Proceedings of the Twenty-Third AAAI Conference on Artificial Intelligence (2008)

Phase Transitions and Complexity of Weighted Satisfiability and Other Intractable Parameterized Problems Yong Gao Department of Computer Science Irving K. Barber School of Arts and Sciences University of British Columbia Okanagan Kelowna, Canada V1V 1V7 [email protected]

Abstract such as backdoor. In particularly, the issue of the worst- case intractability versus the practical hardness of the back- The study of random instances of NP complete and coNP door detection problem has been raised: while the backdoor complete problems has had much impact on our understand- ing of the nature of hard problems. In this work, we initiate detection problem is NP-complete and/or fixed-parameter an effort to extend this line of research to random instances intractable for many types of backdoors, SAT-solvers such of intractable parameterized problems. as SATZ can exploit the existence of small-sized back- We propose random models for a representative intractable doors effectively (Dilkina, Gomes, and Sabharwal 2007; parameterized problem, the weighted d-CNF satisfiability, Szeider 2005). and its generalization to the constraint satisfaction problem. The study of parameterized proof complexity of the The exact threshold for the phase transition of the proposed propositional satisfiability problem has been initiated in models is determined. Lower bounds on the time complex- (Dantchev, Martin, and Szeider 2007), where lower bounds ity of variants of the DPLL algorithm for these parameter- on the parameterized resolution proof are established for ized problems are also established. In particularly, we show CNF formulas that encode some first-order combinatorial that random instances of the weighted 2-CNF satisfiability, principle. already an intractable parameterized problem, are typically easy in both of the satisfiable and unsatisfiable regions by The study of random instances of NP-complete prob- exploiting an interesting connection between the unsatisfia- lems and coNP-complete problems such as SAT has had bility of a weighted 2-CNF formula and the existence of a much impact on our understanding of the nature of hard Hamiltonian-cycle-like global structure. problems (Achlioptas, Beame, and Molloy 2001; Beame et al. 2002; Cheeseman, Kanefsky, and Taylor 1991) as well 1. Introduction as the strength and weakness of well-founded algorithms and heuristics (Gent and Walsh 1998; Gomes et al. 2005; The theory of parameterized complexity and fixed- Gao and Culberson 2007). parameter algorithms is becoming an active research area In this work, we initiate an effort to extend this line of in recent years (Downey and Fellows 1999; Neidermeier research to intractable parameterized problem in the param- 2006). Parameterized complexity provides a new perspec- eterized complexity hierarchy. We propose random models tive on hard algorithmic problems, while fixed-parameter al- for a representative parameterized intractable problem, the gorithms have found applications in a variety of areas such weighted d-CNF satisfiability, and its variants. We establish as computational biology, cognitive modeling, graph theory, the exact threshold of the phase transition of the proposed and various optimization problems. model. This is in contrast to the threshold behavior of most Parameterized algorithmic problems also arise in many of the NP complete problems, such as the satisfiability and areas of artificial intelligence research. See, for example, the graph coloring, where the exact threshold is extremely hard survey of Gottlob and Szeider (Gottlob and Szeider 2007). to nail down. backdoor Recently, some problems related to detecting For the weighted 2-CNF satisfiability problem, which is sets for instances of the propositional satisfiability problem already W[1]-complete, we show that random instances are (SAT) have been studied from the perspective of parameter typically easy in both of the satisfiable and unsatisfiable re- complexity (Szeider 2005; Nishimura, Ragde, and Szeider gions. We prove that random (unsatisfiable) instances of 2004). A backdoor (to instances of a hard problem) is a sub- the weighted 2-CNF satisfiability problem have a “fixed- set of variables that, once fixed, will result in one or more parameter-sized” resolution proof by exploiting an interest- polynomial time sovable subproblems. The existence of ing link between the unsatisfiability of a weighted 2-CNF small sized backdoors naturally leads to fixed-parameter al- formula and the existence of a Hamiltonian cycle in a prop- gorithms. For example, in Bayesian networks and constraint erly defined directed graph. This is somewhat surprising networks, a feedback vertex set of the (directed) network is and indicates that the typical-case behavior of random in- Copyright c 2008, Association for the Advancement of Artificial stances of parameterized problems are quite different from Intelligence (www.aaai.org). All rights reserved. that of NP-complete problems. First, it is the existence of

265 a global structure (a directed Hamiltonian cycle) that leads As in the case of the theory of NP-completeness, the sat- to easy instances, as compared to the case of NP-complete isfiability problem plays an important role in the theory of problems where small structures are usually the signature parameterized complexity (Downey and Fellows 1999). A of easy instances. Secondly, unlike the case of traditional CNF formula (over a set of Boolean variables) is a conjunc- satisfiability, the weighted 2-CNF satisfiability problem is tion of disjunctions of literals. A d-clause is a disjunction of already fixed-parameter intractable. As a comparison, for d-literals. weighted d-CNF satisfiability problem with d > 2, we es- Definition 1. An assignment to a set of n Boolean variables tablish a non-fixed parameter tractable lower bound on the n search tree size of the parameterized version of the ordered is a vector in {0, 1} . The weight of an assignment is the DPLL algorithm. number of the variables that are set to 1 (true) by the as- We observe that there are certain relations between find- signment, i.e., the number of 1’s in the assignment. ing a solution to an instance of the parameterized problem A representative W [1]-complete problem is the following under consideration the tasks of backdoor detection with re- weighted d-CNF satisfiability problem (weighted d-SAT) spect to an (extremely) naive sub-solver that simply checks (Downey and Fellows 1999): whether the all-zero assignment is satisfying, and hope that further investigation along this line may help shed further Problem 1. Weighted d-SAT light on the (typical-case) hardness of the backdoor detec- Input: A CNF formula consisting of d-clauses, and a tion problem and the empirical observation on the effective- positive integer k. ness of practical satisfiability solvers in exploiting small- sized backdoors. Question: Is there a satisfying assignment of weight k? The results reported in the current paper are all of theo- retical nature and the objective is to lay the necessary foun- Unlike the traditional satisfiability problem, weighted dation for a study of the phase transitions and typical-case 2-SAT is already W[1]-complete. The anti-monotone hardness of fixed-parameter intractable problems. We be- weighted d-SAT problem (the problem where each clause lieve that in the future study, empirical studies are indispens- contains negative literatures only) is also W[1]-complete. able, especially for situations where theoretical analyses are The weighted satisfiability problem (weighted SAT) is hard to come by. See, for example, the work in (Gent and similar to weighted d-SAT except that there is no restriction Walsh 1998; Gomes et al. 2005; Gao and Culberson 2007) on the length of a clause in the formula. Weighted SAT is a and the references therein for the many empirical studies on representative W [2]-complete problem. the phase transitions of NP-hard problems. Parameterized Proof Systems and Parameterized 2. Preliminaries DPLL Algorithms Parameterized complexity A formal definition of a parameterized tree-like resolution parameterized decision problem An instance of a is a pair proof system for the weighted d-SAT problem is given in (I, k) where I is a problem instance and k is the problem pa- (Dantchev, Martin, and Szeider 2007). Basically, a param- rameter. Usually, the parameter k either specifies the “size” eterized resolution can be regarded as a classical resolution of the solution or is related to some structural property of that have access (for free) to all clauses with more than k the underlying problem, such as the treewidth of a graph. negated variables, where k is the parameter of the weighted For example, in an instance (I, k) of the parameterized ver- SAT problem. tex cover (VC) problem, I is a graph and k is the size of the vertex cover. The question is to decide whether I has a Accordingly, one can consider the parameterized version vertex cover of size k. of the DPLL algorithm for weighted SAT. It proceeds in the A parameterized problem is fixed-parameter tractable same way as the standard DPLL algorithm with the excep- (FPT) if any instance (I, k) of the problem can be solved tion that a node in the search tree fails if in f(k)|I|O(1) time, where f(k) is a computable function of 1. either a clause has been falsified by the partial assignment, k only. It is known that the parameterized VC problem can be solved in O(1.28k + kn) time where n is the number of 2. or the number of variables assigned to true in the partial vertices of the input graph. assignment has exceeded k. Parameterized problems are inter-related by parameter- In addition to the parameterized version of the general ized reductions, resulting in a classification of parameterized DPLL algorithm, we may also consider the parameterized problems into a hierarchy of complexity classes version of ordered-DPLL and the oblivious-DPLL algo- FPT ⊆ W [1] ⊆ W [2] · · · ⊆ XP. rithms studied in (Beame et al. 2002). In the ordered DPLL, an ordering of the variables is fixed and the next variable se- At the lowest level is the class of FPT problems. The top lected to branch on is always the first one that has not been level XP contains all the problems that can be solved in assigned a value. In the oblivious-DPLL algorithm, the vari- time f(k)ng(k). It is widely believed that the inclusions are able orderings for different branches of the DPLL search tree strict and the notion of completeness can be naturally de- may be different, but these orderings are pre-specified before fined via parameterized reductions. the algorithm starts and are independent of the formula.

266 n,p 3. Main Results Note that the average number of clauses in Fk,d is n
We study the behavior of random instances of weighted d- d p(n). In the literature, there are several slightly different SAT and its variants. In the study of the phase transitions random models for CNF formulas, including the one that of traditional satisfiability problem, a widely-used instance picks a collection of clauses uniformly at random with (or distribution is the random d-CNF formula that consists of without) replacements. For most of the properties studied a set of m clauses selected uniformly at random with (or previously, these models can be regarded as equivalent up to dn
without) replacement from all the 2 d possible d-clauses. a “small” number of clauses so that one is free to choose a However, it is problematic to use the above instance distri- model that is easy to analyze. bution for parameterized problems such as weighted d-SAT. For fixed-parameter problems, the equivalence of these For example, under this distribution, random instances of the models are not obvious, especially for the weighted 2-SAT. anti-monotone weighted 2-SAT (a W[1]-complete problem), This is the chief reason we use the above definition. are always satisfiable and a solution can be found by the We establish the exact threshold of the phase transition of n,p simple algorithm that greedily assigns true to variables. For Fk,d for any d ≥ 2: the case of weighted 2-SAT, random instances can be solved c log n Theorem 1. Let p = nd−1 with c > 0 being a constant and by simply counting the number of “independent” positive ∗ d clauses. (For weighted d-SAT with d > 2, the situation let c = (2 − 1)(d − 1)!. We have  ∗ doesn’t become any better.) h n,p i 0, if c > c ; lim F is satisfiable = ∗ In search for a more interesting model, we notice that n P k,d 1, if c < c . the all-zero assignment {0, ··· , 0} always satisfies an anti- For the case of d = 2 and d = 3, the thresholds are respec- monotone CNF formula (in the traditional sense). Moti- ∗ ∗ vated by this observation and the fact that the weighted tively c = 3 and c = 14. anti-monotone 2-SAT is already fixed-parameter intractable Proof. See Section 4.  (W[1]-complete), we consider a random model of weighted For weighted 2-SAT which is already W[1]-complete, we d-SAT in which clauses that contain positive literals only provide another proof by exploring an interesting connec- n,p are forbidden. Random instances from such a model are al- tion between the unsatisfiability of Fk,2 and the existence ways satisfied by the all-zero assignment in the traditional of a Hamiltonian cycle in a properly-defined directed graph. sense, and thus bear a (superficial) similarity to the model The proof actually shows that with high probability, random n,p of standard satisfiability with hidden solutions studied in the instances of Fk,2 in the unsatisfiable regime have a para- literature. See, for example, (Jia, Moore, and Strain 2007). metric resolution refutation of size O(k)nO(1) which can The difference is that in our case, there is no point to really also be constructed in time O(k)nO(1). In other words, typ- “hide” the planted assignment — what we are looking for ical unsatisfiable instances from F n,p are fixed-parameter are assignments of weight k, i.e., assignments whose Ham- k,2 tractable. ming distance to the hidden solution is k. The problem is n,p fixed-parameter intractable regardless of whether or not one Theorem 2. Consider a random instance from Fk,2 where knows the hidden solution. c(log n+c1 log log n) k is a fixed constant. Let p = n . We have We note that forbidding clauses that contain positive liter- for any c ≥ 3 and c1 > k − 1, with high probability a als only does not make the weighted d-SAT problem trivial parametric resolution proof of size O(k)nO(1) can be con- in the parameterized sense. In fact, based on Marx’s result structed in O(k)nO(1) time (Marx 2005) on a parameterized analog of Schaefer’s Di- Proof. See Section 5. chotomy theorem for parameterized Boolean constraint sat-  The above result still holds even if the parameter k is isfaction problems, we have in Θ(log n). By way of contrast, a lower bound nΩ(k) on Lemma 1. Weighted d-SAT with a hidden solution is W[1]- the the size of a DPLL resolution refutation for random complete for any d ≥ 2 even if clauses containing positive n,p instances of Fk,d with d ≥ 3 can be easily established, literals only are forbidden. as shown in the following theorem 3. The difference be- m,n A formal definition of the random model for weighted d- tween the resolution complexity of weighted 2-SAT Fk,2 m,n SAT is in the following and weighted 3-SAT Fk,3 is interesting. Note that unlike Definition 2. A random instance of the weighted d-SAT with the situation in the standard satisfiability problem, weighted n,p 2-SAT and weighted 3-SAT are both W [1]-complete. a hidden solution is denoted by Fk,d where k is the param- log n eter, n is the number of variables, and p = p(n) defines the Theorem 3. For p > c n2 with c > 14 and any fixed probability that a clause will be included in the CNF for- parameter k > 0, the size of the search tree of an oblivious mula. DPLL algorithm for an instance of the random instances of n,p n,p Ω(k) The clauses of Fk,d are selected in the following way: Fk,3 is n with high probability. each subset {xi1 , ··· , xid } of d variables contributes one Proof. See Section 6. n,p d  clause to Fk,d with probability p(n). There are 2 − 1 pos- Generalizations sible clauses on {xi1 , ··· , xid } (excluding the clause that contains positive literals only), and we pick one of them uni- As a generalization to weighted SAT, we can consider the formly at random. following weighted binary constraint satisfaction problem

267 (weighted-CSP) with domain size d > 2. Let D be the do- Relations to Backdoor Detections main and d0 ∈ D be a fixed domain value. An instance of There are certain relations between the weighted satisfiabil- weighted CSP is a pair (C, k) where k > 0 is an integer ity problems discussed in this paper and the backdoor de- and C is a CSP instance such that for any pair of variables tection problem for the traditional satisfiability of the corre- (xi, xj), the tuple (d0, d0) is always compatible. Note that sponding CNF formula. Consider the naive subsolver A that {d0, ··· , d0} is always a satisfying assignment. The ques- simply checks if the all-zero assignment is satisfying. tion is to decide whether C has a satisfying solution that For the CNF formula defined in Definition 3, it is clear differs from the solution (d0, ··· , d0) by k variables. We that a weight-k satisfying assignment for the parameterized are not aware of any study on this type of CSPs. But in problem specifies a weak-backdoor with respect to A for the terms of the modelling aspect of CSPs, this may be useful in standard satisfiability of the CNF formula. modelling tasks where one has a configuration with a set of For the weighted d-SAT instance where clauses contain- default settings and wants to see if k of the default settings ing only positive literals are forbidden, the relation is subtle can be changed without violating any constraint. — if an instance has no weight-k satisfying assignment but Lemma 2. Weighted CSP is W[1]-complete if the domain the formula is satisfiable in the traditional sense, then the size is a fixed constant. formula has no size-k strong backdoor with respect to A. n,p While more arguments are needed for these observations A random model similar to Fk,d for weighted d-SAT can be considered: with probability p = p(n), there is a con- to materialize, we hope that studies along this line may help straint between a pair of variables. For each constraint, t shed further light on the nature of the complexity of back- pairs of domain values are selected as the no-goods from door detection problems. 2 D \{(d0, d0)}, where t > 0 is the constraint tightness. The expected number of satisfying solutions to the ran- 4. Proof of Theorem 1 dom weighted binary CSP is Proof. Let T be the collection of subsets of d variables and n t let s be an assignment to the variables. We say that a subset (d − 1)k(1 − p(n) )k(n−k) of variables T = {x , ··· , x } ∈ T is s-good if either k d2 − 1 1 d 1. T doesn’t contribute a clause to F n,p, or which tends to 0 if p(n) = c log n such that d,k n n,p t 2. the d-clause in Fd,k contributed by T is satisfied by the c > , d2 − 1 assignment s. giving an immediate upper bound on the threshold of the Let S be the set of assignments of weight k. Recall that phase transition. the weight of an assignment is the number of variables that In the study of the phase transitions of random CSPs, are assigned to true by the assignment. Consider an assign-

much effort has been paid to designing random models to ment s ∈ S where the k variables {yi1 , ··· , yik } are as- n,p avoid flawed variables and constraints. See, e.g., (Gent et al. signed to true. From the definition of Fd,k , the probability 2001; Gao and Culberson 2007). We note that in the context for a subset T of d variables to be s-good is of fixed-parameter complexity, the flawed variable is not a  1, if T ∩ {y , ··· , y } = φ; big issue for the weighted CSP problem we have defined [T ] = i1 ik P is s-good 1 − p(n) 1 , . (1) in the above. This is because the solutions we are looking 2d−1 otherwise for are assignments that differ from the “hidden” solution Let Ts ⊂ T be the collection of subsets of d variables that (d0, ··· , d0) by k variables, while in the tradition CSPs a have a nonempty intersection with {y , ··· , y }. We have single flawed variable will result in the unsatisfiability. i1 ik d Another way to generalize the weighted SAT problem is X n − kk to consider the morphing of instances from weighted 2-SAT |T | = . s d − j j and the general weighted SAT. The former is W[1]-complete j=1 while the latter is W[2]-complete. The model is interesting X S since the structure of its instances bears many similarities to Let be the number of the assignments in that satisfy F n,p. Then, we have the structure of the CNF formulas that encode practical prob- d,k lems such as planning and model-checking. Initial results on X Y the phase transitions and complexity of this model turn out E [X] = P [T is s-good] to be interesting. Details can be found in (Gao 2008). s∈S T ∈T p1,p2,m X Y Definition 3. An instances of Mn,k consists of = P [T is s-good] 1. a collection of anti-monotone 2-clauses. Each of the po- s∈S T ∈Ts n
k tential 2 anti-monotone clauses is included indepen- P (n−k)(k) n  1  d−j j dently with probability p1, and = 1 − p(n) j=1 , (2) 2. m monotone clauses obtained independently in the fol- k 2d − 1 lowing way: for each clause, each of the n variables ap- which is asymptotically equivalent to pear with probability p2. − kc log n The problem parameter of the parameterized problem is k. nke (2d−1)(d−1)!

268 log n (Recall that p(n) = c nd−1 ). For the case of i = 0, it can be shown that lim D0 = The upper bound on the threshold c∗ follows from n ( [X])2. This proves the claim. The theorem follows from Markov’s inequality. To lower bound the threshold c∗, we E equations (2), (3), and (5). use Chebyshev’s inequality   2 E X 5. Proof of Theorem 2 P [X = 0] ≤ − 1. (E [X])2 Theorem 2 is established by exploiting an interesting con- Let Di be the number of pairs of satisfying assignments nection between the parameterized resolution complexity of of weight k that have i overlaps. We have weighted 2-SAT and the existence of a Hamiltonian cycle X Y in a properly defined directed graph. The following lemma [D ] = [T s s ] , E i P is both 1-good and 2-good establishes the connection. s1,s2 T :T ∩Y 6=φ Lemma 3. Consider an instance F of the weighted 2-SAT where the sum is over all (ordered) pairs of weight-k assign-  2 problem with parameter k and consider an arbitrary order- ments with i-overlaps. Write E X as ing {x1, ··· , xn} of the variables. If F contains the follow- k ing cycle of “forcing” clauses,  2 X E X = E [Di] . (3) x ∨ x , x ∨ x , ··· , x ∨ x , x ∨ x , i=1 1 2 2 3 n−1 n n 1 Let  > 0 be any number. We claim that for and c = 1 −  < then there is a parameterized resolution proof of size O(kn) 1 2d−1 d − 1!, We claim that for F. Furthermore, the parameterized version of the DPLL ( 2k−i algorithm constructs such a resolution proof. E [Di] ∈ o(n ) for i > 1, and 2 (4) Proof. Recall that the parameterized resolution proof lim D0 = (E [X]) n system has access to all the clauses that contain more than k Let s1 and s2 be two assignments with i over- negated variables or more than n − k positive variables. For laps. Without loss of generality, assume that each i ≥ 1, resolving the set of variables assigned to true by s1 is xi ∨ xi+1, ··· , xi+k ∨ xi+k+1 {y1, ··· , yk−i, yk−i+1 ··· , yk} and the set of variables as- signed to true by s2 is {yk−i+1, ··· , yk, yk+1, ··· , y2k−i}. and xi+1 ∨ · · · ∨ xi+k+1 results in xi. These xi’s together Let Y = {y1, ··· , yk, ··· , y2k−i}. Consider a subset T with x1 ∨ · · · ∨ xn−k+1 result in a contradiction.  of d variables. The probability that T is both s1-good and We emphasize that a cycle of forcing clauses, if exist, can s2-good can be estimated as follows: be automatically exploited by DPLL-style algorithms. For n,p 1. If T ∩ Y = φ, then P [T is good for s1 ands2] = 1. the random model F2,k of weighted 2-SAT, the existence of a cycle of “forcing” clauses can be shown due to a re- 2. If T ∩ Y 6= φ, then P [T is good for s1 and s2] is at most sult of McDiarmid on the relation between the existence of  1  1 − p(n) . a Hamiltonian cycle (or a long path) in a random graph and 2d − 1 the existence of a directed Hamiltonian cycle (or a directed To see this, note that in this case T has a non- long path) in a “directed random graph”. empty intersection with either {y1, ··· , yk}, or Lemma 4. (McDiarmid 1981) Let 0 < p < 1/2 and {yk−i+1, ··· , y2k−i}, or both. Therefore, either D(n, p) be a random directed graph obtained from the undi-  1  rected graph G(n, 2p) by randomly directing the edges. [T is s -good] = 1 − p(n) , or P 1 2d − 1 Then,   1 P [D(n, p) has a directed Hamiltonian cycle] P [T is s2-good] = 1 − p(n) . 2d − 1 ≥ P [G(n, p) has a Hamiltonian cycle] . Note that the total number of T ∈ T such that T ∩ Y 6= φ is We remark that the above inequality also holds for the d X n − (2k − i)2k − i existence of simple paths of certain length, while in (McDi- . armid 1981) it is shown that for simple paths without length d − j j j=1 constraints, the above inequality becomes equality. It follows that for i > 1, From Lemma 3, it is sufficient to show that there is a      directed Hamiltonian cycle in the following directed graph n n − k Y 1 0 [D ] = 1 − p(n) DF (n, p ): Each vertex corresponds to a variable. There E i k k − i 2d − 1 T :T ∩Y 6=φ is an directed edge from xi to xj if and only if the 2-clause n,p d xi∨xj is in the random formula F . From the definition of P n−(2k−i) 2k−i d,k   ( d−j )( j ) n,p 0 0 1 j=1 F , D (n, p ) is the random directed graph D(n, p ) dis- ∼ n2k−i 1 − p(n) d,k F 2d − 1 0 1 c log n cussed in Lemma 4 with the edge probability p = 3 n . (2k−i)c log n Since c > 3, the result follows from Lemma 4 and the 2k−i − d ∼ n e (2 −1)(d−1)! threshold of the existence of a Hamiltonian cycle in the ran- ∼ n2k−i. (5) dom graph G(n, p) (see, for example, (Bollobas 2001)).

269 6. Proof of Theorem 3 Beame, P.; Karp, R.; Pitassi, T.; and Saks, M. 2002. The ef- Proof. We lower bound the number of nodes that an obliv- ficiency of resolution and Davis-Putnam procedures. SIAM 1 − Journal on Computing 31(4):1048–1075. ious DPLL algorithm has to explore at depth n 2 where  > 0 is small constant. Associated with each node is a par- Bollobas, B. 2001. Random Graphs. Cambridge University tial assignment to the variables on the path from the root to Press. the node. Let v be a node and let x1, ··· , xl be the variables Cheeseman, P.; Kanefsky, B.; and Taylor, W. 1991. Where on the path to v in the search tree. We say v is promising if the really hard problems are. In Proceedings of the 12th 1. exactly k variables in {x , ··· , x } have been set to true; International Joint Conference on Artificial Intelligence, 2 1 l 331–337. Morgan Kaufmann. 2. no clause that contains at least two literals from n,p Dantchev, S.; Martin, B.; and Szeider, S. 2007. Param- {x , ··· , x } is in F . 1 l k,3 eterized proof complexity. In Proceedings of The 48th A promising node will be explored by an oblivious DPLL Annual Symposium on Foundations of Computer Science algorithm since the partial assignment to the variables on (FOCS’07), 150–160. IEEE Press. the path to the promising node neither falsify any clause nor Dilkina, B.; Gomes, C.; and Sabharwal, A. 2007. Tradeoffs produce a unit clause. in the complexity of backdoor detection. In Proceedings of 1 − Let V be the set of nodes at depth n 2 such that for each the 13th International Conference on Principles and Prac- k v ∈ V , 2 variables on the path from the root to v have been tice of Constraint Programming (CP’07), 256–270. set to true. The total number of such nodes is Downey, R., and Fellows, M. 1999. Parameterized Com- 1 −−1 1 −−1 1 − n 2  n 2  n 2  plexity. Springer. |V | = + = k − 1 k k Gao, Y., and Culberson, J. 2007. Consistency and random 2 2 2 constraint satisfaction models. 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