Indian Institute of Technology, Delhi Randomized Algorithms (2-SAT & MAX-3-SAT) (14/02/2008) Sandhya S. Pillai(2007MCS3120) Sunita Sharma(2007MCS2927) Advanced Algorithms Course (CSL 758) Prof. Kavitha Telikepalli Prof. Naveen Garg

Indian Institute of Technology, Delhi Agenda Introduction 2-SAT Problem Monte Carlo vs. Las Vegas methods Analysis of 2-SAT Random Walks and Markov inequality 3-SAT Problem Why 3-SAT is NP-Hard? Max 3-SAT Problem randomization & de- randomization References

Indian Institute of Technology, Delhi Randomized Algorithm In addition to input, algorithm takes a source of random numbers and makes random choices during execution. Behavior can vary even on a fixed input.

Indian Institute of Technology, Delhi SAT Satisfiability (SAT) is the problem of deciding whether a boolean formula in propositional logic has an assignment that evaluates to true. SAT occurs as a problem and is a tool in applications (e.g. Artificial Intelligence and circuit design) and it is considered a fundamental problem in theory, since many problems can be naturally reduced to it and it is the 'mother' of NP-complete problems.

Indian Institute of Technology, Delhi 2-SAT Problem φ = (x1 V~x2) ^ (x1 V ~x3) ^ (~x1 V x3) ^ (~x1 V ~x2) The k-SAT problem is the variant of SAT, in which each clause consists of exactly k distinct literals. For k >= 3, k-SAT is NP hard, but for k = 1 and 2, there are polynomial time solutions. For k = 1, this solution is trivial But for k = 2, it is slightly tricky. There is a much easier random algorithm. For a system with literals Xi, 0 <= i < m, and clauses Cj, 0 <= j < n, it goes as follows: SAT:Given a boolean formula ф in CNF,determine if ф is satisfiable or not.

Indian Institute of Technology, Delhi Example Given a Boolean formula φ in CNF, determine, whether φ is satisfiable or not.! Ex: AND of clauses (A 1 V A 2 V A 3 V A 4 )Λ (Ā 1 V Ā 3 V Ā 4 ) Λ (A 1 V Ā 2 V Ā 3 ) Each clause has at least one literal set to true. If Ā 1 = true, A 2 = true then Clause 1 and 2 are satisfiable but the Clause 3 requires A3 literal set to be false for it to be a satisfiable one.

Indian Institute of Technology, Delhi Input : A Boolean formula φ in conjunctive normal form with exactly two distinct literals in every clause. E.g., φ = (x1 V~x2) ^ (x1 V ~x3) ^ (~x1 V x3) ^ (~x1 V ~x2) 1)/* Start with an arbitrary initial assignment to the literals*/ for (i = 0; i < m; i++) {x i = true;} Let us call this assignment A 2)/* here function number_satisfied() returns the number of satisfied clauses for the current assignment */ for (t = 0; t < T && number_satisfied(x, n) < n; t++) { select an arbitrary non-satisfied clause C J ; randomly and uniformly pick one of the literals xi in CJ ; x i = (x i + 1) mod 2; } 3) If (number_satisfied(x, n) == n) { report that the set of clauses is satisfiable ;} else{ report that the set of clauses is not satisfiable} Algorithm

Indian Institute of Technology, Delhi Function number_satisfied() can be computed in linear time, some savings might be achieved by only keeping track of the changes between rounds. However, computation time is not so much the issue here, the main point is answering the question how large T must be taken to be reasonable sure that this Monte- Carlo algorithm gives the correct answer for satisfiable systems There is a chance of error and we need to bound that.

Indian Institute of Technology, Delhi Monte Carlo vs. Las Vegas methods Definition A Las Vegas algorithm is a randomized algorithms that always return the correct result. The only variant is that it’s running time might change between executions. –An example for a Las Vegas algorithm is the QuickSort algorithm. Definition A Monte Carlo algorithm is a randomized algorithm that might output an incorrect result. However, the probability of error can be diminished by repeated executions of the algorithm. –The MinCut algorithm is an example of a Monte Carlo algorithm.

Indian Institute of Technology, Delhi Bounding the Error in Algorithm This algorithm will always give correct non-satisfiable instances of ф But if ф is satisfiable Then we need to fix value of T such that this algorithm says “not satisfiable “ with a probability <=1/4 we will assume in the following that there is a satisfiable assignment to the x i Let call it S There may even be several such assignments, but we concentrate on a single one, which we will refer to as the correct assignment. The above process can be modeled as a random walk. The graph is the line graph with n nodes: node y is connected to node y - 1 and node y + 1, as far as these indices are at least 1 and at most n.

Indian Institute of Technology, Delhi Random Walk Let G = (V,E) be a connected, undirected graph. A random walk on G, starting from vertex s ∈ V, is the random process defined as follows: u := s repeat for T steps choose a neighbor v of u uniformly at random. u := v Given G, this is obviously trivial to implement on a computer.

Indian Institute of Technology, Delhi Consider a particle moving in an one-dimensional line. At each point in time, the particle will move either1 step to the right with probability p or 1 step to the left with probability 1 − p. Analysis: let A 2 {0,1} n be any satisfying assignment With probability at least ½ distance to A is reduced With probability at most ½ distance to A is increased 0n

Indian Institute of Technology, Delhi Randomized 2-SAT Analysis Distance can never be larger than n –if it starts at some 0 < i < n –Dominated by a walk where –With probability exactly ½ distance to A is reduced –With probability exactly ½ distance to A is increased

Indian Institute of Technology, Delhi Analysis contd.. Let us define a random variable X i X i =# of steps to reach state n starting from state i X i =1 + # of steps to reach state n starting from state i+1 with probably ½ (X i+1 ) X i =1 + # of steps to reach state n starting from state i-1 with probably ½ ( X i-1 ) this is a memory less property There for start from the current state is a fresh start.

Indian Institute of Technology, Delhi Analysis contd.. E[X i ]=1/2 E[1+X i+1 ]+1/2E[1+ X i-1 ] = 1+(E[ X i-1 ]+E[X i+1 ])/2 Let S i =E[ X i ] Then S i =1+( S i-1 +S i+1 )/2 S n-1 =1+( S n-2 + S n )/2

Indian Institute of Technology, Delhi S 0 =S (1) 2S 1 =S 0 +S (2). 2S n-1 =S n (n) adding equations 1 to n we get S n-1 =2(n-1)+1=2n-1 S n-2 =(2n-1)+(2n-3). S 0 =(2n-1)+(2n-3) =n2 Analysis contd..

Indian Institute of Technology, Delhi Analysis contd.. Let us define another random variable Y Y= # of variables whose truth value is same in current assignment(A) & satisfiable assignment(S) Movement of Y is similar to i-1 ← i → i+1 Now we have to fix T

Indian Institute of Technology, Delhi Markov Inequality Markov's inequality gives an upper bound for the probability that a non-negative function of a random variable is greater than or equal to some positive constant Proposition 1:For any non-negative random variable Y and any real number k >0 we have Pr [Y>=k] <= E [Y] / k As an example let k = 2*E[ Y ]. Then the above says Pr [Y>= 2 *E[ Y ] ] <= 1/2. Namely, if you move out to twice the expectation, you can have only half the area under the curve to your right. This is quite intuitive.

Indian Institute of Technology, Delhi Proof for Markov Inequality E[Y] = ∑ Pr[Y=y]*y = ∑ Pr[Y=y]*y +∑ Pr[Y=y]*y y =k >= 0 + k* Pr [ Y>=k ] =Pr [ Y>=k ]<=E [ Y ] / k Therefore T=4n 2

Indian Institute of Technology, Delhi Probability Pr[φ is satisfiable but algorithm returns unsatisfiable] =Pr[our algorithm does not reach state n in 4n 2 steps] = Pr[ Z>=4E[Z] <= ¼ ] Z=# of steps to reach state n from our start state E[Z]<= n 2

Indian Institute of Technology, Delhi Practical Example of Satisfiability Circuit satisfiability is a good example of this problem that we don't know how to solve in polynomial time. In this problem, the input is a boolean circuit: a collection of and, or, and not gates connected by wires. We will assume that there are no loops in the circuit (so no delay lines or ip-ops). The input to the circuit is a set of m boolean (true/false) values x1; : : : ; xm. The output is a single boolean value. Given specific input values, we can calculate the output in polynomial (actually, linear) time using depth-first-search and evaluating the output of each gate in constant time.

Indian Institute of Technology, Delhi Why NP hard ? The circuit satisfiability problem asks, given a circuit, whether there is an input that makes the circuit output True, or conversely, whether the circuit always outputs False. Nobody knows how to solve this problem faster than just trying all 2 m possible inputs to the circuit, but this requires exponential time. On the other hand, nobody has ever proved that this is the best we can do; maybe there's a clever algorithm that nobody has discovered yet! Hence this comes under NP hard problem.

Indian Institute of Technology, Delhi 3 CNF SAT A special case of SAT that is incredibly useful in proving NP- hardness results is 3SAT (or 3-CNF-SAT). A boolean formula is in conjunctive normal form (CNF) if it is a conjunction (and) of several clauses, each of which is the disjunction (or) of several literals, each of which is either a variable or its negation.

Indian Institute of Technology, Delhi For example: (A 1 V A 2 V A 3 )Λ (Ā 1 V Ā 3 V A 4 ) Λ (A 1 V Ā 2 V A 3 ) Given such a boolean formula, can we come up with an algorithm, that is polynomial in time ? The answer to this question is NO!!! Hence this is NP hard problem.

Indian Institute of Technology, Delhi We could prove that 3SAT is NP-hard by a reduction from the more general SAT problem,but it's easier just to start over from scratch, with a boolean circuit. We perform the reduction in several stages. 1. Make sure every and and or gate has only two inputs. If any gate has k > 2 inputs, replace it with a binary tree of k-1 two-input gates. 2. Write down the circuit as a formula, with one clause per gate. This is just the previous reduction. Proof

Indian Institute of Technology, Delhi 3. Change every gate clause into a CNF formula. There are only three types of clauses, one for each type of gate: A 1 = A 2 ΛA 3 -> (A 1 V Ā 2 V Ā 3 ) Λ(Ā 1 V A 2 ) Λ (Ā 1 V A 1 ) A 1 = A 2 V A 3 -> ( Ā 1 V A 2 V A 3 ) Λ (A 1 V Ā 2 ) Λ ( A 1 V Ā 3 ) A 1 = Ā 2 -> (A 1 VA 2 ) Λ ( Ā 1 V Ā 2 ) 4. Make sure every clause has exactly three literals. Introduce new variables into each one- and two-literal clause, and expand it into two clauses as follows: A 1 -> ( A 1 V e V u) Λ (A 1 V ē V u) Λ (A 1 V e V ū) Λ (A 1 V ē V ū ) A 1 V A 2 -> (A 1 V A 2 V e) Λ (A 1 V A 2 V ē )

Indian Institute of Technology, Delhi Example If we start with the above example formula, we obtain the following 3CNF formula.

Indian Institute of Technology, Delhi Although the 3CNF formula is complicated than the original one at first glance, it's actually only a constant factor larger. Even if the formula were larger than the circuit by a polynomial, like n 373, we would have a valid reduction. The formula is satisfiable if and only if the original circuit is satisfiable. As with the more general SAT problem, the formula is only a constant factor larger than any reasonable description of the original circuit, and the reduction can be carried out in polynomial time. Thus, we have a polynomial-time reduction from circuit satisfiability to 3SAT

Indian Institute of Technology, Delhi T CSAT (n) T 3 SAT (n) >= T CSAT (( ᾨ (n)) - O(n) So 3SAT is NP-hard. Finally, since 3SAT is a special case of SAT, it is also in NP, so 3SAT is NP- complete.

Indian Institute of Technology, Delhi Max 3-CNF This problem is to find an assignment which maximizes the number of satisfiable clauses. Let us take for example (A 1 V A 2 V Ā 3 ) ^ (A 1 V Ā 2 VA 4 ) ^ ( A 2 V Ā 3 V A 4 )

Indian Institute of Technology, Delhi Randomized Algorithm for Max 3-CNF Set each variable to true with probability 1/2 independently. (i.e. for instance, toss the coin and if head, set the variable to true and if tail, set the variable to false). If a clause is not satisfied, this means all the 3 variables in the clause are false. Prob. that a clause is not satisfied = 1/2 * 1/2 * 1/2 (since each variable is independently set to false) Prob. that a clause is satisfied = 1 - Prob. that a clause is not satisfied = 1 - 1/8 = 7/8

Indian Institute of Technology, Delhi There may be dependent clauses too. Eg. (A 1 V A 2 V A 3 ) Λ ( Ā 1 V A 2 V A 3 ) In the above two clauses the first clause is true and hence the second clause is bound to be true if x2 = true or x3 = true or both are true. Each clause is satisfied with prob. 7/8 Let m be the no. of clauses. Expected number of satisfiable clauses can be found as follows. Let Xi be a random variable. Xi = 1 if i th clause is satisfied. = 0 otherwise X = ∑ i Xi Expected value of X E[X] = E[X1 + X Xm]

Indian Institute of Technology, Delhi Linearity of expectation: Let r be any real number and let X and X1,..., Xn be random variables on a discrete probability space Ω such that their expectations all exist. Then the expectation of rX and of X Xm exists and we have E [rX] = rE [X] and E [X Xn ] = E [X1 ] E [Xn ]. Since linearity of expectations always holds, E[X] = E[X1] + E[X2] E[Xm] = 7/8 + 7/ /8 = 7/8* m

Indian Institute of Technology, Delhi Approximation Algorithm Approximation algorithms are algorithms used to find approximate solutions to optimization problems. Approximation algorithms are often associated with NP- hard problems; since it is unlikely that there can ever be efficient polynomial time exact algorithms solving NP- hard problems, one settles for polynomial time sub- optimal solutions. This is termed as 7/8 th approximation algorithm because, if the MAX-3SAT is satisfiable, then the expected weight of the assignment found is at least 7/8 of optimal clauses.

Indian Institute of Technology, Delhi Derandomization De randomization : First devise a randomized algorithm then argue that it can be derandomized to yield a deterministic algorithm Consider the following formula. (A 1 V Ā 2 V A 3 ) Λ ( Ā 2 V Ā 3 V A 1 ) Λ( Ā 3 V A 2 V Ā 1 ) The assignment for the literals in the above clauses can be drawn in the form of a binary tree.

Indian Institute of Technology, Delhi Assignment of the Variables A1=1 A1=0 A2=1A2=0 A2=1 A3=0 A3=1 A3=0A3=1 A3=0 A3=1 A3=0A3=1

Indian Institute of Technology, Delhi At each leaf node, the number of clauses satisfied by the assignment of the variables is shown on selection of the particular path towards that leaf. If n variables, then the tree will consist of n levels. If n levels are there, 2 n leaves are there. Thus 2 n assignments are possible. The average of the numbers on the leaves = 7/8 * m In 7/8 of the leaves, the clause will be contributing for appearing in the leaf.There are 2 n leaves. Each clause contributes 1 to 7/8 * 2 n leaves Sum of the numbers on leaves = 7/8 * 2 n * m = 21 (Since m = 3 in the example here) Avg. of the numbers on leaves = (7/8 * 2 n * m)/ 2 n = 21/8=2.65

Indian Institute of Technology, Delhi Here, starting from the root, first compute the average at each node. Then pick the path having greater average. The process for computing the average is Consider the root assignment A 1 =0, A 1 = 1 if A 1 = 1, compute the satisfiable clauses probability. i.e. (A 1 V Ā 2 V A 3 )... satisfied. ( Ā 2 V Ā 3 V A 1 )... satisfied ( Ā 3 V A 2 V Ā 1 )... not satisfied. -> By discarding Ā 1 from the clause we obtain ( A 2 V Ā 3 ). The probability that this clause is not satisfied is 1/4. So the probabilitythat this clause is satisfied is ¾. Thus the total probability turns out to be ( ¾) =1+1/4. This is the average when A 1 = 1 Similarly compute the average for A 1 = 0. which comes out to be 10/4. Choose the greater average and move towards that path. So we move towards A 1 = 1 path.

Indian Institute of Technology, Delhi Now compute the average by taking the assignment A 2 =0 and A 2 =1 and move towards the path with greater average. Thus at each level i, we have to compute the average, with assignment true, and with false. Thus, on reaching the leaf, we would have the average number of clauses which will be satisfiable with the assignment on the chosen path.

Indian Institute of Technology, Delhi In the above algorithm greedy approach is been followed, at each level we check which sub tree will give the best average & we take decision according to the current maximum. Thus algorithm may result in an sub-optimal result.

Indian Institute of Technology, Delhi References Randomized Algorithmes: R.Motwani Chapters 1,5,6,10 Introducton to Algorithms: CLRS Chapter 5 A compendium of NP webpage

Indian Institute of Technology, Delhi Thank You -Sandhya S. Pillai (2007MCS3120) -Sunita Sharma (2007MCS2927)