P and NP.

minimum number  (G) of colors needed to color a graph in such a way that no two adjacent vertices have the same color. 10 Traveling Salesman Given a finite set C={c 1,...,c m } of cities, a distance function d(c i, c j ) of nonnegative/leads to an answer of no for a certain guess, that it does not accept it? 18 The Class NP Definition: The class of decision problems NP consists of those decision problems for which there exists a nondeterministic algorithm that runs in polynomial time. –The running time of /

##### Pattern Recognition and Image Analysis Dr. Manal Helal – Fall 2014 Lecture 10 Non-Linear Classifiers 2: SVM.

of ( + ) 10 for some values of and. By The Binomial Theorem:Binomial Theorem 10 +10 9 +45 82 +120 73 +210 64 +252 55 +210 46 +120 3 7 +45 28 +10 9 + 10 The trick: we can evaluate the dot product between degree 10 polynomials without explicitly forming them./lies on the margin, that is i ( T i + ) = 1 a training sample i with ( α i ≠0) is called a support vector The class of is h() = sign( T + ), substituting gives: We only need to remember the few training samples where ( α i ≠0). 47 Recovering the bias/

##### CS 583: Algorithms All pairs shortest path Ch. 25 NP Completeness Ch. 34 Review for final.

interesting class of problems whose status is unknown No polynomial-time algorithm has been discovered for an NP-Complete problem No one has been able to prove that no-polynomial algorithm exists for any of them No super-polynomial (e.g. exponential) low bound has been shown for /problem cannot be found with the same greedy strategy Greedy strategy: take in order of dollars/pound Example: 3 items weighing 10, 20, and 30 pounds, knapsack can hold 50 pounds Suppose item 2 is worth \$100. Assign values to the/

##### 1/15 Agnostically learning halfspaces FOCS 2005. 2/15 Set X, F class of functions f: X! {0,1}. Efficient Agnostic Learner w.h.p. h: X! {0,1} poly(1/ )

y) 2 X £ {0,1} f* = argmin f 2F P [f(x) y] 3/15 n n nn Set X n µ R n, F n class of functions f: X n ! {0,1}. Efficient Agnostic Learner w.h.p. n h: X n ! {0,1} n, poly(n,1/ ) samples/free for halfspaces Useful properties of logconcave dists: projection is logconcave, …, q(x) ¼ I (x ¸ 0) degree d=10 q(w ¢ x) ¼ I (w ¢ x ¸ 0) degree d=10 14/15 Approximating I(x ¸ ) (1 dimension) Bound min deg(q) · d E [(q(x) – I (x ¸ )) 2 ] Continuous distributions: orthogonal polynomials Normal: Hermite polynomials Logconcave /

##### Reductions Complexity ©D.Moshkovitz.

If f(w)HAMCYCLE then wHAMPATH  easy to verify    Complexity ©D.Moshkovitz Closeness Under Reductions Definition: A complexity class C is closed under reductions if, whenever L is reducible to L’ and L’C, then L is also in C. C / “black box” (procedure) that decides B, to construct a polynomial-time machine which decides A. polynomial time algorithm for B polynomial time algorithm for A Complexity ©D.Moshkovitz Cook reduction of HAMCYCLE to HAMPATH. For each edge (u,v)E, if there’s a Hamiltonian path/

##### 1 Bart Jansen Polynomial Kernels for Hard Problems on Disk Graphs Accepted for presentation at SWAT 2010.

Not closed under edge deletion Not closed under edge contraction 9 TRIANGLE PACKING AND H-MATCHING Structure theory and kernels 10 Triangle Packing Input:Graph G, integer k Question:Are there k vertex-disjoint triangles in G? Parameter:k NP/ a subset of what another red vertex sees Same for the blue vertices Structural theorems show that in such colored graphs the sizes of the color classes are polynomially related So size of the largest class is polynomial in the size of smallest class Hence |V| = |R| + |B|/

##### David Evans CS200: Computer Science University of Virginia Computer Science Class 38: Intractable Problems (Smiley Puzzles.

6 (8.9/10), 7 (8.6/10) and 8 (8.1/10) We’ve learned how to show some problems are undecidable –Harder (average on question 4 = 7.7) Today: reasoning about the complexity of some problems 21 April 2004CS 200 Spring 20043 Complexity Classes Class P: problems that can be solved in polynomial time by a deterministic TM. O (n k ) for some constant k/

##### Based on Powerpoint slides by Giorgi Japaridze, Villanova University Space Complexity and Interactive Proof Systems Sections 8.0, 8.1, 8.2, 8.3, 10.4.

] = Pr[(V  P)(w,r) = accept], where r is a randomly selected string of length p(n). The class IP 10.4.f Definition 10.28 Say that a language A is in IP if some polynomial time function V and arbitrary function P exist, where for every function P’ and string w: 1. w  A implies Pr[V  P accepts w] > 2/3, and/

##### Redefining Developmental Math for Non-Algebra Core Math Courses Dr. Daryl Stephens Murray Butler East Tennessee State.

, precalculus, calculus) for graduation. These students take MATH 1530, Probability and Statistics, as their core math class. About 90% of/10.1 Radicals and Radical Functions Appendices D. An Introduction to Using a Graphing Utility D. An Introduction to Using a Graphing Utility G. Mean, Median, and Mode G. Mean, Median, and Mode New DSPM 0850 5. Exponents and Polynomials 5. Exponents and Polynomials 5.1 Exponents 5.1 Exponents 5.2 Polynomial Functions and Adding and Subtracting Polynomials 5.2 Polynomial/

##### 1 NP-completeness Lecture 2: Jan 11. 2 P The class of problems that can be solved in polynomial time. e.g. gcd, shortest path, prime, etc. There are many.

i.e. a path which visits every vertex exactly once? HAMILTONIAN CYCLE HAMILTONIAN PATH 10 NP Two equivalent definitions: Given a solution, we can check in polynomial time whether it is correct. There is a non-deterministic algorithm (a magic /if it is correct in polynomial time. P is the class of problems that we can find a solution in polynomial time. NP (Non-deterministic polynomial time): A class of decision problems whose solutions can be “verified” in polynomial time. For each “yes” instance, there/

##### 15-251 Great Theoretical Ideas in Computer Science for Some.

Class “P” of Decision Problems What is an efficient algorithm? polynomial time O(n c ) for some constant c non-polynomial time Is an O(n) algorithm efficient? How about O(n log n)? O(n 2 ) ? O(n 10 ) ? O(n log n ) ? O(2 n ) ? O(n!) ? We consider non-polynomial time algorithms to be inefficient. And hence a necessary condition for/ HAM  NP The Class NP The class of sets L for which there exist “short” proofs of membership (of polynomial length) that can be “quickly” verified (in polynomial time). Recall: A /

##### Preference Elicitation in Combinatorial Auctions: An Overview Tuomas Sandholm [For an overview, see review article by Sandholm & Boutilier in the textbook.

e.g. side constraints –New query policies New polynomially elicitable valuation classes Using models of how costly it is to answer different queries [Hudson & S. AMEC-02] Decision-theoretic elicitation using priors Elicitors for markets beyond combinatorial auctions –(Combinatorial) reverse auctions & /queries with order queries Integrated as before –Computationally more expensive agents items 345678 9 2 10 23456 queries 1 10 100 1000 40 80 120 160 queries Full revelation Total cost Value cost Order cost /

##### 88- 1 Chapter 8 The Theory of NP-Completeness. 88- 2 P: the class of problems which can be solved by a deterministic polynomial algorithm. NP : the class.

13 Semantic tree In a semantic tree, each path from the root to a leaf node represents a class of assignments. If each leaf node is attached with a clause, then it is unsatisfiable. 88-/(6) -FAILURE v -SUCCESS (7) SUCCESS (8) x(1)=7 (9) x(2)≠7 (10) 88- 24 Satisfiable at the following assignment : i=1 satisfying (1) i≠2 satisfying (2), /NP-hard: Every NP algorithm for problem A can be transformed in polynomial time to SAT [Horowitz 1998] such that SAT is satisfiable if and only if the answer for A is “ YES ”. /

##### CS Master – Introduction to the Theory of Computation Jan Maluszynski - HT 200710.1 Lecture 10-11 NP-Completeness Jan Maluszynski, IDA, 2007

Jan Maluszynski - HT 200710.2 Outline Lecture 9-10 : NP-completeness (Sipser 7.4 – 7.5) 1.Motivation P vs. NP 2.NP-complete problems definition & intuition 3.3-SAT 4.Polynomial time reducibility 5.Cook-Levin theorem 6.A survey / Importance of time-complexity analysis Polynomial-time complexity preserved by realistic models of computations. P is a class of problems that can be realistically solved on a computer. The notion of NP-completeness based on P; relevant for practical computing. CS Master –/

##### Kernelization for a Hierarchy of Structural Parameters Bart M. P. Jansen Third Workshop on Kernelization 2-4 September 2011, Vienna.

for fixed k (Graph Coloring) Because it is compositional (Long Path) The natural parameter might not admit polynomial kernels Change the parameter instead of the class of inputs Alternative direction to kernels for restricted graph classes Guide the search for reduction/vertices on the outer face Outerplanar Deletion number Distance to planar with all vertices on the outer face 10 Vertex Cover Distance to linear forest Distance to Cograph Distance to Chordal Treewidth Chromatic Number Odd Cycle Transversal/

##### 1 Polynomial Space The classes PS and NPS Relationship to Other Classes Equivalence PS = NPS A PS-Complete Problem.

most c p(n) steps on input of length n, for some constant c and polynomial p. uSay L(M) is in the class EP. uIf M is an NTM instead, say L(M) is in the class NEP (nondeterministic exponential polytime ). 6 More Class Relationships uP  NP  PS  EP, and at least/t p(n). wIf M accepts, it does so without repeating an ID. uThus, M’ is exponential-polytime bounded, proving L(M) is in EP. 10 Savitch’s Theorem: PS = NPS uKey Idea: a polyspace NTM has “only” c p(n) different ID’s it can enter.  Implement a /

##### 1 The Theory of NP-Completeness 2 NP P NPC NP: Non-deterministic Polynomial P: Polynomial NPC: Non-deterministic Polynomial Complete P=NP? X = P.

Polynomial P: Polynomial NPC: Non-deterministic Polynomial Complete P=NP? X = P 3 P: the class of problems which can be solved by a deterministic polynomial algorithm. NP : the class of decision problem which can be solved by a non-deterministic polynomial algorithm. NP-hard: the class/ as not in P No known deterministic algorithms that are polynomially bounded for many problems in NP So, “ does P = NP/ (7) SUCCESS (8) x(1)=7 (9) x(2)≠7 (10) 30 Satisfiable with the following assignment: i=1 satisfying (1) i≠2/

##### Elementary Algebra Exam 4 Material Exponential Expressions & Polynomials.

use commutative and associative properties to rearrange and regroup the factors so as to group the “a” factors and powers of 10 separately and to use rules of exponents to end up with an answer in scientific notation It is also common to round/ 1 – 55, 59 – 69, 73 – 77 MyMathLab Section 4.4 for practice MyMathLab Homework Quiz 4.4 is due for a grade on the date of our next class meeting Multiplying Polynomials To multiply polynomials: –Get rid of parentheses by multiplying every term of the first by every /

##### College Algebra Prerequisite Topics Review Quick review of basic algebra skills that you should have developed before taking this class 18 problems that.

for squaring a binomial: Problem 3 Perform the indicated operation: Answer: Dividing a Polynomial by a Polynomial First write each polynomial /polynomials as being prime We will also completely factor polynomials by writing them as a product of prime polynomials Importance of Factoring If you don’t learn to factor polynomials you can’t pass college algebra or more advanced math classes/ and bottom: Divide out common factors to get: Problem 10 Reduce to lowest terms: Answer: Finding the Least Common /

##### Nonlinear Regression Functions.  General Ideas  Nolinear in a Single Variable Polynomials Logarithm  Interactions Between Independent Variables Between.

EL = 0  When Hi EL = 1,  Two regression lines: one for each Hi STR group.  Class size reduction is estimated to have a larger effect when the percent of English learners / the sample, especially large values of income. Question #1: Investigate by considering a polynomial in STR(column 5) Interpreting the regression function via plots (preceding regression is labeled/H 1 : nonzero interaction t = -0.58/0.50 = -1.17 not significant at the 10% level.  H 0 : both coeffs involving STR = 0 vs. H 1 : at least /

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