Exercises Computational Complexity
|
|
- Dana Hodge
- 5 years ago
- Views:
Transcription
1 Exercises Computational Complexity March 22, 2017 Exercises marked with a are more difficult. 1 Chapter 7, P and NP Exercise 1. Suppose some pancakes are stacked on a surface such that no two pancakes have the same size. For convenience denote the smallest pancake by 1, the second smallest by 2, etc. The only permissible operation on the stack of pancakes is as follows: Insert a spatula between 2 pancakes or between the bottom pancake and the surface, then rotate the pancakes above the spatula. Give an algorithm that sorts any stack of n pancakes (smallest at the top largest at the bottom) with at most poly(n) many flips = A solution for this example: Exercise 2. Show that TIME(f(n)) TIME(n + εf(n)) for every f : N N and every ε > 0. We use multitape Turing machines. Why do we need to add n to εf(n)? (Hint: simulate any machine by an other machine with a large enough alphabet that encodes several tape-cells in one cell.) Exercise 3. Show that P and NP are closed under intersection, union and star operator. (Hint: the hardest part is to show that P is closed under the star operator. Use dynamic programming, i.e., the technique in the proof of Theorem 7.16.) Ex and Exercise 4. Show that subset sum in unary notation is in P, i.e., show that Unary SUBSET- SUM= {1 t #1 w 1 #... #1 wn : x {0, 1} n : t = x i w i } is in P. Ex Exercise 5. Show that if P = NP, then all languages A P, except A = and A = Σ, are NP-complete. Ex Date: March 22, Many exercises are copied or copied with small variations from the book Introduction to the theory of computation by Michael Sipser. 1
2 Exercise 6. Show that if P = NP, then there exists an algorithm that on input a Boolean formula φ decides whether φ is satisfiable, and if it is satisfiable, returns an assignment of the variables that makes φ true. Moreover, the algorithm runs in time polynomial in the size of φ. (Note that with this technique, one can compute certificates for any problem in NP.) A variant of ex Exercise 7. Let CNF k be the set { φ : φ is a satisfiable cnf-formula where each variable appears in at most k places}. (This is not the same kcnf, where each term has at most k literals.) Show that CNF 2 P. Show that CNF 3 is NP-complete. Exercise* 8. Let OneSAT be the set of cnf-formula for which an assignment exists for which each clause contains exactly one true literal. Prove that OneSAT is NP-complete. Let One3SAT be the set of 3cnf-formula for which such an assignment exists. Show that also One3SAT is NP-complete. Exercise 9. A colouring of a graph is an assignment of colours to nodes so that no two nodes that are connected by an edge have the same colour. Let 3COLOUR be the set of such graphs. Show that 3COLOUR is NP-complete. (Hint: Let the 3 colours be T, F and N. Interpret the first as True and False, and represent them on a palette.) Ex F T palette N variable OR-gadget Exercise 10. Consider the following scheduling problem. You are given a list of final exams F 1,..., F k to be scheduled, and a list of students S 1,..., S l. Each student is taking some specified subset of these exams. You must schedule each exam into exactly one slot such that no student is required to take two exams in the same slot. The problem is to determine if such a schedule exists that uses only h slots. Formulate this problem as a language and show that it is NP-complete. Ex Exercise 11. In the following solitaire game, you are given an m m board. On each of its m 2 positions lies either a green stone, a red stone, or nothing at all. You play by removing stones from the board. You win if each column contains only stones of a single colour and each row contains at least one stone. Winning may or may not be possible depending upon the initial configuration. Prove that the set of winnable game configurations is NP-complete. Ex
3 Exercise* 12. Let f be a function for which f(n) o(n log n). Show that all languages that can be decided in time f(n) are regular. Ex Exercise 13. This problem investigates resolution, a method for proving the unsatisfiability of cnf-formulas. Let φ = C 1 C 2 C m be a formula in cnf, where C i are its clauses. Let C = {C 1,..., C m }. In a resolution step we take two clauses C a and C b in C such that some variable occurs positively in C a and negatively in C b. Thus C a = (x y 1 y 2 y k ) and C b = (x z 1 z 2 z l ). We form the new clause (y 1 y k z 1 z l ) and remove repeated literals. Add this new clause to C. Repeat the resolution steps until no additional clauses can be obtained. If the empty clause () is in C, then declare φ unsatisfiable. Show that resolution is sound, i.e., it never declares satisfiable formulas to be unsatisfiable. Show that resolution is complete, i.e., every unsatisfiable formula is declared to be unsatisfiable A 2CNF -formula is a conjunction of clauses of 2 variables. Let 2SAT be the set of satisfiable 2CNF -formulas. Show that 2SAT P. 2 Chapter 8: Space complexity Exercise 14. A ladder is a sequence in of strings s 1, s 2,..., s k such that each strings differs from the previous one by replacing exactly one character. For example: Let LADDER DFA be head, hear, near, bear, beer, deer, deed, feed, feet, fret, free { M, s, e : M is a DFA and L(M) contains a ladder starting in s and ending in e}. Show that this set is in PSPACE. Ex Exercise 15. Show that if A PSPACE, then A PSPACE. Part of Ex Exercise 16. Show that if every NP-hard language is also PSPACE-hard, then NP = PSPACE. Ex Exercise 17. Show that TQBF restricted to the formulas where the part following the quantifiers is in 3CNF form is still PSPACE-complete. Ex Exercise 18. Let A LBA = { M, w : M is a linear bounded automaton that accepts w}. Show that A LBA is PSPACE-complete. Ex Exercise 19. Two Boolean formulas are equivalent if they have the same set of variables and are true on the same set of assignments to those variables (i.e., they describe the same Boolean function). A Boolean formula is minimal if no shorter Boolean function is equivalent to it. Let MIN -FORMULA be the collection of minimal Boolean formulas. 3
4 Show that MIN -FORMULA PSPACE. Explain why this argument fails to show that MIN -FORMULA conp: If φ MIN -FORMULA, then φ has a smaller equivalent formula. A non-deterministic Turing machine can verify that φ MIN -FORMULA. Ex Exercise 20. In exercise 17 you showed that the set of TQBF in 3CNF -form is PSPACEcomplete. Now, consider Boolean formula that have no negations, i.e., formula that are constructed with only AND and OR operations. Such formula are called monotone. { } x 1 x 2... x 2n 1 x 2n Φ(x 1,..., x 2n ) : Φ is a monotone Boolean formula Do one of the following things: (a) show that this set is in P or (b) show that it is PSPACEcomplete. Hint: think about why these formula are called monotone? Exercise 21. Exercise Exercise* 22. In the game generalized geography, two players alternate turns and construct a single path in a directed graph. A player looses if he can not extend the end of the path with a vertex that does not belong to the path. In a variant of this game, both players alternate turns and each player constructs his own path. A player looses if in his turn he can not extend his path with a vertex that does not appear in his and the other players path. Let GG be the set of triples (G, a, b) where the first player has a winning strategy in the modified geography game when he starts in vertex a and the second player starts in vertex b. Show that GG is PSPACE-complete. Hint: There are several solutions. It might help to note that the reduction for GG works with arbitrary Boolean formula. Rewrite a given Boolean formula such that it is build from literals, OR operations, and binary AND operations. The basic ideas are similar from the construction of GG, but the forall player can one of the options while the exist player goes through an AND-gate. Exercise 23. During the lectures, it was shown that (directed) Generalized Geography is PSPACE-complete. Surprisingly, the undirected variant is in P. The goal of the exercise is to prove this. In the game undirected vertex geography two players alternate moves and construct together a single path in an undirected graph. Initially, the beginning and the end of the path coincide with a given start vertex s. At his turn a player should select a vertex that is connected with the end of the path and that does not already appear in the path. This vertex becomes the new end of the path. A player looses if he can not select such a neighbouring vertex. Let UVG be the set of pairs (G, s) where G is a graph and the first player has a winning strategy when he starts playing. A matching in a graph is a set of edges without common neighbours, i.e., no pair of edges is incident on a common vertex. A maximum matching is a matching of maximal cardinality. Show that the first player has a winning strategy in the game (G, s) if and only if every maximum matching has an edge that is incident on s. There exists an algorithm, called Blossom algorithm, that computes the maximal size of a matching in a graph in polynomial time. Assume you can use this algorithm. Use it to construct an algorithm that decides UVG in polynomial time 4
5 Exercise 24. The cat-and-mouse game is played by two players, Cat and Mouse, on an arbitrary undirected graph. At a given point each player occupies a vertex of the graph. The players take turns moving to a vertex adjacent to the one that they currently occupy. A special vertex of the graph is called Hole. Cat wins if the two players ever occupy the same vertex. He looses if Mouse reaches the Hole before the preceding happens. He also looses if a situation repeats (i.e., the two players simultaneously occupy positions that they simultaneously occupied previously and it is the same player s turn to move). Let HAPPY -CAT be the set of quadruples (G, c, m, h) such that G is a graph, and c, m and h are the positions of Cat, Mouse and Whole, for which Cat has a winning strategy if he moves first. Show that HAPPY -CAT P. Hint: the solution is not too hard. Ex Chapter 9: Intractability Exercise 25. What is the error in the following fallacious argument that P NP? Assume that P = NP, then SAT P and so for some k, SAT TIME(n k ). Therefore P TIME(n k ). But by the time hierarchy theorem, TIME(n k+1 ) contains a language not in TIME(n k ), which contradicts P TIME(n k ). Therefore P NP. Ex Exercise 26. Consider the function pad : Σ N (Σ {#}) defined by pad(s, l) = s# j with j = max(0, l s ). For any language A and any function f, let pad(a, f) be the language of elements pad(x, f( x )) for all x A. Prove that A TIME(n 6 ), then pad(a, n 2 ) TIME(n 3 ). If NEXPTIME EXPTIME, then P NP. P SPACE(n 2 ). Ex Exercise 27. Let L be the language over the alphabet A = {a, b} that contains all strings that start with at most N symbols a, in other words L = ɛ ba aba aaba a N ba. This is a regular expression with exponentiation. Give a more compact regular expression with exponentiation that uses a single union operation. Exercise 28. Define the USAT problem to be { φ : φ is a Boolean formula that has a single satisfying assignment} Show that USAT P SAT. Ex Exercise 29. Consider the set { φ : φ SAT and the lexicographically first x 1... x n for which φ(x 1,..., x n ) is true, has x n = 0}. Show that this set is in P SAT. 5
6 Exercise 30. Let MIN -FORMULA be the language defined in exercise 19. Show that the complement of this language is in NP SAT. Exercise 31. Show that if P SAT = NP then NP = conp. Ex Exercise 32. Let P A,1 be the class of languages that can be decided in polynomial time with oracle A, but for which for each input, the machine makes at most one query on the oracle. Show that NP conp P SAT,1, If NP conp, then NP conp P SAT,1. Ex Exercise 33. Define a language L to be downward-self-reducible if there is a polynomial-time algorithm R that for any n and x Σ n, decides whether x L using the oracle L n 1, which solves membership of L for inputs of size at most n 1. In other words, R L n 1 (x) = L(x) for all n and all x of length n. Prove that if L is downward-self-reducible, then L PSPACE. (From the previous edition of Sipser s book.) Exercise 34. Imagine you are given two oracles and one of them is the set TQBF. You don t know which one. Design an algorithm that can access the two oracles A and B and that decides TQBF in polynomial time. Ex Exercise 35. Adapt the proof of Theorem 9.20 to show that there exists a decidable set B such that conp B NP B. Ex Exercise* 36. Suppose we generate a random language B in the following way: for every n, with probability 1/2, B has no strings of length n, and with probability 1/2 we pick a random string to be in B. Show that for each ε > 0 we have that P B NP B with probability at least 1 ε. 4 Chapter 10: Advanced topics Exercise 37. Show that if NP BP P, then NP = RP. Ex Exercise 38. Show that: The parity function for n-bit strings can be computed by a branching program of size O(n). Ex The majority function for n-bit strings, i.e., the function x n i=1 x i > n/2, can be computed by a branching program of size O(n 2 ). Ex Any function on n-bit strings can be computed by branching programs of size 2 n. Ex Exercise 39. Show that if A is a regular language, then a family of branching programs B 1, B 2,... exists such that B n accepts exactly the n-bit strings in A and the size of B n is bounded by a constant times n. Ex Exercise 40. Let EQ BP be the set of pairs (B 1, B 2 ) of equivalent branching programs. Show that EQ BP is conp-complete. Ex
8.1 Polynomial-Time Reductions
8.1 Polynomial-Time Reductions Classify Problems According to Computational Requirements Q. Which problems will we be able to solve in practice? A working definition. Those with polynomial-time algorithms.
More informationCS 125 Section #10 Midterm 2 Review 11/5/14
CS 125 Section #10 Midterm 2 Review 11/5/14 1 Topics Covered This midterm covers up through NP-completeness; countability, decidability, and recognizability will not appear on this midterm. Disclaimer:
More informationMore on Polynomial Time and Space
CpSc 8390 Goddard Fall15 More on Polynomial Time and Space 20.1 The Original NP-Completeness Proof A configuration/snapshot of a machine is a representation of its current state (what info would be needed
More information9.1 Cook-Levin Theorem
CS787: Advanced Algorithms Scribe: Shijin Kong and David Malec Lecturer: Shuchi Chawla Topic: NP-Completeness, Approximation Algorithms Date: 10/1/2007 As we ve already seen in the preceding lecture, two
More informationChapter 8. NP and Computational Intractability. Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved.
Chapter 8 NP and Computational Intractability Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved. 1 Algorithm Design Patterns and Anti-Patterns Algorithm design patterns.
More information! Greed. O(n log n) interval scheduling. ! Divide-and-conquer. O(n log n) FFT. ! Dynamic programming. O(n 2 ) edit distance.
Algorithm Design Patterns and Anti-Patterns Chapter 8 NP and Computational Intractability Algorithm design patterns. Ex.! Greed. O(n log n) interval scheduling.! Divide-and-conquer. O(n log n) FFT.! Dynamic
More informationNP-Completeness. Algorithms
NP-Completeness Algorithms The NP-Completeness Theory Objective: Identify a class of problems that are hard to solve. Exponential time is hard. Polynomial time is easy. Why: Do not try to find efficient
More informationNP-Hardness. We start by defining types of problem, and then move on to defining the polynomial-time reductions.
CS 787: Advanced Algorithms NP-Hardness Instructor: Dieter van Melkebeek We review the concept of polynomial-time reductions, define various classes of problems including NP-complete, and show that 3-SAT
More informationComputability Theory
CS:4330 Theory of Computation Spring 2018 Computability Theory Other NP-Complete Problems Haniel Barbosa Readings for this lecture Chapter 7 of [Sipser 1996], 3rd edition. Sections 7.4 and 7.5. The 3SAT
More information! Greed. O(n log n) interval scheduling. ! Divide-and-conquer. O(n log n) FFT. ! Dynamic programming. O(n 2 ) edit distance.
Algorithm Design Patterns and Anti-Patterns 8. NP and Computational Intractability Algorithm design patterns. Ex.! Greed. O(n log n) interval scheduling.! Divide-and-conquer. O(n log n) FFT.! Dynamic programming.
More information8 NP-complete problem Hard problems: demo
Ch8 NPC Millennium Prize Problems http://en.wikipedia.org/wiki/millennium_prize_problems 8 NP-complete problem Hard problems: demo NP-hard (Non-deterministic Polynomial-time hard), in computational complexity
More informationCS 151 Complexity Theory Spring Final Solutions. L i NL i NC 2i P.
CS 151 Complexity Theory Spring 2017 Posted: June 9 Final Solutions Chris Umans 1. (a) The procedure that traverses a fan-in 2 depth O(log i n) circuit and outputs a formula runs in L i this can be done
More information/633 Introduction to Algorithms Lecturer: Michael Dinitz Topic: Approximation algorithms Date: 11/27/18
601.433/633 Introduction to Algorithms Lecturer: Michael Dinitz Topic: Approximation algorithms Date: 11/27/18 22.1 Introduction We spent the last two lectures proving that for certain problems, we can
More informationNP and computational intractability. Kleinberg and Tardos, chapter 8
NP and computational intractability Kleinberg and Tardos, chapter 8 1 Major Transition So far we have studied certain algorithmic patterns Greedy, Divide and conquer, Dynamic programming to develop efficient
More information8.1 Polynomial-Time Reductions
Algorithm Design Patterns and Anti-Patterns Analysis of Algorithms Algorithm design patterns. Ex. Greed. O(n 2 ) Dijkstra s SSSP (dense) Divide-and-conquer. O(n log n) FFT. Dynamic programming. O(n 2 )
More informationUSING QBF SOLVERS TO SOLVE GAMES AND PUZZLES. Zhihe Shen. Advisor: Howard Straubing
Boston College Computer Science Senior Thesis USING QBF SOLVERS TO SOLVE GAMES AND PUZZLES Zhihe Shen Advisor: Howard Straubing Abstract There are multiple types of games, such as board games and card
More informationW4231: Analysis of Algorithms
W4231: Analysis of Algorithms 11/23/99 NP-completeness of 3SAT, Minimum Vertex Cover, Maximum Independent Set, Boolean Formulae A Boolean formula is an expression that we can build starting from Boolean
More informationALGORITHMS EXAMINATION Department of Computer Science New York University December 17, 2007
ALGORITHMS EXAMINATION Department of Computer Science New York University December 17, 2007 This examination is a three hour exam. All questions carry the same weight. Answer all of the following six questions.
More informationReductions and Satisfiability
Reductions and Satisfiability 1 Polynomial-Time Reductions reformulating problems reformulating a problem in polynomial time independent set and vertex cover reducing vertex cover to set cover 2 The Satisfiability
More information1 Definition of Reduction
1 Definition of Reduction Problem A is reducible, or more technically Turing reducible, to problem B, denoted A B if there a main program M to solve problem A that lacks only a procedure to solve problem
More informationComplexity Classes and Polynomial-time Reductions
COMPSCI 330: Design and Analysis of Algorithms April 19, 2016 Complexity Classes and Polynomial-time Reductions Lecturer: Debmalya Panigrahi Scribe: Tianqi Song 1 Overview In this lecture, we introduce
More informationNP versus PSPACE. Frank Vega. To cite this version: HAL Id: hal https://hal.archives-ouvertes.fr/hal
NP versus PSPACE Frank Vega To cite this version: Frank Vega. NP versus PSPACE. Preprint submitted to Theoretical Computer Science 2015. 2015. HAL Id: hal-01196489 https://hal.archives-ouvertes.fr/hal-01196489
More informationBest known solution time is Ω(V!) Check every permutation of vertices to see if there is a graph edge between adjacent vertices
Hard Problems Euler-Tour Problem Undirected graph G=(V,E) An Euler Tour is a path where every edge appears exactly once. The Euler-Tour Problem: does graph G have an Euler Path? Answerable in O(E) time.
More informationIntroduction to Algorithms / Algorithms I Lecturer: Michael Dinitz Topic: Approximation algorithms Date: 11/18/14
600.363 Introduction to Algorithms / 600.463 Algorithms I Lecturer: Michael Dinitz Topic: Approximation algorithms Date: 11/18/14 23.1 Introduction We spent last week proving that for certain problems,
More informationP and NP (Millenium problem)
CMPS 2200 Fall 2017 P and NP (Millenium problem) Carola Wenk Slides courtesy of Piotr Indyk with additions by Carola Wenk CMPS 2200 Introduction to Algorithms 1 We have seen so far Algorithms for various
More informationarxiv: v2 [cs.cc] 29 Mar 2010
On a variant of Monotone NAE-3SAT and the Triangle-Free Cut problem. arxiv:1003.3704v2 [cs.cc] 29 Mar 2010 Peiyush Jain, Microsoft Corporation. June 28, 2018 Abstract In this paper we define a restricted
More information6.045J/18.400J: Automata, Computability and Complexity Final Exam. There are two sheets of scratch paper at the end of this exam.
6.045J/18.400J: Automata, Computability and Complexity May 18, 2004 6.045 Final Exam Prof. Nancy Lynch Name: Please write your name on each page. This exam is open book, open notes. There are two sheets
More informationThe Complexity of Camping
The Complexity of Camping Marzio De Biasi marziodebiasi [at] gmail [dot] com July 2012 Version 0.04: a re-revised sketch of the proof Abstract We prove that the tents puzzle game is NP -complete using
More informationSome Hardness Proofs
Some Hardness Proofs Magnus Lie Hetland January 2011 This is a very brief overview of some well-known hard (NP Hard and NP complete) problems, and the main ideas behind their hardness proofs. The document
More informationUnit 8: Coping with NP-Completeness. Complexity classes Reducibility and NP-completeness proofs Coping with NP-complete problems. Y.-W.
: Coping with NP-Completeness Course contents: Complexity classes Reducibility and NP-completeness proofs Coping with NP-complete problems Reading: Chapter 34 Chapter 35.1, 35.2 Y.-W. Chang 1 Complexity
More informationCS 580: Algorithm Design and Analysis
CS 580: Algorithm Design and Analysis Jeremiah Blocki Purdue University Spring 2018 Homework 4: Due tomorrow (March 9) at 11:59 PM Recap Linear Programming Very Powerful Technique (Subject of Entire Courses)
More information3/7/2018. CS 580: Algorithm Design and Analysis. 8.1 Polynomial-Time Reductions. Chapter 8. NP and Computational Intractability
Algorithm Design Patterns and Anti-Patterns CS 580: Algorithm Design and Analysis Jeremiah Blocki Purdue University Spring 2018 Algorithm design patterns. Ex. Greedy. O(n log n) interval scheduling. Divide-and-conquer.
More informationThe Satisfiability Problem [HMU06,Chp.10b] Satisfiability (SAT) Problem Cook s Theorem: An NP-Complete Problem Restricted SAT: CSAT, k-sat, 3SAT
The Satisfiability Problem [HMU06,Chp.10b] Satisfiability (SAT) Problem Cook s Theorem: An NP-Complete Problem Restricted SAT: CSAT, k-sat, 3SAT 1 Satisfiability (SAT) Problem 2 Boolean Expressions Boolean,
More informationUniversity of Nevada, Las Vegas Computer Science 456/656 Fall 2016
University of Nevada, Las Vegas Computer Science 456/656 Fall 2016 The entire examination is 925 points. The real final will be much shorter. Name: No books, notes, scratch paper, or calculators. Use pen
More information1 Introduction. 1. Prove the problem lies in the class NP. 2. Find an NP-complete problem that reduces to it.
1 Introduction There are hundreds of NP-complete problems. For a recent selection see http://www. csc.liv.ac.uk/ ped/teachadmin/comp202/annotated_np.html Also, see the book M. R. Garey and D. S. Johnson.
More informationNP-Complete Reductions 2
x 1 x 1 x 2 x 2 x 3 x 3 x 4 x 4 12 22 32 CS 447 11 13 21 23 31 33 Algorithms NP-Complete Reductions 2 Prof. Gregory Provan Department of Computer Science University College Cork 1 Lecture Outline NP-Complete
More information6.045 Final Exam. Name: Please write your name on each page. This exam is open book, open notes.
6.045J/18.400J: Automata, Computability and Complexity May 18, 2004 6.045 Final Exam Prof. Nancy Lynch, Nati Srebro Susan Hohenberger Please write your name on each page. This exam is open book, open notes.
More informationNP Completeness. Andreas Klappenecker [partially based on slides by Jennifer Welch]
NP Completeness Andreas Klappenecker [partially based on slides by Jennifer Welch] Overview We already know the following examples of NPC problems: SAT 3SAT We are going to show that the following are
More informationReductions. Linear Time Reductions. Desiderata. Reduction. Desiderata. Classify problems according to their computational requirements.
Desiderata Reductions Desiderata. Classify problems according to their computational requirements. Frustrating news. Huge number of fundamental problems have defied classification for decades. Desiderata'.
More information1. [5 points each] True or False. If the question is currently open, write O or Open.
University of Nevada, Las Vegas Computer Science 456/656 Spring 2018 Practice for the Final on May 9, 2018 The entire examination is 775 points. The real final will be much shorter. Name: No books, notes,
More informationFINAL EXAM SOLUTIONS
COMP/MATH 3804 Design and Analysis of Algorithms I Fall 2015 FINAL EXAM SOLUTIONS Question 1 (12%). Modify Euclid s algorithm as follows. function Newclid(a,b) if a
More informationCS521 \ Notes for the Final Exam
CS521 \ Notes for final exam 1 Ariel Stolerman Asymptotic Notations: CS521 \ Notes for the Final Exam Notation Definition Limit Big-O ( ) Small-o ( ) Big- ( ) Small- ( ) Big- ( ) Notes: ( ) ( ) ( ) ( )
More informationLecture 14: Lower Bounds for Tree Resolution
IAS/PCMI Summer Session 2000 Clay Mathematics Undergraduate Program Advanced Course on Computational Complexity Lecture 14: Lower Bounds for Tree Resolution David Mix Barrington and Alexis Maciel August
More informationFinding a winning strategy in variations of Kayles
Finding a winning strategy in variations of Kayles Simon Prins ICA-3582809 Utrecht University, The Netherlands July 15, 2015 Abstract Kayles is a two player game played on a graph. The game can be dened
More informationFixed Parameter Algorithms
Fixed Parameter Algorithms Dániel Marx Tel Aviv University, Israel Open lectures for PhD students in computer science January 9, 2010, Warsaw, Poland Fixed Parameter Algorithms p.1/41 Parameterized complexity
More informationSolutions for the Exam 6 January 2014
Mastermath and LNMB Course: Discrete Optimization Solutions for the Exam 6 January 2014 Utrecht University, Educatorium, 13:30 16:30 The examination lasts 3 hours. Grading will be done before January 20,
More informationCMPSCI611: The SUBSET-SUM Problem Lecture 18
CMPSCI611: The SUBSET-SUM Problem Lecture 18 We begin today with the problem we didn t get to at the end of last lecture the SUBSET-SUM problem, which we also saw back in Lecture 8. The input to SUBSET-
More informationNP-Complete Problems
1 / 34 NP-Complete Problems CS 584: Algorithm Design and Analysis Daniel Leblanc 1 1 Senior Adjunct Instructor Portland State University Maseeh College of Engineering and Computer Science Winter 2018 2
More informationPCP and Hardness of Approximation
PCP and Hardness of Approximation January 30, 2009 Our goal herein is to define and prove basic concepts regarding hardness of approximation. We will state but obviously not prove a PCP theorem as a starting
More information2SAT Andreas Klappenecker
2SAT Andreas Klappenecker The Problem Can we make the following boolean formula true? ( x y) ( y z) (z y)! Terminology A boolean variable is a variable that can be assigned the values true (T) or false
More informationCS 512, Spring 2017: Take-Home End-of-Term Examination
CS 512, Spring 2017: Take-Home End-of-Term Examination Out: Tuesday, 9 May 2017, 12:00 noon Due: Wednesday, 10 May 2017, by 11:59 am Turn in your solutions electronically, as a single PDF file, by placing
More informationPSPACE Problems. The Class PSPACE. PSPACE Problems are interesting since: They form the first interesting class potentially
PSPACE Problems Space Complexity: If an algorithm A solves a problem X by using O(f(n)) bits of memory where n is the size of the input we say that X SPACE(f(n)). The Class PSPACE Def: X PSPACE if and
More informationW[1]-hardness. Dániel Marx 1. Hungarian Academy of Sciences (MTA SZTAKI) Budapest, Hungary
W[1]-hardness Dániel Marx 1 1 Institute for Computer Science and Control, Hungarian Academy of Sciences (MTA SZTAKI) Budapest, Hungary School on Parameterized Algorithms and Complexity Będlewo, Poland
More informationMassachusetts Institute of Technology Department of Electrical Engineering and Computer Science Algorithms For Inference Fall 2014
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.438 Algorithms For Inference Fall 2014 Recitation-6: Hardness of Inference Contents 1 NP-Hardness Part-II
More informationWhere Can We Draw The Line?
Where Can We Draw The Line? On the Hardness of Satisfiability Problems Complexity 1 Introduction Objectives: To show variants of SAT and check if they are NP-hard Overview: Known results 2SAT Max2SAT Complexity
More informationLecture 21: Other Reductions Steven Skiena. Department of Computer Science State University of New York Stony Brook, NY
Lecture 21: Other Reductions Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Problem of the Day Show that the Dense
More informationW[1]-hardness. Dániel Marx. Recent Advances in Parameterized Complexity Tel Aviv, Israel, December 3, 2017
1 W[1]-hardness Dániel Marx Recent Advances in Parameterized Complexity Tel Aviv, Israel, December 3, 2017 2 Lower bounds So far we have seen positive results: basic algorithmic techniques for fixed-parameter
More informationFinal Exam May 8, 2018
Real name: CS/ECE 374 A Spring 2018 Final Exam May 8, 2018 NetID: Gradescope name: Gradescope email: Don t panic! If you brought anything except your writing implements and your two double-sided 8½" 11"
More informationTheory of Computations Spring 2016 Practice Final Exam Solutions
1 of 8 Theory of Computations Spring 2016 Practice Final Exam Solutions Name: Directions: Answer the questions as well as you can. Partial credit will be given, so show your work where appropriate. Try
More informationLecture 21: Other Reductions Steven Skiena
Lecture 21: Other Reductions Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.stonybrook.edu/ skiena Problem of the Day Show that the dense
More informationComplexity of unique list colorability
Complexity of unique list colorability Dániel Marx 1 Department of Computer Science and Information Theory, Budapest University of Technology and Economics, Budapest H-151, Hungary. Abstract Given a list
More informationWe ve studied the main models and concepts of the theory of computation:
CMPSCI 601: Summary & Conclusions Lecture 27 We ve studied the main models and concepts of the theory of computation: Computability: what can be computed in principle Logic: how can we express our requirements
More informationP and NP CISC5835, Algorithms for Big Data CIS, Fordham Univ. Instructor: X. Zhang
P and NP CISC5835, Algorithms for Big Data CIS, Fordham Univ. Instructor: X. Zhang Efficient Algorithms So far, we have developed algorithms for finding shortest paths in graphs, minimum spanning trees
More informationChapter 8. NP-complete problems
Chapter 8. NP-complete problems Search problems E cient algorithms We have developed algorithms for I I I I I finding shortest paths in graphs, minimum spanning trees in graphs, matchings in bipartite
More informationCOMP260 Spring 2014 Notes: February 4th
COMP260 Spring 2014 Notes: February 4th Andrew Winslow In these notes, all graphs are undirected. We consider matching, covering, and packing in bipartite graphs, general graphs, and hypergraphs. We also
More informationOnline Ramsey Theory
Charles University in Prague Faculty of Mathematics and Physics MASTER THESIS Pavel Dvořák Online Ramsey Theory Computer Science Institute of Charles University Supervisor of the master thesis: Study programme:
More informationUmans Complexity Theory Lectures
Introduction Umans Complexity Theory Lectures Lecture 5: Boolean Circuits & NP: - Uniformity and Advice, - NC hierarchy Power from an unexpected source? we know P EXP, which implies no polytime algorithm
More information(p 300) Theorem 7.27 SAT is in P iff P=NP
pp. 292-311. The Class NP-Complete (Sec. 7.4) P = {L L decidable in poly time} NP = {L L verifiable in poly time} Certainly all P is in NP Unknown if NP is bigger than P (p. 299) NP-Complete = subset of
More information1. Suppose you are given a magic black box that somehow answers the following decision problem in polynomial time:
1. Suppose you are given a magic black box that somehow answers the following decision problem in polynomial time: Input: A CNF formula ϕ with n variables x 1, x 2,..., x n. Output: True if there is an
More informationExamples of P vs NP: More Problems
Examples of P vs NP: More Problems COMP1600 / COMP6260 Dirk Pattinson Australian National University Semester 2, 2017 Catch Up / Drop in Lab When Fridays, 15.00-17.00 Where N335, CSIT Building (bldg 108)
More information14.1 Encoding for different models of computation
Lecture 14 Decidable languages In the previous lecture we discussed some examples of encoding schemes, through which various objects can be represented by strings over a given alphabet. We will begin this
More information1 Undirected Vertex Geography UVG
Geography Start with a chip sitting on a vertex v of a graph or digraph G. A move consists of moving the chip to a neighbouring vertex. In edge geography, moving the chip from x to y deletes the edge (x,
More informationThe Resolution Algorithm
The Resolution Algorithm Introduction In this lecture we introduce the Resolution algorithm for solving instances of the NP-complete CNF- SAT decision problem. Although the algorithm does not run in polynomial
More informationRandomness and Computation March 25, Lecture 5
0368.463 Randomness and Computation March 25, 2009 Lecturer: Ronitt Rubinfeld Lecture 5 Scribe: Inbal Marhaim, Naama Ben-Aroya Today Uniform generation of DNF satisfying assignments Uniform generation
More informationLimitations of Algorithmic Solvability In this Chapter we investigate the power of algorithms to solve problems Some can be solved algorithmically and
Computer Language Theory Chapter 4: Decidability 1 Limitations of Algorithmic Solvability In this Chapter we investigate the power of algorithms to solve problems Some can be solved algorithmically and
More informationSteven Skiena. skiena
Lecture 22: Introduction to NP-completeness (1997) Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Among n people,
More informationExample of a Demonstration that a Problem is NP-Complete by reduction from CNF-SAT
20170926 CNF-SAT: CNF-SAT is a problem in NP, defined as follows: Let E be a Boolean expression with m clauses and n literals (literals = variables, possibly negated), in which - each clause contains only
More informationCS154, Lecture 18: PCPs, Hardness of Approximation, Approximation-Preserving Reductions, Interactive Proofs, Zero-Knowledge, Cold Fusion, Peace in
CS154, Lecture 18: PCPs, Hardness of Approximation, Approximation-Preserving Reductions, Interactive Proofs, Zero-Knowledge, Cold Fusion, Peace in the Middle East There are thousands of NP-complete problems
More informationCopyright 2000, Kevin Wayne 1
Guessing Game: NP-Complete? 1. LONGEST-PATH: Given a graph G = (V, E), does there exists a simple path of length at least k edges? YES. SHORTEST-PATH: Given a graph G = (V, E), does there exists a simple
More informationP -vs- NP. NP Problems. P = polynomial time. NP = non-deterministic polynomial time
P -vs- NP NP Problems P = polynomial time There are many problems that can be solved correctly using algorithms that run in O(n c ) time for some constant c. NOTE: We can say that an nlogn algorithm is
More informationTheory Bridge Exam Example Questions Version of June 6, 2008
Theory Bridge Exam Example Questions Version of June 6, 2008 This is a collection of sample theory bridge exam questions. This is just to get some idea of the format of the bridge exam and the level of
More informationOn Covering a Graph Optimally with Induced Subgraphs
On Covering a Graph Optimally with Induced Subgraphs Shripad Thite April 1, 006 Abstract We consider the problem of covering a graph with a given number of induced subgraphs so that the maximum number
More informationVertex Cover Approximations
CS124 Lecture 20 Heuristics can be useful in practice, but sometimes we would like to have guarantees. Approximation algorithms give guarantees. It is worth keeping in mind that sometimes approximation
More informationSAT-CNF Is N P-complete
SAT-CNF Is N P-complete Rod Howell Kansas State University November 9, 2000 The purpose of this paper is to give a detailed presentation of an N P- completeness proof using the definition of N P given
More informationCSE 105 THEORY OF COMPUTATION
CSE 105 THEORY OF COMPUTATION Spring 2016 http://cseweb.ucsd.edu/classes/sp16/cse105-ab/ Today's learning goals Sipser Ch 3.2, 3.3 Define variants of TMs Enumerators Multi-tape TMs Nondeterministic TMs
More informationLinear Time Unit Propagation, Horn-SAT and 2-SAT
Notes on Satisfiability-Based Problem Solving Linear Time Unit Propagation, Horn-SAT and 2-SAT David Mitchell mitchell@cs.sfu.ca September 25, 2013 This is a preliminary draft of these notes. Please do
More informationChapter 10 Part 1: Reduction
//06 Polynomial-Time Reduction Suppose we could solve Y in polynomial-time. What else could we solve in polynomial time? don't confuse with reduces from Chapter 0 Part : Reduction Reduction. Problem X
More informationAn IPS for TQBF Intro to Approximability
An IPS for TQBF Intro to Approximability Outline Proof that TQBF (the language of true (valid) quantified Boolean formulas) is in the class IP Introduction to approximation algorithms for NP optimization
More informationNP-complete Reductions
NP-complete Reductions 1. Prove that 3SAT P DOUBLE-SAT, i.e., show DOUBLE-SAT is NP-complete by reduction from 3SAT. The 3-SAT problem consists of a conjunction of clauses over n Boolean variables, where
More informationDefinition: A context-free grammar (CFG) is a 4- tuple. variables = nonterminals, terminals, rules = productions,,
CMPSCI 601: Recall From Last Time Lecture 5 Definition: A context-free grammar (CFG) is a 4- tuple, variables = nonterminals, terminals, rules = productions,,, are all finite. 1 ( ) $ Pumping Lemma for
More information6.006 Introduction to Algorithms. Lecture 25: Complexity Prof. Erik Demaine
6.006 Introduction to Algorithms Lecture 25: Complexity Prof. Erik Demaine Today Reductions between problems Decision vs. optimization problems Complexity classes P, NP, co NP, PSPACE, EXPTIME, NP completeness
More information1. (10 points) Draw the state diagram of the DFA that recognizes the language over Σ = {0, 1}
CSE 5 Homework 2 Due: Monday October 6, 27 Instructions Upload a single file to Gradescope for each group. should be on each page of the submission. All group members names and PIDs Your assignments in
More informationP and NP CISC4080, Computer Algorithms CIS, Fordham Univ. Instructor: X. Zhang
P and NP CISC4080, Computer Algorithms CIS, Fordham Univ. Instructor: X. Zhang Efficient Algorithms So far, we have developed algorithms for finding shortest paths in graphs, minimum spanning trees in
More informationEND-TERM EXAMINATION
(Please Write your Exam Roll No. immediately) Exam. Roll No... END-TERM EXAMINATION Paper Code : MCA-205 DECEMBER 2006 Subject: Design and analysis of algorithm Time: 3 Hours Maximum Marks: 60 Note: Attempt
More informationBoolean Functions (Formulas) and Propositional Logic
EECS 219C: Computer-Aided Verification Boolean Satisfiability Solving Part I: Basics Sanjit A. Seshia EECS, UC Berkeley Boolean Functions (Formulas) and Propositional Logic Variables: x 1, x 2, x 3,, x
More informationName: CS 341 Practice Final Exam. 1 a 20 b 20 c 20 d 20 e 20 f 20 g Total 207
Name: 1 a 20 b 20 c 20 d 20 e 20 f 20 g 20 2 10 3 30 4 12 5 15 Total 207 CS 341 Practice Final Exam 1. Please write neatly. You will lose points if we cannot figure out what you are saying. 2. Whenever
More information5. Lecture notes on matroid intersection
Massachusetts Institute of Technology Handout 14 18.433: Combinatorial Optimization April 1st, 2009 Michel X. Goemans 5. Lecture notes on matroid intersection One nice feature about matroids is that a
More information1. Provide two valid strings in the languages described by each of the following regular expressions, with alphabet Σ = {0,1,2}.
1. Provide two valid strings in the languages described by each of the following regular expressions, with alphabet Σ = {0,1,2}. (a) 0(010) 1 (b) (21 10) 0012 (c) 1 (200) 100 01 Examples: 1, 200, 111,
More informationA New Reduction from 3-SAT to Graph K- Colorability for Frequency Assignment Problem
A New Reduction from 3-SAT to Graph K- Colorability for Frequency Assignment Problem Prakash C. Sharma Indian Institute of Technology Survey No. 113/2-B, Opposite to Veterinary College, A.B.Road, Village
More informationProblem set 2. Problem 1. Problem 2. Problem 3. CS261, Winter Instructor: Ashish Goel.
CS261, Winter 2017. Instructor: Ashish Goel. Problem set 2 Electronic submission to Gradescope due 11:59pm Thursday 2/16. Form a group of 2-3 students that is, submit one homework with all of your names.
More information