# Generating models of a matched formula with a polynomial delay

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Generating models of a matched formula with a polynomial delay Petr Savicky Institute of Computer Science, Academy of Sciences of Czech Republic, Pod Vodárenskou Věží 2, Praha 8, Czech Republic Petr Kučera Department of Theoretical Computer Science and Mathematical Logic, Faculty of Mathematics and Physics, Charles University, Malostranské nám. 25, Praha 1, Czech Republic Abstract A matched formula is a CNF formula, such that the system of the sets of the variables, which appear in individual clauses, has a system of distinct representatives. Such a formula is always satisfiable. Matched formulas are used, for example, in the area of parametrized complexity. We prove that the problem of counting the number of the models (satisfying assignments) of a matched formula is #P-complete. On the other hand, we define a class of formulas generalizing the matched formulas and prove that for a formula in this class, one can choose in polynomial time a variable suitable for splitting the tree for the search of the models of the formula. As a consequence, the models of a formula from this class, in particular of any matched formula, can be generated sequentially with a delay polynomial in the size of the input. On the other hand, we prove that this task cannot be performed efficiently for the linearly satisfiable formulas, which is a generalization of matched formulas containing the class considered above. Introduction In this paper we consider the problem of counting the models (satisfying assignments) and generating subsets of the models of a given formula in conjunctive normal form (CNF). It is well known that a problem of counting the models of a general CNF is #P-complete (Sipser 2006). The problem of generating the models of a general CNF formula is clearly also hard, because checking whether there is at least one satisfying assignment of the formula, the SAT problem, is NP-complete (Garey and Johnson 1979). That is why we consider subclasses of formulae for which the satisfiability problem can be solved in polynomial time. In particular we consider the class of matched formulas introduced in (Franco and Van Gelder 2003). Given a CNF formula ϕ we consider its incidence graph I(ϕ) defined as follows. I(ϕ) is a bipartite graph with one partity consisting of clauses of ϕ and the other partity containing the variables of ϕ. An edge {x, C} for a variable x and a clause C is in I(ϕ) if x or x appears in C. It was observed in (Aharoni and Linial 1986; Tovey 1984) that if I(ϕ) admits a matching (i.e. a set of pairwise disjoint edges) of size m (where m is the number of clauses in ϕ), then ϕ is satisfiable. Later in (Franco and Van Gelder 2003) the formulas satisfying this condition were called matched formulas. Since a matching of maximum size in a bipartite graph can be found in polynomial time (see e.g. (Lovász and Plummer 1986)), one can check efficiently whether a given formula is matched. Given a general formula ϕ we can measure how far it is from being matched by considering its maximum deficiency δ (ϕ), the number of clauses which remain unmatched in a maximum matching of I(ϕ). A formula ϕ is thus matched iff δ (ϕ) = 0. A weaker notion of deficiency δ(ϕ) = m n, where m is the number of clauses and n the number of the variables in ϕ, is also often being considered. Matched formulas play a significant role in the theory of satisfiability solving. Since their introduction, matched formulas were considered as a base class in parameterized algorithms for satisfiability, see e.g. (Flum and Grohe 2006) for an overview of parameterized algorithms theory. In particular, the authors of (Fleischner, Kullmann, and Szeider 2002) show that satisfiability of the formulas whose maximum deficiency is bounded by a constant can be decided in polynomial time. This result was later improved in (Szeider 2003) to an algorithm for satisfiability parameterized with maximum deficiency of a formula. Parameterization based on backdoor sets with respect to matched formulas were considered in (Szeider 2007). Several generalizations of matched formulas were considered in the literature, too. In (Kullmann 2000), matched formulas were generalized into the class of linearly satisfiable formulas. Autarkies based on matchings were studied in (Kullmann 2003). Another generalization was considered in (Szeider 2005) as classes of bi-clique satisfiable and varsatisfiable formulas. Unfortunately, for both bi-clique and var-satisfiable formulas it is hard to check if a formula falls into one of these classes (Szeider 2005). Since all matched formulas are trivially satisfiable, we ask a stronger question: How hard is it to count or enumerate the models of a matched formula? We prove that counting the models of a matched formula is a #P-complete problem, and turn our attention to generating models of a matched formula. Since the number of the models can be large, we consider the efficiency with respect to the input and output size. More specifically, one can consider two variants of the algorithm. In one of them, the input of the algorithm is a formula and a natural number k. The output is a set of k different models of the formula, if such models exist, or all

2 models of the formula, if their number is less than k. Such an algorithm is considered polynomially output-efficient, if its running time is at most k ϕ O(1), where ϕ is the length of the input formula ϕ. For simplicity, we present our algorithms in another setting, when the algorithm receives as input a formula and generates a sequence of all its models. The time needed for generating the first model and the time between generating any two consecutive models in the sequence is polynomial in the length of the formula. This type of complexity bound will be called a polynomial delay in this paper, although the notion of a polynomial delay is sometimes understood in a weaker sense. The main result of the paper is an algorithm, which generates models of a matched formula with a polynomial delay. The algorithm constructs a splitting tree, whose nodes correspond to either a matched or an unsatisfiable formula. However, in some cases, this strategy is not sufficient, since some nodes of the tree cannot be split in this way. We prove that such a node corresponds to a formula, which can be satisfied by iterated elimination of pure literals. Formulas with this property will be called pure literal satisfiable. These formulas were studied in (Kullmann 2000) as a subclass of linearly satisfiable formulas. If a node with a pure literal satisfiable formula is reached, the algorithm switches to a simpler strategy. We prove that the models of a pure literal satisfiable formula can be generated with a delay linear in the length of the formula. On the other hand, the #SAT problem for pure literal satisfiable formulas is #P-complete, because this problem is #P-complete for monotone 2CNFs (Valiant 1979a; 1979b), which are pure literal satisfiable. Several classes generalizing matched formulas were considered in the literature such as biclique satisfiable and varsatisfiable formulas (Szeider 2005) and linearly satisfiable formulas (Kullmann 2000). We show in this paper that our result does not transfer to the class of linearly satisfiable formulas by demonstrating that it is not possible to generate models of a linearly satisfiable formula with a polynomial delay unless P=NP. Note that this is an extended abstract and most of the proofs are therefore omitted. Definitions In this section we give the necessary definitions and recall the results we use in this paper. Boolean functions A Boolean function of n variables is a mapping f : {0, 1} n {0, 1}. A literal is either a variable, called positive literal, or its negation, called negative literal. The negation of the variable x will be denoted x or x. A clause is a disjunction of a set of literals, which contains at most one literal for each variable. Formula ϕ is in conjunctive normal form (CNF) or, equivalently, ϕ is a CNF formula, if it is a conjuction of clauses. We often treat a clause as a set of its literals and a CNF formula as a set of its clauses. It is a well known fact that every Boolean function can be represented by a CNF formula (see e.g. (Genesereth and Nilsson 1987)). The size of a formula ϕ is the number of the clauses in ϕ and will be denoted as ϕ. The length of a formula ϕ is the total number of occurrences of literals in ϕ, i.e. the sum of the sizes of the clauses in ϕ, and will be denoted as ϕ. Given a variable x and a value a {0, 1}, ϕ[x = a] denotes a formula originating from ϕ by substituting x with value a and the obvious simplifications consisting in removing falsified literals and satisfied clauses. We extend this notation to negative literals as well by setting ϕ[x = a] = ϕ[x = a]. The formula obtained from ϕ by assigning the values a 1,..., a k {0, 1} to the variables x 1,..., x k is denoted as ϕ[x 1 = a 1, x 2 = a 2,..., x k = a k ]. We say that a literal l is pure in a CNF formula, if it occurs in the formula and the negated literal l does not. A literal is irrelevant in a formula, if neither the literal nor its negation occurs in the formula. A variable is pure, if it appears only positively, or only negatively in ϕ, i.e. it appears in a literal, which is pure in ϕ. Let ϕ be a formula defining a Boolean function f on n variables. An assignment of values v {0, 1} n is a model of ϕ (also a satisfying assignment, or a true point of ϕ), if it satisfies f, i.e. if f(v) = 1. The set of models of ϕ is denoted as T (ϕ). The models in T (ϕ) are defined on the variables, which have an occurrence in ϕ. The set of the variables of the function defined by a formula can be larger, however, we do not introduce a special notation for this more general case. For algorithmic purposes, this is also not necessary, since adding an irrelevant variable to a formula changes the set of the models by adding this variable with both possible values to each element of the original set of models. A partial assignment assigns values only to a subset of the variables. For a formula of the variables x 1,..., x n, it can be represented as a ternary vector v {0, 1, } n, where v i = denotes the fact that x i is not assigned a value by v. Note that an empty clause does not admit a satisfying assignment and an empty CNF is satisfied by any assignment. Matched formulas In this paper we use standard graph terminology, see e.g. (Bollobás 1998). Given an undirected graph G = (V, E), a subset of edges M E is a matching in G if the edges in M are pairwise disjoint. A bipartite graph G = (A, B, E) is an undirected graph with disjoint sets of vertices A and B, and the set of edges E satisfying E A B. For a set W of vertices of G, let Γ(W ) denote the neighbourhood of W in G, i.e. the set of all vertices adjacent to some element of W. We shall use the following well-known result on matchings in bipartite graphs: Theorem 1 (Hall s Theorem). Let G = (A, B, E) be a bipartite graph. A matching M of size M = A exists if and only if for every subset S of A we have that S Γ(S). Let ϕ = C 1... C m be a CNF formula on n variables X = {x 1,..., x n }. We associate a bipartite graph I(ϕ) = (ϕ, X, E) with ϕ (also called the incidence graph of ϕ), where the vertices correspond to clauses in ϕ and the variables X. A clause C i is connected to a variable x j (i.e. {C i, x j } E) if C i contains x j or x j. A CNF formula ϕ is matched if I(ϕ) has a matching of size m, i.e. if there is a matching which pairs each clause with a unique 2

3 variable, this is called perfect matching of I(ϕ). Note, that a matched CNF is trivially satisfiable, since each clause can be satisfied by the literal containing the variable matched to the given clause. A variable which is matched to some clause in a given matching M is called matched in M, it is free in M otherwise. Generating models with a polynomial delay The main goal of this paper is to describe an algorithm which given a matched formula ϕ generates the set T (ϕ) of models of ϕ with a polynomial delay. Let us state more formally what we require of such an algorithm. We say that an algorithm generates the models of a Boolean formula ϕ with a polynomial delay, if there is a polynomial p, such that the algorithm given a formula ϕ as an input satisfies the following properties. 1. It works in steps, each of which takes time O(p( ϕ )). 2. In each step, it either finds a model of ϕ different from the models obtained in the previous steps (in particular, any model in the first step) or determines that that there is no such model, so the previous steps already found all the models of ϕ. If an algorithm with the properties above exists, it follows that we can construct the set T (ϕ) of all models in time O(( T (ϕ) + 1) p( ϕ )), which means that the algorithm is polynomial with respect to the size of the input and output. Note that since T (ϕ) may be of exponential size with respect to ϕ, efficiency with respect to the size of the input and output is the best we can hope for when constructing T (ϕ). Note also that often in the literature algorithms working with polynomial delay are allowed to spend time O(p( T, ϕ )) to generate a new model, where T is the set of models found in the previous steps. Our definition is more restrictive than this. Efficient splitting tree algorithm The idea of the algorithm is to construct a decision tree for the function represented by a given satisfiable CNF, such that every subtree larger than a single leaf contains a 1- leaf. The depth of the tree is at most the number of the variables. If this tree is searched in a DFS order, then the time needed in an arbitrary moment to reach a 1-leaf is at most n times the time needed to split a node. In the following, we show that for some classes of formulas including the matched formulas, it is possible to find a splitting procedure, which yields a tree as described above. A decision tree for a Boolean function f is a labeled binary tree, where each inner node is labeled with a variable, while leaves and edges have labels 0 or 1. A decision tree computes f(x) for a given assignment x by a process, which starts at the root and in each visited node follows the edge labeled by the value of the variable, which is the label of the node. The output is the label of the leaf reached by this process. If a computation path tests a variable, which was tested in the previous part of the path, then this test is redundant. We consider only trees without such redundant tests. A decision tree representing the same function as a given CNF formula ϕ can be constructed top down as follows. The root of the tree is assigned to ϕ. For each non-leaf node of the tree assigned to a formula ψ, we choose an arbitrary split variable x, which has an occurrence in ψ, and assign the restricted formulas ψ[x = 0] and ψ[x = 1] to the successors. A node assigned to an empty formula becomes a 1-leaf and a node assigned to a formula, which contains an empty clause, becomes a 0-leaf. The resulting decision tree represents the function given by ϕ, although it can be too large for practical purposes. Each path from the root to an inner node u of the tree corresponds to a partial assignment which changes ϕ to a formula representing the function computed by the subtree, whose root is u. The depth of a tree for a function of n variables is at most n. Each leaf node labeled with 1 represents a set of models of ϕ, more precisely, a leaf in depth d represents 2 n d models of ϕ. Moreover, different leaves of the tree represent disjoint sets of models. Given a decision tree for the function represented by ϕ we can, by traversing it, generate all models of ϕ in time proportional to its size. This process leads to a large delay between generating successive models, if the tree contains large subtrees with only 0-leaves. The following condition on a class of formulas describes a situation, when this can be avoided. Definition 2. Let U be a class of formulas, let ϕ U and let x be a variable with an occurrence in ϕ. We say that x is a splitting variable for ϕ relative to U, if for every a {0, 1}, such that ϕ[x = a] is satisfiable, we have ϕ[x = a] U. A class of formulas U has the splitting property, if every formula in U containing a variable contains a splitting variable relative to U. Definition 3. Let U be a class of formulas having the splitting property. The splitting complexity of a formula ϕ U, which contains a variable, is the time needed to find some of the splitting variables x for ϕ relative to U and the results of the satisfiability tests for the formulas ϕ[x = 0] and ϕ[x = 1]. We say that the class U has a splitting complexity c(ϕ) if every nontrivial formula ϕ U and every nontrivial formula in U obtained from ϕ by a partial assignment has splitting complexity at most c(ϕ). If moreover c(ϕ) p( ϕ ) where p is a polynomial, then we say that U has a polynomial splitting complexity. If ϕ U, then its splitting complexity is also an upper bound on the time of a satisfiability test for ϕ itself, since ϕ is satisfiable, if and only if at least one of the formulas ϕ[x = 0] and ϕ[x = 1], where x is a splitting variable, is satisfiable. Theorem 4. If a class of formulas U has the splitting property with the splitting complexity c(ϕ), where c(ϕ) ϕ for each formula ϕ U, then the models of a formula ϕ U with n variables can be generated with a delay O(n c(ϕ)). Remark 5. If ϕ contains a unit clause and U is closed under the unit propagation, then a variable x contained in a unit clause is a splitting variable, which can be identified efficiently. The reason is that if ϕ is known to be satisfiable, then one of the formulas ϕ[x = a] contains an empty clause and, hence, the other is satisfiable. Remark 6. Any class U satisfying that 3

4 1. the satisfiability of formulas in U can be tested in polynomial time, and 2. U is closed under partial assignments has a polynomial splitting complexity. Indeed, in this case any variable in a formula ϕ from U is a splitting variable and the satisfiability tests for the corresponding restrictions can be obtained in polynomial time. Examples of such classes are Horn formulas, SLUR formulas, 2CNFs, q-horn formulas etc. Hence, as an immediate corollary of Theorem 4, it is possible to generate the models of formulas in these classes with a polynomial delay. The main result of this paper is that a slight generalization of matched formulas also has polynomial splitting complexity although the class of matched formulas is not closed under partial assignments. Pure literal satisfiable formulas Before considering matched formulas, let us make a small detour to the class of formulas which are satisfiable by iterated elimination of pure literals, which we call pure literal satisfiable. These formulas have already been considered in (Kullmann 2000) as a special case of linearly satisfiable formulas. A set of literals is called consistent, if it does not contain contradictory literals. If l is a literal, let assign(l) be the assignment to the variable contained in l, which satisfies l. For a consistent set or sequence of literals L, let assign(l) be the partial assignment of the variables satisfying the literals in L. For a formula ϕ, ϕ[l] is an abbreviation for ϕ[assign(l)]. Definition 7. A pure literal sequence for a formula ϕ is a consistent sequence of literals (l 1,..., l k ), such that for every i = 1,..., k, the literal l i is either pure or irrelevant in the formula ϕ[l 1,..., l i 1 ]. In particular, l 1 is pure or irrelevant in ϕ. A pure literal sequence is called strict, if each of the literals l i is pure in ϕ[l 1,..., l i 1 ]. If L is a pure literal sequence for ϕ, the formula ϕ[l] will be called the reduced formula corresponding to ϕ and L. If ϕ[l] does not contain a pure literal, L will be called a maximal pure literal sequence for ϕ. Definition 8. A formula ϕ is pure literal satisfiable, if there is a pure literal sequence L for ϕ, such that the reduced formula ϕ[l] is empty or, equivalently, assign(l) is a satisfying assignment of ϕ. An autarky for a formula ϕ is a partial assignment v of the variables, such that every clause is either satisfied or unchanged by v. For more on autarkies see e.g. (Kullmann 2000). Note that every initial segment of a pure literal sequence defines an assignment to the variables, which is an autarky. Moreover, one can easily verify that this property characterizes pure literal sequences. Let us note that pure literal satisfiable formulas are not closed under partial assignments. Consider a formula ψ, which is either unsatisfiable or satisfiable and does not contain a pure literal. Let ϕ be the formula obtained from ψ by adding a new variable x as a positive literal to every clause. Formula ϕ is pure literal satisfiable, but ϕ[x = 0] = ψ is not pure literal satisfiable. It follows that pure literal satisfiable formulas do not satisfy the second property required in Remark 6 and we have to put more effort into showing that pure literal satisfiable formulas have the splitting property and a polynomial splitting complexity. For every CNF formula, it may be tested in polynomial time, whether it is pure literal satisfiable. In order to find a pure literal sequence witnessing this fact, the procedure FindPLS in Figure 1 uses a greedy approach, which at each step chooses and satisfies any pure literal in the current formula. This approach is meaningful, since if a literal is pure at some stage of the procedure, it either remains pure or becomes irrelevant in the following stages. The pure literal sequence obtained by the procedure depends on the nondeterministic choices made by the procedure, however, by Lemma 9, the resulting reduced formula is uniquely determined by the input. Procedure FindPLS Input: A CNF formula ϕ. Output: A maximal strict pure literal sequence L for ϕ and the corresponding reduced formula. Definition: For a formula ϕ, let Pure(ϕ) denote the set of the literals, which are pure in it. initialize ψ with the input formula ϕ initialize L with an empty list while Pure(ψ) is not empty choose any literal l from Pure(ψ) add l to L and apply assign(l) to ψ Output the list L and the formula ψ Figure 1: Constructing pure literal sequence Lemma 9. Let ϕ be a CNF formula and let L be a pure literal sequence obtained by FindPLS for ϕ. 1. The formula ϕ[l] is uniquely determined by ϕ. 2. The formula ϕ is pure literal satisfiable, if and only if ϕ[l] is empty. Since the running time of FindPLS procedure is polynomial in the length of the input formula, a maximal pure literal sequence for a formula can be constructed in polynomial time. The complexity of constructing a maximal pure literal sequence for a formula ϕ is, in fact, O( ϕ ) by Lemma 23. Lemma 10. Let L = (l 1,..., l n ) be a pure literal sequence for a formula ϕ, which contains a literal for each variable of ϕ. For i = 1,..., n, denote by x i the variable contained in l i. If x i is the variable with the largest index i among the variables, which have an occurence in ϕ, then x i is a splitting variable for ϕ and each of the formulas ϕ[x i = 0] and ϕ[x i = 1] is satisfiable, if and only if it does not contain an empty clause. The following lemma is sufficient for this and the next section, however, the splitting complexity is, in fact, O( ϕ ) by Theorem 24. Lemma 11. The class of pure literal satisfiable formulas has a polynomial splitting complexity. 4

5 Proof. If ϕ is pure literal satisfiable, then a pure literal sequence, which satisfies it, can be obtained by FindPLS in polynomial time. If the sequence does not contain literals for all variables, it is extended in polynomial time by appending arbitrary literals for the missing variables to obtain a pure literal sequence satisfying the assumption of Lemma 10. Then, this lemma implies a method to select a splitting variable and obtain the results of the satisfiability test for the corresponding restrictions in polynomial time. If a pure literal sequence satisfies the assumption of Lemma 10 for a formula ϕ, then the same sequence can be used to find a splitting variable for all formulas in a splitting tree for ϕ. Using this, the models of a pure literal satisfiable formula can be generated with a delay smaller than the general bound from Theorem 4, see Corollary 24. Remark 12. The sign of a literal for a given variable, which occurs in a strict pure literal sequence, is not uniquely determined. Each of the variables y 1 and y 2 can occur both positively and negatively in a strict pure literal sequence for the formula (x 1 y 1 ) (x 2 y 1 ) (x 3 y 2 ) (x 4 y 2 ) (y 1 y 2 ). For example, (x 2, y 1, x 3, y 2 ) and (x 4, y 2, x 1, y 1 ) are strict pure literal sequences for this formula. Matched formulas In this section we concentrate on matched formulas. Let us start with showing that the problem of determining the number of models of a matched formula ϕ, i.e. the size T (ϕ), is as hard as a general #SAT problem. Theorem 13. The problem of determining T (ϕ) given a matched formula ϕ is #P-complete. Our goal is to show that we can generate the models of a matched formula with a polynomial delay. Theorem 4 cannot be used for this directly, since the class of the matched formulas does not have the splitting property as can be seen from the following example. Consider the formula (x 1 x 2 ) (x 1 x 3 ) (x 2 x 3 ). This formula is matched, but it has no splitting variable. Indeed, setting x 1 to 0 leads to a satisfiable, yet not matched formula (x 2 )(x 3 )(x 2 x 3 ) and by symmetry this is true for variables x 2 and x 3 as well. In order to achieve our objective, we have to consider a richer class of formulas. The class we consider generalizes matched and pure literal satisfiable formulas as follows. Note that an empty formula is matched, since it corresponds to an empty graph and we can formally assume that an empty graph possesses the required matching. Definition 14. A formula ϕ is called pure literal matched, if the reduced formula obtained by FindPLS procedure for ϕ is matched. Elimination of a pure literal preserves the property of being matched, since a pure literal is an autarky. Hence, a matched formula is pure literal matched. Clearly, every pure literal satisfiable formula is pure literal matched, since its reduced formula is empty and, hence, matched. The basic idea of an efficient splitting algorithm for the matched formulas is presented in the following theorem. The splitting complexity of the pure literal matched formulas can be bounded as O(n 2 ϕ ) by Theorem 26. Theorem 15. The class of pure literal matched formulas has the splitting property with a polynomial splitting complexity. In order to prove Theorem 15, we have to show several statements concernig the structure of a matched formula. If V is a set of variables, we say that a clause is restricted to V, if it contains only literals with variables from V. Definition 16. Let V be a subset of the variables of a matched formula ϕ. The set V will be called a critical block, if the number of clauses of ϕ, which are restricted to V, is equal to V. Formally, if V is empty, it is also a critical block. Note that if ϕ is a matched formula, V is a subset of its variables, and C is the set of the clauses in ϕ restricted to V, then by Hall s theorem (Theorem 1 above) we have C Γ(C) V. Critical blocks are those achieving the equality. Basic property of critical blocks with respect to the possible perfect matchings of I(ϕ) is as follows. Lemma 17. Let V be a critical block of a matched formula ϕ. Then, in every perfect matching of I(ϕ), the variables from V are matched to clauses restricted to V. Another useful property of the set of the critical blocks is as follows. Lemma 18. The set of the critical blocks of a matched formula is closed under intersection. If ϕ is a formula and x is a variable contained in at least one critical block, then Lemma 18 implies that there is a unique inclusion minimal critical block of ϕ containing x, which is equal to the intersection of all critical blocks of ϕ containing x. If a matched formula has the same number of clauses and variables, then every variable is contained in a critical block, since the set of all the variables of the formula is a critical block. Definition 19. If ϕ is a matched formula with the same number of clauses and variables and x is some of its variables, then let B x denote the inclusion minimal critical block containing x. The notation B x does not specify the formula, since it will always be clear from the context. Lemma 20. Let ϕ be a matched formula with the same number of clauses and variables. Let l be a literal containing a variable x and let us assume that the formula ϕ[ l] is not matched. Then 1. the literal l is pure or irrelevant in the clauses of ϕ restricted to B x, 2. if a clause C of ϕ contains l, then in every matching for ϕ, C is matched to a variable y, such that B x B y (where denotes strict inclusion). The structure of the critical blocks will be used to show the following proposition needed to prove Theorem 15. 5

6 Theorem 21. Let ϕ be a matched formula. If for every variable x, which has an occurence in ϕ, there is a {0, 1}, such that ϕ[x = a] is not matched, then ϕ is pure literal satisfiable. Proof of Theorem 15. Assume, ϕ is a pure literal matched formula. Let L be a pure literal sequence obtained by Find- PLS procedure for ϕ and let ψ = ϕ[l], which is, by the assumption, a matched formula. Since L is maximal, ψ does not contain a pure literal. If ψ is empty, then ϕ is itself a pure literal satisfiable formula and we can find a splitting variable for ϕ by the method from Lemma 11. If ψ is not empty, then it is matched and not pure literal satisfiable. Hence, by Theorem 21, there is a variable x of ψ, such that ψ[x = 0] and ψ[x = 1] are both matched. Since L does not contain a literal with the variable x, the application of assign(l) and x = a commute for each a {0, 1}. Hence, L is a pure literal sequence for the formula ϕ[x = a] and the application of assign(l) to ϕ[x = a] leads to ψ[x = a], which is matched. Hence, for each a {0, 1}, the formula ϕ[x = a] is pure literal matched and the variable x is a splitting variable for the formula ϕ. A time polynomial in the length of the formula is sufficient to select a splitting variable x as in the proof above. If ψ is nonempty, the satisfiability of ϕ[x = 0] and ϕ[x = 1] is guaranteed by the choice of x. If ψ is empty, ϕ is pure literal satisfiable and the method from Lemma 11 is used. Hence, a splitting variable and the results of the required satisfiability tests can be obtained in a polynomial time. Similarly as the class of matched formulas, also the class of pure literal matched formulas is closed under the unit propagation. This implies that the unit propagation can be used as part of the construction of the splitting tree by Remark 5. Proposition 22. The class of pure literal matched formulas is closed under the unit propagation. Algorithms and complexity In this section, we prove specific complexity bounds for the algorithms presented in the previous sections. The complexity bounds are derived for the RAM model with the unit cost measure and the word size O(log ϕ ), where ϕ is the input formula. Let us first concentrate on the pure literal satisfiable formulas. Lemma 23. A maximal pure literal sequence L for a CNF formula ϕ can be constructed in time O( ϕ ). Theorem 24. A pure literal satisfiable formula ϕ has splitting complexity O( ϕ ). Moreover, the set T (ϕ) of the models of ϕ can be generated with a delay O( ϕ ). Now let us concentrate on the time complexity of selecting a splitting variable of a pure literal matched formula. Lemma 25. The splitting complexity of a pure literal matched formula ϕ of n variables is O(n ϕ ). As a corollary of Lemma 25 and the general bound from Theorem 4, we get the following. Corollary 26. Models of a pure literal matched formula ϕ on n variables can be generated with a delay O(n 2 ϕ ). Linearly satisfiable formulas In this section we consider the class of linearly satisfiable formulas. By results of (Kullmann 2000), this class generalizes both the matched formulas and the pure literal satisfiable formulas and, by combining the proofs, also the class of pure literal matched formulas. In this section, we show that it is not possible to generate models of linearly satisfiable formulas with a polynomial delay unless P=NP. As a consequence, linearly satisfiable formulas do not have a polynomial splitting complexity unless P=NP. This consequence follows also unconditionally from a fact that there is a linearly satisfiable formula which does not have a splitting variable with respect to the class of linearly satisfiable formulas. We omit the details due to space limitations. Conclusion and directions for further research In this paper we have shown that it is possible to generate the models of a matched formula ϕ of n variables with delay O(n 2 ϕ ). As a byproduct we have shown that the models of a pure literal satisfiable formula ϕ (i.e. a formula satisfiable by iterated pure literal elimination) can be generated with delay O( ϕ ). We have also shown that this result cannot be generalized for the class of linearly satisfiable formulas, since it is not possible to generate models of linearly satisfiable formulas with a polynomial delay unless P=NP. Let us mention that the procedure for generating the models with a bounded delay can be extended to formulas, for which a small strong backdoor set with respect to the class of matched formulas with empty clause detection can be found. Assume, B is such a backdoor set for a formula ϕ, i.e. B is a set of variables satisfying that any partial assignment to variables in B leads to a matched formula, or to a formula containing an empty clause. Then we can generate the decision tree for ϕ (and thus generate its models) in time O(2 B ϕ + T (f) n 2 ϕ ). Unfortunately, searching for strong backdoor sets with respect to the class of matched formulas is hard (Szeider 2007). An interesting question is whether our approach could be used with the parameterized satisfiability algorithm based on maximum deficiency (see (Szeider 2003)) in order to get a parameterized algorithm for generating the models of a general formula. Acknowledgement Petr Savický was supported by CE-ITI and GAČR under the grant number GBP202/12/G061 and by the institutional research plan RVO: Petr Kučera was supported by the Czech Science Foundation (grant GA S). References Aharoni, R., and Linial, N Minimal non-twocolorable hypergraphs and minimal unsatisfiable formulas. Journal of Combinatorial Theory, Series A 43(2): Bollobás, B Modern Graph Theory, volume 184 of Graduate Texts in Mathematics. Springer. 6

7 Fleischner, H.; Kullmann, O.; and Szeider, S Polynomial-time recognition of minimal unsatisfiable formulas with fixed clause-variable difference. Theoretical Computer Science 289(1): Flum, J., and Grohe, M Parameterized complexity theory, volume 3. Springer. Franco, J., and Van Gelder, A A perspective on certain polynomial-time solvable classes of satisfiability. Discrete Appl. Math. 125(2-3): Garey, M., and Johnson, D Computers and Intractability: A Guide to the Theory of NP-Completeness. San Francisco: W.H. Freeman and Company. Genesereth, M., and Nilsson, N Logical Foundations of Artificial Intelligence. Los Altos, CA: Morgan Kaufmann. Kullmann, O Investigations on autark assignments. Discrete Applied Mathematics 107(1 3): Kullmann, O Lean clause-sets: generalizations of minimally unsatisfiable clause-sets. Discrete Applied Mathematics 130(2): The Renesse Issue on Satisfiability. Lovász, L., and Plummer, M. D Matching Theory. North-Holland. Sipser, M Introduction to the Theory of Computation, volume 2. Thomson Course Technology Boston. Szeider, S Minimal unsatisfiable formulas with bounded clause-variable difference are fixed-parameter tractable. In Warnow, T., and Zhu, B., eds., Computing and Combinatorics, volume 2697 of Lecture Notes in Computer Science. Springer Berlin Heidelberg Szeider, S Generalizations of matched CNF formulas. Annals of Mathematics and Artificial Intelligence 43(1-4): Szeider, S Matched formulas and backdoor sets. In Marques-Silva, J., and Sakallah, K., eds., Theory and Applications of Satisfiability Testing SAT 2007, volume 4501 of Lecture Notes in Computer Science. Springer Berlin Heidelberg Tovey, C. A A simplified NP-complete satisfiability problem. Discrete Applied Mathematics 8(1): Valiant, L. 1979a. The complexity of computing the permanent. Theoretical Computer Science 8(2): Valiant, L. 1979b. The complexity of enumeration and reliability problems. SIAM Journal on Computing 8(3):

### Lecture 7: NP-Complete Problems

IAS/PCMI Summer Session 2000 Clay Mathematics Undergraduate Program Basic Course on Computational Complexity Lecture 7: NP-Complete Problems David Mix Barrington and Alexis Maciel July 25, 2000 1. Circuit

### R u t c o r Research R e p o r t. Boolean functions with a simple certificate for CNF complexity. Ondřej Čepeka Petr Kučera b Petr Savický c

R u t c o r Research R e p o r t Boolean functions with a simple certificate for CNF complexity Ondřej Čepeka Petr Kučera b Petr Savický c RRR 2-2010, January 24, 2010 RUTCOR Rutgers Center for Operations

### On the Relationship between Classes P and NP

Journal of Computer Science 8 (7): 1036-1040, 2012 ISSN 1549-3636 2012 Science Publications On the Relationship between Classes P and NP Anatoly D. Plotnikov Department of Computer Systems and Networks,

### Introduction to Logic in Computer Science: Autumn 2006

Introduction to Logic in Computer Science: Autumn 2006 Ulle Endriss Institute for Logic, Language and Computation University of Amsterdam Ulle Endriss 1 Plan for Today Now that we have a basic understanding

### Exponential time algorithms for graph coloring

Exponential time algorithms for graph coloring Uriel Feige Lecture notes, March 14, 2011 1 Introduction Let [n] denote the set {1,..., k}. A k-labeling of vertices of a graph G(V, E) is a function V [k].

### 1 Definitions. Supplementary Material for: Digraphs. Concept graphs

Supplementary Material for: van Rooij, I., Evans, P., Müller, M., Gedge, J. & Wareham, T. (2008). Identifying Sources of Intractability in Cognitive Models: An Illustration using Analogical Structure Mapping.

### JUST-IN-TIME SCHEDULING WITH PERIODIC TIME SLOTS. Received December May 12, 2003; revised February 5, 2004

Scientiae Mathematicae Japonicae Online, Vol. 10, (2004), 431 437 431 JUST-IN-TIME SCHEDULING WITH PERIODIC TIME SLOTS Ondřej Čepeka and Shao Chin Sung b Received December May 12, 2003; revised February

### 1. Nondeterministically guess a solution (called a certificate) 2. Check whether the solution solves the problem (called verification)

Some N P problems Computer scientists have studied many N P problems, that is, problems that can be solved nondeterministically in polynomial time. Traditionally complexity question are studied as languages:

### Counting Falsifying Assignments of a 2-CF via Recurrence Equations

Counting Falsifying Assignments of a 2-CF via Recurrence Equations Guillermo De Ita Luna, J. Raymundo Marcial Romero, Fernando Zacarias Flores, Meliza Contreras González and Pedro Bello López Abstract

### On the k-path cover problem for cacti

On the k-path cover problem for cacti Zemin Jin and Xueliang Li Center for Combinatorics and LPMC Nankai University Tianjin 300071, P.R. China zeminjin@eyou.com, x.li@eyou.com Abstract In this paper we

### Page 1. CSCE 310J Data Structures & Algorithms. CSCE 310J Data Structures & Algorithms. P, NP, and NP-Complete. Polynomial-Time Algorithms

CSCE 310J Data Structures & Algorithms P, NP, and NP-Complete Dr. Steve Goddard goddard@cse.unl.edu CSCE 310J Data Structures & Algorithms Giving credit where credit is due:» Most of the lecture notes

### princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora

princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora Scribe: One of the running themes in this course is the notion of

### The Classes P and NP

The Classes P and NP We now shift gears slightly and restrict our attention to the examination of two families of problems which are very important to computer scientists. These families constitute the

### The Goldberg Rao Algorithm for the Maximum Flow Problem

The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }

### OHJ-2306 Introduction to Theoretical Computer Science, Fall 2012 8.11.2012

276 The P vs. NP problem is a major unsolved problem in computer science It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a \$ 1,000,000 prize for the

### Every tree contains a large induced subgraph with all degrees odd

Every tree contains a large induced subgraph with all degrees odd A.J. Radcliffe Carnegie Mellon University, Pittsburgh, PA A.D. Scott Department of Pure Mathematics and Mathematical Statistics University

### Nan Kong, Andrew J. Schaefer. Department of Industrial Engineering, Univeristy of Pittsburgh, PA 15261, USA

A Factor 1 2 Approximation Algorithm for Two-Stage Stochastic Matching Problems Nan Kong, Andrew J. Schaefer Department of Industrial Engineering, Univeristy of Pittsburgh, PA 15261, USA Abstract We introduce

### Bounded Treewidth in Knowledge Representation and Reasoning 1

Bounded Treewidth in Knowledge Representation and Reasoning 1 Reinhard Pichler Institut für Informationssysteme Arbeitsbereich DBAI Technische Universität Wien Luminy, October 2010 1 Joint work with G.

### Optimal Binary Space Partitions in the Plane

Optimal Binary Space Partitions in the Plane Mark de Berg and Amirali Khosravi TU Eindhoven, P.O. Box 513, 5600 MB Eindhoven, the Netherlands Abstract. An optimal bsp for a set S of disjoint line segments

### Cycle transversals in bounded degree graphs

Electronic Notes in Discrete Mathematics 35 (2009) 189 195 www.elsevier.com/locate/endm Cycle transversals in bounded degree graphs M. Groshaus a,2,3 P. Hell b,3 S. Klein c,1,3 L. T. Nogueira d,1,3 F.

### P versus NP, and More

1 P versus NP, and More Great Ideas in Theoretical Computer Science Saarland University, Summer 2014 If you have tried to solve a crossword puzzle, you know that it is much harder to solve it than to verify

### NP-Completeness I. Lecture 19. 19.1 Overview. 19.2 Introduction: Reduction and Expressiveness

Lecture 19 NP-Completeness I 19.1 Overview In the past few lectures we have looked at increasingly more expressive problems that we were able to solve using efficient algorithms. In this lecture we introduce

### Chapter 1. NP Completeness I. 1.1. Introduction. By Sariel Har-Peled, December 30, 2014 1 Version: 1.05

Chapter 1 NP Completeness I By Sariel Har-Peled, December 30, 2014 1 Version: 1.05 "Then you must begin a reading program immediately so that you man understand the crises of our age," Ignatius said solemnly.

### Boolean Representations and Combinatorial Equivalence

Chapter 2 Boolean Representations and Combinatorial Equivalence This chapter introduces different representations of Boolean functions. It then discuss the applications of these representations for proving

### Tree-representation of set families and applications to combinatorial decompositions

Tree-representation of set families and applications to combinatorial decompositions Binh-Minh Bui-Xuan a, Michel Habib b Michaël Rao c a Department of Informatics, University of Bergen, Norway. buixuan@ii.uib.no

### Introduction to Algorithms. Part 3: P, NP Hard Problems

Introduction to Algorithms Part 3: P, NP Hard Problems 1) Polynomial Time: P and NP 2) NP-Completeness 3) Dealing with Hard Problems 4) Lower Bounds 5) Books c Wayne Goddard, Clemson University, 2004 Chapter

### Cloud Computing is NP-Complete

Working Paper, February 2, 20 Joe Weinman Permalink: http://www.joeweinman.com/resources/joe_weinman_cloud_computing_is_np-complete.pdf Abstract Cloud computing is a rapidly emerging paradigm for computing,

### NP-Completeness and Cook s Theorem

NP-Completeness and Cook s Theorem Lecture notes for COM3412 Logic and Computation 15th January 2002 1 NP decision problems The decision problem D L for a formal language L Σ is the computational task:

### Determination of the normalization level of database schemas through equivalence classes of attributes

Computer Science Journal of Moldova, vol.17, no.2(50), 2009 Determination of the normalization level of database schemas through equivalence classes of attributes Cotelea Vitalie Abstract In this paper,

### What s next? Reductions other than kernelization

What s next? Reductions other than kernelization Dániel Marx Humboldt-Universität zu Berlin (with help from Fedor Fomin, Daniel Lokshtanov and Saket Saurabh) WorKer 2010: Workshop on Kernelization Nov

### Lecture 1: Course overview, circuits, and formulas

Lecture 1: Course overview, circuits, and formulas Topics in Complexity Theory and Pseudorandomness (Spring 2013) Rutgers University Swastik Kopparty Scribes: John Kim, Ben Lund 1 Course Information Swastik

### Approximation Algorithms

Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NP-Completeness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms

### Graphs without proper subgraphs of minimum degree 3 and short cycles

Graphs without proper subgraphs of minimum degree 3 and short cycles Lothar Narins, Alexey Pokrovskiy, Tibor Szabó Department of Mathematics, Freie Universität, Berlin, Germany. August 22, 2014 Abstract

### CMSC 858T: Randomized Algorithms Spring 2003 Handout 8: The Local Lemma

CMSC 858T: Randomized Algorithms Spring 2003 Handout 8: The Local Lemma Please Note: The references at the end are given for extra reading if you are interested in exploring these ideas further. You are

### Why? A central concept in Computer Science. Algorithms are ubiquitous.

Analysis of Algorithms: A Brief Introduction Why? A central concept in Computer Science. Algorithms are ubiquitous. Using the Internet (sending email, transferring files, use of search engines, online

### Partitioning edge-coloured complete graphs into monochromatic cycles and paths

arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edge-coloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and Political

### GRAPH THEORY LECTURE 4: TREES

GRAPH THEORY LECTURE 4: TREES Abstract. 3.1 presents some standard characterizations and properties of trees. 3.2 presents several different types of trees. 3.7 develops a counting method based on a bijection

### CSC 373: Algorithm Design and Analysis Lecture 16

CSC 373: Algorithm Design and Analysis Lecture 16 Allan Borodin February 25, 2013 Some materials are from Stephen Cook s IIT talk and Keven Wayne s slides. 1 / 17 Announcements and Outline Announcements

### A Working Knowledge of Computational Complexity for an Optimizer

A Working Knowledge of Computational Complexity for an Optimizer ORF 363/COS 323 Instructor: Amir Ali Ahmadi TAs: Y. Chen, G. Hall, J. Ye Fall 2014 1 Why computational complexity? What is computational

### Discuss the size of the instance for the minimum spanning tree problem.

3.1 Algorithm complexity The algorithms A, B are given. The former has complexity O(n 2 ), the latter O(2 n ), where n is the size of the instance. Let n A 0 be the size of the largest instance that can

### 5.1 Bipartite Matching

CS787: Advanced Algorithms Lecture 5: Applications of Network Flow In the last lecture, we looked at the problem of finding the maximum flow in a graph, and how it can be efficiently solved using the Ford-Fulkerson

### NP-complete? NP-hard? Some Foundations of Complexity. Prof. Sven Hartmann Clausthal University of Technology Department of Informatics

NP-complete? NP-hard? Some Foundations of Complexity Prof. Sven Hartmann Clausthal University of Technology Department of Informatics Tractability of Problems Some problems are undecidable: no computer

### Offline 1-Minesweeper is NP-complete

Offline 1-Minesweeper is NP-complete James D. Fix Brandon McPhail May 24 Abstract We use Minesweeper to illustrate NP-completeness proofs, arguments that establish the hardness of solving certain problems.

### 2.3 Scheduling jobs on identical parallel machines

2.3 Scheduling jobs on identical parallel machines There are jobs to be processed, and there are identical machines (running in parallel) to which each job may be assigned Each job = 1,,, must be processed

### On the Fixed-Parameter Tractability of the Equivalence Test of Monotone Normal Forms

On the Fixed-Parameter Tractability of the Equivalence Test of Monotone Normal Forms Matthias Hagen Friedrich-Schiller-Universität Jena, Institut für Informatik, D 07737 Jena Abstract We consider the problem

### COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the Higman-Sims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact

### A Graph-Theoretic Network Security Game. Marios Mavronicolas, Vicky G. Papadopoulou, Anna Philippou and Paul G. Spirakis

Int. J. Autonomous and Adaptive Communications Systems, Vol. x, No. x, xx 1 A Graph-Theoretic Network Security Game Marios Mavronicolas, Vicky G. Papadopoulou, Anna Philippou and Paul G. Spirakis Department

### Lecture 17 : Equivalence and Order Relations DRAFT

CS/Math 240: Introduction to Discrete Mathematics 3/31/2011 Lecture 17 : Equivalence and Order Relations Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last lecture we introduced the notion

### On the independence number of graphs with maximum degree 3

On the independence number of graphs with maximum degree 3 Iyad A. Kanj Fenghui Zhang Abstract Let G be an undirected graph with maximum degree at most 3 such that G does not contain any of the three graphs

### Large induced subgraphs with all degrees odd

Large induced subgraphs with all degrees odd A.D. Scott Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, England Abstract: We prove that every connected graph of order

### Fairness in Routing and Load Balancing

Fairness in Routing and Load Balancing Jon Kleinberg Yuval Rabani Éva Tardos Abstract We consider the issue of network routing subject to explicit fairness conditions. The optimization of fairness criteria

### Full and Complete Binary Trees

Full and Complete Binary Trees Binary Tree Theorems 1 Here are two important types of binary trees. Note that the definitions, while similar, are logically independent. Definition: a binary tree T is full

### 3. Linear Programming and Polyhedral Combinatorics

Massachusetts Institute of Technology Handout 6 18.433: Combinatorial Optimization February 20th, 2009 Michel X. Goemans 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the

### The chromatic spectrum of mixed hypergraphs

The chromatic spectrum of mixed hypergraphs Tao Jiang, Dhruv Mubayi, Zsolt Tuza, Vitaly Voloshin, Douglas B. West March 30, 2003 Abstract A mixed hypergraph is a triple H = (X, C, D), where X is the vertex

### An Approximation Algorithm for Bounded Degree Deletion

An Approximation Algorithm for Bounded Degree Deletion Tomáš Ebenlendr Petr Kolman Jiří Sgall Abstract Bounded Degree Deletion is the following generalization of Vertex Cover. Given an undirected graph

### Optimizing Description Logic Subsumption

Topics in Knowledge Representation and Reasoning Optimizing Description Logic Subsumption Maryam Fazel-Zarandi Company Department of Computer Science University of Toronto Outline Introduction Optimization

### Combinatorial 5/6-approximation of Max Cut in graphs of maximum degree 3

Combinatorial 5/6-approximation of Max Cut in graphs of maximum degree 3 Cristina Bazgan a and Zsolt Tuza b,c,d a LAMSADE, Université Paris-Dauphine, Place du Marechal de Lattre de Tassigny, F-75775 Paris

### ARTICLE IN PRESS. European Journal of Operational Research xxx (2004) xxx xxx. Discrete Optimization. Nan Kong, Andrew J.

A factor 1 European Journal of Operational Research xxx (00) xxx xxx Discrete Optimization approximation algorithm for two-stage stochastic matching problems Nan Kong, Andrew J. Schaefer * Department of

### Satisfiability Checking

Satisfiability Checking SAT-Solving Prof. Dr. Erika Ábrahám Theory of Hybrid Systems Informatik 2 WS 10/11 Prof. Dr. Erika Ábrahám - Satisfiability Checking 1 / 40 A basic SAT algorithm Assume the CNF

### Degree Hypergroupoids Associated with Hypergraphs

Filomat 8:1 (014), 119 19 DOI 10.98/FIL1401119F Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Degree Hypergroupoids Associated

### SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

### Polytope Examples (PolyComp Fukuda) Matching Polytope 1

Polytope Examples (PolyComp Fukuda) Matching Polytope 1 Matching Polytope Let G = (V,E) be a graph. A matching in G is a subset of edges M E such that every vertex meets at most one member of M. A matching

### Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs

CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like

### On Integer Additive Set-Indexers of Graphs

On Integer Additive Set-Indexers of Graphs arxiv:1312.7672v4 [math.co] 2 Mar 2014 N K Sudev and K A Germina Abstract A set-indexer of a graph G is an injective set-valued function f : V (G) 2 X such that

### Max-Min Representation of Piecewise Linear Functions

Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 43 (2002), No. 1, 297-302. Max-Min Representation of Piecewise Linear Functions Sergei Ovchinnikov Mathematics Department,

### COMS4236: Introduction to Computational Complexity. Summer 2014

COMS4236: Introduction to Computational Complexity Summer 2014 Mihalis Yannakakis Lecture 17 Outline conp NP conp Factoring Total NP Search Problems Class conp Definition of NP is nonsymmetric with respect

### Analysis of Algorithms, I

Analysis of Algorithms, I CSOR W4231.002 Eleni Drinea Computer Science Department Columbia University Thursday, February 26, 2015 Outline 1 Recap 2 Representing graphs 3 Breadth-first search (BFS) 4 Applications

### CS510 Software Engineering

CS510 Software Engineering Propositional Logic Asst. Prof. Mathias Payer Department of Computer Science Purdue University TA: Scott A. Carr Slides inspired by Xiangyu Zhang http://nebelwelt.net/teaching/15-cs510-se

### Introduction to NP-Completeness Written and copyright c by Jie Wang 1

91.502 Foundations of Comuter Science 1 Introduction to Written and coyright c by Jie Wang 1 We use time-bounded (deterministic and nondeterministic) Turing machines to study comutational comlexity of

### Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson

Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement

### Best Monotone Degree Bounds for Various Graph Parameters

Best Monotone Degree Bounds for Various Graph Parameters D. Bauer Department of Mathematical Sciences Stevens Institute of Technology Hoboken, NJ 07030 S. L. Hakimi Department of Electrical and Computer

### Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques. My T. Thai

Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques My T. Thai 1 Overview An overview of LP relaxation and rounding method is as follows: 1. Formulate an optimization

### ON THE COMPLEXITY OF THE GAME OF SET. {kamalika,pbg,dratajcz,hoeteck}@cs.berkeley.edu

ON THE COMPLEXITY OF THE GAME OF SET KAMALIKA CHAUDHURI, BRIGHTEN GODFREY, DAVID RATAJCZAK, AND HOETECK WEE {kamalika,pbg,dratajcz,hoeteck}@cs.berkeley.edu ABSTRACT. Set R is a card game played with a

### Chapter. NP-Completeness. Contents

Chapter 13 NP-Completeness Contents 13.1 P and NP......................... 593 13.1.1 Defining the Complexity Classes P and NP...594 13.1.2 Some Interesting Problems in NP.......... 597 13.2 NP-Completeness....................

### Degrees that are not degrees of categoricity

Degrees that are not degrees of categoricity Bernard A. Anderson Department of Mathematics and Physical Sciences Gordon State College banderson@gordonstate.edu www.gordonstate.edu/faculty/banderson Barbara

### Applied Algorithm Design Lecture 5

Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design

### CS 598CSC: Combinatorial Optimization Lecture date: 2/4/2010

CS 598CSC: Combinatorial Optimization Lecture date: /4/010 Instructor: Chandra Chekuri Scribe: David Morrison Gomory-Hu Trees (The work in this section closely follows [3]) Let G = (V, E) be an undirected

### Theoretical Computer Science (Bridging Course) Complexity

Theoretical Computer Science (Bridging Course) Complexity Gian Diego Tipaldi A scenario You are a programmer working for a logistics company Your boss asks you to implement a program that optimizes the

### 8.1 Min Degree Spanning Tree

CS880: Approximations Algorithms Scribe: Siddharth Barman Lecturer: Shuchi Chawla Topic: Min Degree Spanning Tree Date: 02/15/07 In this lecture we give a local search based algorithm for the Min Degree

### High degree graphs contain large-star factors

High degree graphs contain large-star factors Dedicated to László Lovász, for his 60th birthday Noga Alon Nicholas Wormald Abstract We show that any finite simple graph with minimum degree d contains a

### Lecture 7: Approximation via Randomized Rounding

Lecture 7: Approximation via Randomized Rounding Often LPs return a fractional solution where the solution x, which is supposed to be in {0, } n, is in [0, ] n instead. There is a generic way of obtaining

### Measuring Rationality with the Minimum Cost of Revealed Preference Violations. Mark Dean and Daniel Martin. Online Appendices - Not for Publication

Measuring Rationality with the Minimum Cost of Revealed Preference Violations Mark Dean and Daniel Martin Online Appendices - Not for Publication 1 1 Algorithm for Solving the MASP In this online appendix

### The degree, size and chromatic index of a uniform hypergraph

The degree, size and chromatic index of a uniform hypergraph Noga Alon Jeong Han Kim Abstract Let H be a k-uniform hypergraph in which no two edges share more than t common vertices, and let D denote the

### One last point: we started off this book by introducing another famously hard search problem:

S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani 261 Factoring One last point: we started off this book by introducing another famously hard search problem: FACTORING, the task of finding all prime factors

### Regular Languages and Finite Automata

Regular Languages and Finite Automata 1 Introduction Hing Leung Department of Computer Science New Mexico State University Sep 16, 2010 In 1943, McCulloch and Pitts [4] published a pioneering work on a

### POWER SETS AND RELATIONS

POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty

### 24. The Branch and Bound Method

24. The Branch and Bound Method It has serious practical consequences if it is known that a combinatorial problem is NP-complete. Then one can conclude according to the present state of science that no

### System BV is NP-complete

System BV is NP-complete Ozan Kahramanoğulları 1,2 Computer Science Institute, University of Leipzig International Center for Computational Logic, TU Dresden Abstract System BV is an extension of multiplicative

### Discrete Applied Mathematics. The firefighter problem with more than one firefighter on trees

Discrete Applied Mathematics 161 (2013) 899 908 Contents lists available at SciVerse ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam The firefighter problem with

Computing and Informatics, Vol. 20, 2001,????, V 2006-Nov-6 UPDATES OF LOGIC PROGRAMS Ján Šefránek Department of Applied Informatics, Faculty of Mathematics, Physics and Informatics, Comenius University,

### No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics

No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results

### Outline. NP-completeness. When is a problem easy? When is a problem hard? Today. Euler Circuits

Outline NP-completeness Examples of Easy vs. Hard problems Euler circuit vs. Hamiltonian circuit Shortest Path vs. Longest Path 2-pairs sum vs. general Subset Sum Reducing one problem to another Clique

### Mathematics for Algorithm and System Analysis

Mathematics for Algorithm and System Analysis for students of computer and computational science Edward A. Bender S. Gill Williamson c Edward A. Bender & S. Gill Williamson 2005. All rights reserved. Preface

### ON FUNCTIONAL SYMBOL-FREE LOGIC PROGRAMS

PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical and Mathematical Sciences 2012 1 p. 43 48 ON FUNCTIONAL SYMBOL-FREE LOGIC PROGRAMS I nf or m at i cs L. A. HAYKAZYAN * Chair of Programming and Information

### A 2-factor in which each cycle has long length in claw-free graphs

A -factor in which each cycle has long length in claw-free graphs Roman Čada Shuya Chiba Kiyoshi Yoshimoto 3 Department of Mathematics University of West Bohemia and Institute of Theoretical Computer Science

### A class of on-line scheduling algorithms to minimize total completion time

A class of on-line scheduling algorithms to minimize total completion time X. Lu R.A. Sitters L. Stougie Abstract We consider the problem of scheduling jobs on-line on a single machine and on identical

### Scheduling. Open shop, job shop, flow shop scheduling. Related problems. Open shop, job shop, flow shop scheduling

Scheduling Basic scheduling problems: open shop, job shop, flow job The disjunctive graph representation Algorithms for solving the job shop problem Computational complexity of the job shop problem Open

### COUNTING INDEPENDENT SETS IN SOME CLASSES OF (ALMOST) REGULAR GRAPHS

COUNTING INDEPENDENT SETS IN SOME CLASSES OF (ALMOST) REGULAR GRAPHS Alexander Burstein Department of Mathematics Howard University Washington, DC 259, USA aburstein@howard.edu Sergey Kitaev Mathematics