A Fixed-Parameter Algorithm for Random Instances of Weighted d-CNF Satisfiability

A Fix ed-P arameter Algorithm for Random Instances of W eighted d -CNF Satisfiability Y ong Gao ∗ Department of Computer Science Irving K. Barber School of Arts and Sciences Uni versity of British Columbia Okanag an K elo wna, Canada V1V 1V7 Nov ember 28, 2021 Abstract W e study random instances of the weighted d -CNF satisfiability problem (WEIGHTED d - SA T), a generic W[1]-complete problem. A random instance of the problem consists of a fixed parameter k and a random d -CNF formula F n,p k,d generated as follows: for each subset of d variables and with probability p , a clause ov er the d variables is selected uniformly at random from among the 2 d − 1 clauses that contain at least one negated literals. W e show that random instances of WEIGHTED d -SA T can be solved in O ( k 2 n + n O (1) ) - time with high probability , indicating that typical instances of WEIGHTED d -SA T under this instance distribution are fixed-parameter tractable. The result also hold for random instances from the model F n,p k,d ( d 0 ) where clauses containing less than d 0 (1 < d 0 < d ) neg ated literals are forbidden, and for random instances of the renormalized (miniaturized) version of WEIGHTED d -SA T in certain range of the random model’ s parameter p ( n ) . This, together with our previous results on the threshold behavior and the resolution complexity of unsatisfiable instances of F n,p k,d , provides an almost complete characterization of the typical-case behavior of random instances of WEIGHTED d -SA T . 1 Intr oduction The theory of parameterized complexity and fixed-parameter algorithms is becoming an activ e re- search area in recent years [8, 16]. Parameterized complexity provides a new perspectiv e on hard algorithmic problems, while fixed-parameter algorithms hav e found applications in a variety of ar - eas such as artificial intelligence, computational biology , cognitive modeling, graph theory , and v arious optimization problems. The study of the typical-case behavior of random instances of NP-complete problems and coNP- complete problems such as satisfiability (SA T) and graph coloring has had much impact on our understanding of the nature of hard problems as well as the strength and weakness of algorithms and well-founded heuristics [1, 3, 5, 7]. Designing polynomial-time algorithms that solve random ∗ W ork supported by NSERC Discovery Grant RGPIN 327587-06 1 instances of NP-complete problems under v arious random distributions has also been an activ e research area. In this work, we extend this line of research to intractable parameterized problems. W e study random instances of the weighted d -CNF satisfiability problem (WEIGHTED d -SA T), a generic W[1]-complete parameterized problem. An instance of WEIGHTED d -SA T consists of a d -CNF formula F and a fixed parameter k > 0 . The question is to decide if there is a satisfying assignment with Hamming distance k to the all-zero assignment. A variant of WEIGHTED d -SA T is MINI- WEIGHTED d -SA T that asks if there is a satisfying assignment with Hamming distance k log n to the all-zero assignment. W e sho w that there is an O ( k 2 n + n O (1) ) -time algorithm that solves random instances of WEIGHTED d -SA T with high probability for any p ( n ) = c log n n d − 1 . The result also hold for random instances from the more general model F n,p k,d ( d 0 ) where clauses containing less than d 0 (1 < d 0 < d ) negated literals are forbidden, and for random instances of MINI-WEIGHTED d -SA T with the ran- dom model’ s parameter p ( n ) being in a certain range. This, together with our pre vious results on the threshold behavior and resolution complexity of unsatisfiable instances of F n,p k,d in [11], provides a nearly complete characterization of the typical-case behavior of random instances of WEIGHTED d -SA T . T o the best knowledge of the author, this is the first work in the literature on the fixed- parameter tractability of random instances of a W[1]-complete problem. The main result of this paper is that instances from the random distribution F n,p k,d (and its general- ization F n,p k,d ( d 0 ) ) of WEIGHTED d -SA T are “typically” fixed-parameter tractable for any p = c log n n d − 1 with c > 0 . Theorem 1 There is an O ( k 2 n + n O (1) ) -time algorithm that with high pr obability , either finds a satisfying assignment of weight k or reports that no such assignment exists for a random instance ( F n,p k,d , k ) of WEIGHTED d -SA T for any p = c log n n d − 1 with c > 0 . In the appendices, we show that the same algorithm can be extended to solve random instances from the more general model F n,p k,d ( d 0 ) and random instances of MINI-WEIGHTED d-SA T for certain range of the probability parameter p ( n ) . The next section contains necessary preliminaries and a detailed description of the random model. In Section 3, we present the algorithm W -SA T together with a discussion on its time com- plexity . In Section 4, we prove that W -SA T succeeds with high probability for random instances of WEIGHTED d -SA T . In the last section, we discuss directions for future work. 2 Pr eliminaries and Random Models f or WEIGHTED d -SA T An instance of a parameterized decision pr oblem is a pair ( I , k ) where I is a problem instance and k is the problem parameter [8, 16]. Usually , the parameter k either specifies the “size” of the solution or is related to some structural property of the underlying problem, such as the tree width of a graph. A parameterized problem is fixed-parameter tractable (FPT) if any instance ( I , k ) of the problem can be solved in f ( k ) | I | O (1) time, where f ( k ) is a computable function that depends only on k . Parameterized problems are inter-related by parameterized reductions, resulting in a classification of parameterized problems into a hierarchy of complexity classes F P T ⊆ W [1] ⊆ W [2] · · · ⊆ X P . It is belie ved that the inclusions are strict and the notion of completeness can be naturally defined via parameterized reductions. 2 2.1 W eighted CNF Satisfiability and its Random Model As with the theory of NP-completeness, the satisfiability problem plays an important role in the theory of parameterized comple xity . A CNF formula (o ver a set of Boolean variables) is a conjunc- tion of disjunctions of literals. A d -clause is a disjunction of d -literals. A d -CNF formula is a CNF formula that consists of d -clauses only . An assignment to a set of n Boolean v ariables is a v ector in { TR UE, F ALSE } n . The weight of an assignment is the number of variables that are set to TR UE by the assignment. It is conv enient to identify TRUE with 1 and F ALSE with 0 . Thus, an assignment can also be regarded as a vector in { 0 , 1 } n and the weight of an assignment is just its Hamming distance to the all-zero assignment. A representativ e W [1] -complete problem is the following weighted d-CNF satisfiability (WEIGHTED d-SA T) problem: Problem 1 WEIGHTED d-SA T Instance: A CNF formula consisting of d -clauses, and a positive inte ger k . Question: Is ther e a satisfying assignment of weight k ? In [14], Marx studied the parameterized complexity of the more general parameterized Boolean constraint satisfaction problem. One of the results of Marx ([14], Lemma 4.1), when applied to CNF formulas, is that any instance of WEIGHTED d -SA T can be reduced to at most d k instances each of which is a conjunction of clauses that contain at least one negated literal. Marx further proved that WEIGHTED d -SA T is W[1]-complete e ven when restricted to CNF formulas that consist of clauses of the form x ∨ y . W e use G ( n, p ) to denote the Erd ¨ os-Renyi random graph where n is the number of vertices and p is the edge probability [4]. In G ( n, p ) , each of the possible  n 2  edges appears independently with probability p . A random hyper-graph G ( n, p, d ) is a hypergraph where each of the  n d  pos- sible hyperedges appears independently with probability p . Throughout the paper, by “with high probability” we mean that the probability of the e vent under consideration is 1 − o (1) . W e will be working with the following random model of WEIGHTED d -SA T , which is basically similar in spirit to random CNF formulae with a planted solution studied in traditional (constraint) satisfiability (See, e.g., [2, 9, 10, 12, 13, 15] and the references therein). Definition 2.1 Let X = { x 1 , · · · , x n } be a set of Boolean variables and p = p ( n ) be a function of n . Let k and d be two positive constants. W e define a random model F n,p k,d for WEIGHTED d-SA T parameterized by k as follows: T o gener ate an instance F fr om F n,p k,d , we first construct a random hyper graph G ( n, p, d ) using X as the verte x set. F or eac h hyper edge { x i 1 , · · · , x i d } , we include in F a d -clause selected uniformly at random fr om the set of 2 d − 1 non-monotone d -clauses defined over the variables { x i 1 , · · · , x i d } . (A monotone clause is a clause that contains positive literals only). The model F n,p k,d can be generalized to F n,p k,d ( d 0 ) as follo ws: instead of from the set of non- monotone clauses, we select uniformly at random from the set of clauses over { x i 1 , · · · , x i d } that contain at least d 0 negated literals. Note that F n,p k,d is just F n,p k,d (1) . In the rest of this paper, we will be focusing on F n,p k,d , but will discuss how the algorithm and the results can be adapted to F n,p k,d ( d 0 ) in Appendix A. 3 Note that since monotone clauses are excluded, the all-zero assignment always satisfies a ran- dom instance of F n,p k,d in the traditional sense. On the other hand, in vie w of Marx’ s results we mentioned earlier in this subsection, forbidding monotone clauses is not really a restriction. As a matter of fact, our study begins with a random model that doesn’ t pose any restriction on the type of clauses that can appear in a formula. Such a model, howe ver , turns out to be trivially unsatisfiable since unless the model parameter p ( n ) is e xtremely small, a random instance will contain more than 2 k independent monotone clauses. 2.2 Residual Graphs of CNF F ormulas and Induced Formulas Associated with a CNF formula is its r esidual graph over the set of variables in volved in the formula. There is an edge between two v ariables if they both occur in some common clause. The residual graph of a random instance of F n,p k, 2 is the random graph G ( n, p ) . The residual graph of a random instance of F n,p k,d is the primal graph of the random hyper graph G ( n, p, d ) . Let F be a d -CNF formula and V ⊂ X be a subset of v ariables. The induced formula F V of F over V is defined to be the CNF formula F V that consists of the follo wing two types of clauses: 1. the clauses in F that only in volv e the variables in V ; 2. the clauses of size at least 2 obtained by removing any literal whose corresponding variables are in X \ V . 3 A Fixed-Parameter Algorithm f or Instances of F n,p k ,d In this section, we describe the details of the fixed-parameter algorithm designed for random in- stances of F n,p k,d and show that its time comple xity is O ( k 2 n + n O (1) ) . The results in this section and in the next section together establish Theorem 1. 3.1 General Idea W e describe the general idea in terms of WEIGHTED 2-SA T . A detailed description of the algorithm for F n,p k,d is given in the next subsection. The generalization of the algorithm to the more general random model F n,p k,d ( d 0 ) is presented in Appendix A. The algorithm W -SA T considers all the v ariables x that appears in more than k + 1 clauses of the form x ∨ y . Any such variable cannot be assigned to TR UE. By assigning these forced variables to F ALSE, we get a reduced formula. W -SA T then checks to see if the reduced formula can be decomposed into connected components of size at most log n . If no such decomposition is possible, W -SA T gi ves up. Otherwise, let {F i , 1 ≤ i ≤ m } be the collection of connected components in the reduced formula. F or each connected component F i , use brute-force to find the set of integers L i such that for each k 0 ∈ L i , there is an assignment of weight k 0 to the v ariables in F i that satisfies F i . Finally , a dynamic programming algorithm is applied to find in time O ( k 2 n ) a collection of at 4 most k positiv e integers { k i j , 1 ≤ j ≤ k } such that  k i j ∈ L i j , and k i 1 + k i 2 + · · · + k i k = k Combining the weight- k i j solutions to the subproblems index ed by i j , a weight- k solution can be found. If on the other hand, no such { k i j , 1 ≤ j ≤ k } can be found, we can safely report that the original instance has no weight- k satisfying assignment. 3.2 Details of the Algorithm W -SA T W e first introduce the following concept that is essential to the algorithm: Definition 3.1 Let ( F , k ) be an instance of WEIGHTED d -SA T wher e F is a d -CNF formula and k is the parameter . Consider a variable x and a collection of subsets of variables Y = { Y i , 1 ≤ i ≤ k } wher e Y i = { y ij , 1 ≤ j ≤ ( d − 1) } is a subset of X \ { x } . W e say that the collection Y freezes x if the following two conditions are satisfied: 1. Y i ∩ Y j = ∅ , ∀ i, j . 2. for each 1 ≤ i ≤ k , the clause x ∨ y i 1 ∨ · · · ∨ y i ( d − 1) is in the formula F . A variable x is said to be k-frozen with r espect to a subset of variables V if it is fr ozen by a collection of subsets of variables { Y i , 1 ≤ i ≤ k } such that Y i ⊂ V , ∀ 1 ≤ i ≤ k . A variable that is k -fr ozen with r espect to the set of all variables is simply called a k -frozen variable . It is obvious that a k -frozen v ariable cannot be assigned to TR UE without forcing more than k other v ariables to be TRUE. W e also need the following concept to describe the algorithm: Definition 3.2 Let F be a CNF formula. W e use L F to denote the set of inte gers between 0 and k such that for eac h k 0 ∈ L F , ther e is a satisfying assignment of weight k 0 for F . The algorithm W -SA T is described in Algorithm 1. W e explain in the following the purpose of the subroutine REDUCE(). The subroutine REDUCE( F , U ) simplifies the formula F after the v ariables in U hav e been set to 0. It works in the same way as the unit-propagation based inference in the well-kno wn DPLL procedure for traditional satisfiability search: It remov es any clause that is satisfied by the assignment to the v ariables in U ; deletes all the occurrences of a literal that has become F ALSE due to the assignment; and assigns a proper value to the v ariables that are forced due to the literal-deletion. The procedure terminates when there is no more forced v ariable. It is easy to see the follo wing lemma holds for the subroutine REDUCE(): Lemma 3.1 REDUCE() ne ver assigns TR UE to a variable. If F 0 = REDUCE ( F , U ) is empty , then F has a weight- k satisfying assignment if and only if at least k variables have not been assigned by REDUCE(). 5 Algorithm 1 W -SA T Input: An instance ( F , k ) of WEIGHTED d -SA T Output: A satisfying assignment of weight k , or UNSA T , or F AILURE 1: Find the set of k -frozen variables U and assign them to F ALSE. 2: Let F 0 = REDUCE ( F , U ) be the reduced formula . 3: Find the connected components {F 1 , · · · , F m } of F 0 . 4: If there is a connected component of size larger than log n , return “F AILURE”. 5: Otherwise, for each connected component F i , use brute force to find L F i . 6: Find a set of at most k indices { i j , 1 ≤ j ≤ k } and a set integers { k i j , 1 ≤ j ≤ k } such that k i j ∈ L F i j and k P j =1 k i j = k . Return “UNSA T” if there is no such index set. 7: For each F i j , use brute-force to find a weight- k i j assignment to the variables in F i j that satisfies F i j . 8: Combine the assignments found in the abov e to form a weight- k satisfying assignment to the formula F . 3.3 Correctness and T ime Complexity of W -SA T The correctness follo ws directly from the previous discussion. For the time complexity , we have the follo wing Proposition 3.1 The running time of W-SA T is in O ( k 2 n + n O (1) ) . Pr oof. Since Lines 1 through 4, Line 5, and Line 7 together take n O (1) time, we only need to sho w that Line 6 can be done in O ( k 2 n ) time using dynamic programming. Consider an integer k and a collection { L i , 1 ≤ i ≤ m } where each L i is a subset of integers in { 0 , 1 , · · · , k } . W e say that an integer a is achievable by { L i , 1 ≤ i ≤ m } if there is a set of indices I a = { i j , 1 ≤ j ≤ l } such that for each i j , there is a k i j ∈ L i j so that l P j =1 k i j = k . W e call any such an index set I a a repr esentative set of a . The purpose of Line 6 is to check to see if the integer k is achievable , and if YES, return a representati ve set of k . The Proposition follows from the follo w lemma.  Lemma 3.2 Given a collection { L i , 1 ≤ i ≤ m } and an inte ger k wher e each L i is a subset of inte gers in { 0 , 1 , · · · , k } , ther e is a dynamic pr ogramming algorithm that finds a r epresentative set of k if k is achie vable, or r eports that k is not achie vable. It runs in time O ( k 2 m ) . Pr oof. Let A ( t ) = { ( a, I a ) : 0 ≤ a ≤ k } be the set of pairs ( a, I a ) where 0 ≤ a ≤ k is an integer achie vable by { L i , 1 ≤ i ≤ t } and I a is a representati ve set of a . Let A (0) = ∅ . W e see that A ( t + 1) consists of the pairs of the form (( a + b ) , I a ) satisfying    ( a, I a ) ∈ A ( t ) , b ∈ L t +1 such that b ≤ k − a, and I a = I a ∪ { t } . A typical application of dynamic programming builds A (0) , A (1) , · · · , and A ( m ) . The value k is achie vable by { L i , 1 ≤ i ≤ m } if and only if there is a pair ( k , I k ) in A ( m ) . Since the size of A ( t ) is at most k , the abov e algorithm runs in O ( k 2 m ) time.  6 4 Algorithm W -SA T Succeeds W ith High Probability In this section, we prov e that the algorithm W -SA T succeeds with high probability on random in- stances of F n,p k,d . Due to Proposition 3.1, we only need to show that W -SA T reports “F AILURE” with probability asymptotic to zero. Recall that W -SA T fails only when the reduced formula F 0 ob- tained in Line 2 has a connected component of size at least log n . The rest of this section is de voted to the proof of the follo wing Proposition: Proposition 4.1 Let F = F n,p k,d be the input r andom CNF formula to W -SA T . W ith high pr obability , the r esidual graph of the induced formula F V on V decomposes into a collection of connected components of size at most log n , wher e V is the set of variables that are not k -fr ozen. Pr oof. Let X = { x 1 , · · · , x n } be the set of Boolean variables, and let U be the set of k -fr ozen v ariables so that V = X \ U . Since p = c log n n d − 1 with c > 0 , there will be many k -frozen v ariables so that the size of U is large. If U were a randomly-selected subset of v ariables, the proposition is easy to prov e. The difficulty in our case is that U is not randomly-selected, and consequently F V cannot be assumed to be distributed in the same manner as the input formula F . T o get around this difficulty , we instead directly upper bound the probability P ∗ that the residual graph of F n,p k,d contains as its subgraph a tree T ov er a gi ven set V T of log n variables such that every v ariable x ∈ V T is not k -frozen. Since the variables in F V are not k -frozen, an upper bound on P ∗ is also an upper bound on the probability that the residual graph of F V contains as its subgraph a tree of the size log n . W e then use this upper bound together with Markov’ s inequality to show that the probability that the residual graph of F V has a connected component of size at least log n tends to zero. Let T be a fixed tree over a subset V T of log n v ariables. The difficulty in estimating P ∗ is that the e vent that the residual graph of F n,p k,d contains T as its subgraph and the ev ent that no variable in T is k -frozen are not independent of each other . T o decouple the dependency , we consider the follo wing two ev ents: 1. A : the event that the residual graph of F n,p k,d contains the tree T as its subgraph; and 2. B : the event that none of the v ariables in V T is k -frozen with respect to X \ V T . Since by definition, being k -frozen with respect to a subset of variables implies being k -frozen with respect to all v ariables, we have P ∗ ≤ P {A ∩ B } . (4.1) W e now claim that Lemma 4.1 The two events A and B ar e independent, i.e., P {A|B } = P {B } (4.2) Pr oof. Note that the ev ent A depends only on those d -clauses that contain at least two variables in V T and that the ev ent B depends only on those d -clauses that contain exactly one variable in V T . Due to the definition of the random model F n,p k,d , the appearance of a clause defined ov er a d -tuple of v ariables is independent from the appearance of the other clauses. The lemma follo ws.  Based on Equation (4.1) and Lemma 4.1, we only need to estimate P {A} and P {B } . The follo wing lemma bounds the probability that a variable is not k -frozen. 7 Lemma 4.2 Let x be a variable and W ⊂ X such that x ∈ W and | W | > n − log n . W e have P { x is not k -fr ozen with respect to W } ≤ O (1) max( 1 n δ , log 2 n n ) wher e 0 < δ < c 3(2 d − 1)( d − 1)! . Pr oof. Let N x be the number of clauses of the form x ∨ y 1 ∨ · · · ∨ y d − 1 with { y 1 , · · · , y d − 1 } ⊂ X \ V T . Due to the definition of F n,p k,d , the random v ariable N x follo ws the binomial distribution B in ( p, m ) where p = 1 2 d − 1 c log n n d − 1 and m =  n − log n d − 1  . Write α = c (2 d − 1)( d − 1)! . By the Chernoff bound (see Appendix B), we ha ve P { N x < k } ≤ 2 e − ( pm − k ) 2 3 pm ≤ O ( k ) e − α 3 log n ∈ O ( n − δ ) ( where 0 < δ < α 3 ) . (4.3) Let D be the ev ent that in the random formula F , there are two clauses  x ∨ y 11 ∨ · · · ∨ y 1( d − 1) , and x ∨ y 12 ∨ · · · ∨ y 2( d − 1) such that { y 11 , · · · , y 1( d − 1) } ∩ { y 12 , · · · , y 2( d − 1) } 6 = ∅ . The total number of such pairs of clauses is at most ( d − 1)  n − log n d − 1  n − log n d − 2  . The probability for a specific pair to be in the random formula is  1 2 d − 1 c log n n d − 1  2 . By Marko v’ s inequality , we have P {D } ∈ O ( log 2 n n ) . Since the probability that the v ariable x is not k -frozen is at most P {{ N x < k } ∪ D } , the lemma follo ws.  From Lemma 4.2, we hav e Lemma 4.3 F or sufficiently lar ge n , P {B } < O (1)  n − δ  log n (4.4) for some 0 < δ < min( c 3(2 d − 1)( d − 1)! , 1) . 8 Pr oof. Let E x be the event that a variable x ∈ V T is not k -frozen with respect to X \ V T . Since | V T | = log n , the bound obtained in Lemma 4.2 applies to W = X \ V T . Since for any x ∈ V T , the e vent E x only depends on the existence of clauses of the form x ∨ y i 1 ∨ · · · ∨ y i ( d − 1) with { y i 1 , · · · , y i ( d − 1) } ⊂ X \ V T , we see that the collection of the ev ents { E x , x ∈ V T } are mutually independent. The Lemma follows from Lemma 4.2.  Next, we ha ve the follo wing bound on the probability P {A} . Lemma 4.4 P {A} ≤ O (1)(log n ) log n n − log n . Pr oof. Recall that A is the ev ent that a random instance of F n,p k,d induces all the edges of a fixed tree T with vertex set V T of size log n . W e follow the approach de veloped in [6, 10, 13] and extend the counting argument from 3-clauses to the general case of d -clauses with d > 2 . Let F T be a set of clauses such that ev ery edge of T is induced by some clause in F T . W e say that F T is minimal if deleting any clause from it lea ves at least one edge of T uncov ered. Consider the different ways in which we can cov er the edges of T by clauses. Treat the clauses in F T as being grouped into d − 1 dif ferent groups { S i , 1 ≤ i ≤ ( d − 1) } . A clause in the group S i is in charge of covering exactly i edges of T . Note that a clause in the group S i may “accidently” cov er other edges that are not its responsibility . As long as each clause has its own dedicated set of edges to cov er , there won’ t be any risk of under-counting. Let s i = | S i | , 1 ≤ i ≤ d − 1 . W e see that 0 ≤ s i ≤ log n/i . Since each clause in S i is dedicated to i edges and there are in total log n − 1 edges, we ha ve d − 1 X i =1 is i = log n − 1 . (4.5) Counting very crudely , there are at most  log n i  s i ways to pick the dedicated sets of i edges for the s i clauses in group S i . Since T is a tree, for each set of i edges there are at most  n d − ( i +1)  (2 d − 1) ways to select the corresponding clauses. Therefore, by Marko v’ s inequality , we hav e that P {A} can be upper bounded by X 0 ≤ s i ≤ log n  (log n ) P i is i (2 d − 1) P i ( d − i − 1) s i n P i ( d − i − 1) s i  c log n 2 d − 1 1 n d − 1  P i s i # < O (1) X 0 ≤ s i ≤ log n (log n ) log n n P i ( − is i ) , and due to Equation (4.5), we hav e P {A} ≤ O (1)(log n ) d (log n ) log n n − log n +1 ≤ O (1)(log n ) 2 log n n − log n . 9 This prov es Lemma 4.4.  Continuing the proof of Proposition 4.1, we combine Lemma 4.3 and Lemma 4.4 to get P {A ∩ B } ≤ O (1)(log n ) 2 log n n − log n  n − δ  log n . Since the total number of trees of size log n is at most n log n (log n ) log n − 2 , the probability that the residual graph of F V contains a tree of size log n is n log n (log n ) log n − 2 P {A ∩ B } < O (1)(log n ) 3 log n  n − δ  log n (4.6) Proposition 4.1 follo ws.  Pr oof. [ Pr oof of Theorem 1 ] T o use Proposition 4.1 to prov e that the algorithm W -SA T succeeds with high probability , we note that the reduced formula F 0 in Line 2 of the algorithm W -SA T is sparser than the induced formula F V . In fact, it is easy to see that F 0 is an induced sub-formula of F V ov er the set of variables that hav e not been assigned by the subroutine REDUCE(). Therefore by Proposition 4.1, with high probability F 0 decomposes into a collection of connected components, each of size at most log n . It follows that W -SA T succeeds with high probability . Combining all the above, we conclude that the algorithm W -SA T is a fixed-paramter algorithm and succeeds with high probability on random instances of F n,p k,d . This proves Theorem 1.  5 Discussions The results presented in this paper, together with our pre vious results on the threshold behavior and the resolution complexity of unsatisfiable instances of F n,p d,k in [11], provides a first probabilis- tic analysis of W[1]-complete problems. For WEIGHTED 2-SA T and MINI-WEIGHTED 2-SA T , the behavior of random instances from the studied instance distribution is fully characterized. For WEIGHTED d-SA T with d > 2 , the characterization is almost complete except for a small range of the probability parameter where the parametric resolution complexity is missing. In summary , random instances of WEIGHTED d-SA T from the random model under consideration are “typi- cally” fix ed-parameter tractable, and hard instances (in the sense of fix ed-parameter tractability) are expected only for MINI-WEIGHTED d-SA T . While we belie ve the random model F n,p k,d ( d 0 ) is very natural, we feel that it is challenging to come up with any alternativ e and natural instance distributions for weighted d -CNF satisfiability that are interesting and hard in terms of the complexity of typical instances. On the other hand, there are still many interesting questions with the model F n,p k,d . First, the behavior of random instances of MINI-WEIGHTED d-SA T with d > 2 is interesting due to the relation between such parameterized problems and the exponential time hypothesis of the satisfi- ability problem. Second, for p = c log n n d − 1 with c small enough, there will be sufficient number of “isolated” v ariables and by simply setting k of these variables to TRUE and the rest of the variables to F ALSE, we obtain a weight- k satisfying assignment. It is interesting to see what will happen if these isolated v ariables have been remo ved. 10 Refer ences [1] D. Achlioptas, P . Beame, and M. Molloy . 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Solving random satisfiable 3CNF formulas in expected polynomial time. In Pr oc. of 17th A CM-SIAM Symposium on Discr ete Algorithms , pages 454– 463, 2006. [14] D. Marx. Parameterized complexity of constraint satisfaction problems. Computational Com- plexity , (2):153–183, 2005. [15] M. Molloy . Models and thresholds for random constraint satisfaction problems. In Pr oceed- ings of STOC’02 , pages 209 – 217. A CM Press, 2002. [16] R. Neidermeier . In vitation to F ixed-P arameter Algorithms . Oxford University Press, 2006. 11 6 A ppendix A - Generalization to the Model F n,p k ,d ( d 0 ) Consider the model F n,p k,d ( d 0 ) , d 0 < d, that generalizes the model F n,p k,d . T o generate a random instance F of F n,p k,d ( d 0 ) , we first construct a random hypergraph G ( n, p, d ) in the same way as with the random model F n,p k,d . For each hyperedge { x i 1 , · · · , x i d } , we include in F a d -clause selected uniformly at random from the set of the d -clauses ov er { x i 1 , · · · , x i d } that contain at least d 0 negated literals. Note that with the above definition, the original model F n,p k,d is just F n,p k,d (1) . Similar to the analysis for F n,p k,d presented in [11], the following threshold beha vior of the solution probability can be established Lemma 6.1 Consider a random instance ( F n,p k,d ( d 0 ) , k ) of WEIGHTED d-SAT . Let p = c log n n d − d 0 with c > 0 being a constant and let c ∗ = a d ( d − d 0 )! with a d being the number of d -clauses over a fixed set of d variables that contain at least d 0 ne gated literals. W e have lim n P n F n,p k,d ( d 0 ) is satisfiable o =  1 , if c < c ∗ , 0 , if c > c ∗ For p = c log n n d − d 0 , the algorithm W -SA T can be adapted to solve a random instance of F n,p k,d ( d 0 ) in O ( k 2 n + n O (1) ) n ( d 0 − 1) time by using the follo wing generalization of a k -frozen variable: Definition 6.1 Let ( F , k ) be an instance of WEIGHTED d-SA T wher e F is a d-CNF formula and k is the parameter . Let 2 ≤ d 0 ≤ d be a fixed inte ger . Consider a variable x , a set of ( d 0 − 1) variable S = { x 1 , · · · , x d 0 − 1 } , and a collection of subsets of variables Y = { Y i , 1 ≤ i ≤ k } wher e Y i = { y ij , 1 ≤ j ≤ ( d − d 0 ) } is a subset of X \ ( { x } ∪ S ) . W e say that the collection Y of subsets of variables freeze x on S if 1. Y i ∩ Y j = φ, ∀ i, j . 2. for each 1 ≤ i ≤ k , the clause x 1 ∨ · · · ∨ x d 0 − 1 ∨ x ∨ y i 1 ∨ · · · ∨ y i ( d − d 0 ) is in the formula F . Lemma 6.2 If x is k -fr ozen on S = { x 1 , · · · , x d 0 − 1 } , then assigning all the variables in S to TRUE for ces x to be F ALSE. The modification of W -SA T to solve random instances of F n,p k,d ( d 0 ) is as follo ws: For each of the  n d 0 − 1  possible sets of ( d 0 − 1) v ariables S = ( x 1 , · · · , x d 0 − 1 ) , set them to TR UE and all the v ariables that are k -frozen on S to F ALSE; Apply the subroutine R E D U C E () to obtain a reduced formula F 0 ; Use the same technique in W -SA T to check to see if F 0 has a satisfying assignment of weight k − ( d 0 − 1) . The ov erall running time is O ( k 2 n + n O (1) ) n ( d 0 − 1) . 12 7 A ppendix B - Random Instances of MINI-WEIGHTED d -SA T In the proof in Section 4 and in this section, we use the follo wing Chernoff bound Lemma 7.1 Let I be a binomial random variable with e xpectation µ . W e have P {| I − µ | > t } ≤ 2 e − t 2 3 µ . As a variant of WEIGHTED d -SA T , the problem MINI-WEIGHTED d -SA T with parameter k asks if for a given d -CNF formula, there is a satisfying assignment of weight k log n . For random d -CNF formula F n,p k,d , the algorithm W -SA T for MINI-WEIGHTED d -SA T needs to be adapted to make use of the existence of k log n -frozen variables. T o guarantee that W -SA T still succeeds with high probability , a result similar to Proposition 4.1 is needed. This amounts to showing that the probability for a v ariable x to be k log n -frozen is small enough. For p = c log n n d − 1 with c > k 2 d − 1 ( d − 1)! , this is the case. Theorem 2 There is an O ( k 2 n + n O (1) ) -time algorithm that solves with high pr obability a random instance ( F n,p k,d , k ) of MINI-WEIGHTED d -SA T for any p = c log n n d − 1 with c > k (2 d − 1)( d − 1)! . Pr oof. The proof is almost the same as the proof of Proposition 4.1 except that we need to establish an upper bound on the probability that a variable is not k log n -frozen. For c > k (2 d − 1)( d − 1)! , Lemma 7.1 on the tail probability of a binomial random variable is still effecti ve and the ar guments made in the second half of the proof of Lemma 4.2 and in the proof of Lemma 4.3 are still valid. The only difference is the accuracy of the upper bound. In this case, we have P {B } ≤ O (1) 1 n δ log n where 0 < δ < min( ( k − c ) 2 c 3(2 d − 1)( d − 1)! , 1) , and this is sufficient for the result to hold.  13