Factoring and recognition of read-once functions using cographs and normality and the readability of functions associated with partial k-trees

An approach for factoring general boolean functions was described in Golumbic and Mintz [Factoring logic functions using graph partitioning, in: Proceedings of IEEE/ACM International Conference on Computer Aided Design, November 1999, pp. 195-198] and Mintz and Golumbic [Factoring Boolean functions using graph partitioning, Discrete Appl. Math. 149 (2005) 131-153] which is based on graph partitioning algorithms. In this paper, we present a very fast algorithm for recognizing and factoring read-once functions which is needed as a dedicated factoring subroutine to handle the lower levels of that factoring process. The algorithm is based on algorithms for cograph recognition and on checking normality.For non-read-once functions, we investigate their factoring based on their corresponding graph classes. In particular, we show that if a function F is normal and its corresponding graph is a partial k-tree, then F is a read 2^k function and a read 2^k formula for F can be obtained in polynomial time.     

An improvement on the complexity of factoring read-once Boolean functions

Read-once functions have gained recent, renewed interest in the fields of theory and algorithms of Boolean functions, computational learning theory and logic design and verification. In an earlier paper [M.C. Golumbic, A. Mintz, U. Rotics, Factoring and recognition of read-once functions using cographs and normality, and the readability of functions associated with partial k-trees, Discrete Appl. Math. 154 (2006) 1465-1677], we presented the first polynomial-time algorithm for recognizing and factoring read-once functions, based on a classical characterization theorem of Gurvich which states that a positive Boolean function is read-once if and only if it is normal and its co-occurrence graph is P₄-free.In this note, we improve the complexity bound by showing that the method can be modified slightly, with two crucial observations, to obtain an O(n|f|) implementation, where |f| denotes the length of the DNF expression of a positive Boolean function f, and n is the number of variables in f. The previously stated bound was O(n²k), where k is the number of prime implicants of the function. In both cases, f is assumed to be given as a DNF formula consisting entirely of the prime implicants of the function.     

On covering graphs by complete bipartite subgraphs

We prove that, if a graph with e edges contains m vertex-disjoint edges, then m²/e complete bipartite subgraphs are necessary to cover all its edges. Similar lower bounds are also proved for fractional covers. For sparse graphs, this improves the well-known fooling set lower bound in communication complexity. We also formulate several open problems about covering problems for graphs whose solution would have important consequences in the complexity theory of boolean functions.     

Algebraic and topological closure conditions for classes of pseudo-Boolean functions

We examine classes of real-valued functions of 0-1 variables closed under algebraic operations as well as topological convergence, and having a certain local characteristic (requiring that any function not in the class should have a k-variable minor not belonging to this class). It is shown that for k=2, the only 4 maximal classes with these properties are those of submodular, supermodular, monotone increasing and monotone decreasing functions. All the 13 locally defined closed classes are determined and shown to be intersections of the 4 maximal ones. All maximal classes for k≥3 are determined and characterized by the sign of higher order derivatives of the functions in the class.     

Self-duality of bounded monotone boolean functions and related problems

In this paper we examine the problem of determining the self-duality of a monotone boolean function in disjunctive normal form (DNF). We show that the self-duality of monotone boolean functions with n disjuncts such that each disjunct has at most k literals can be determined in O(2^k2k²n) time. This implies an O(n²logn) algorithm for determining the self-duality of -DNF functions. We also consider the version where any two disjuncts have at most c literals in common. For this case we give an O(n^4(c+1)) algorithm for determining self-duality.     

Graphs with the maximum or minimum number of 1-factors

Recently Alon and Friedland have shown that graphs which are the union of complete regular bipartite graphs have the maximum number of 1-factors over all graphs with the same degree sequence. We identify two families of graphs that have the maximum number of 1-factors over all graphs with the same number of vertices and edges: the almost regular graphs which are unions of complete regular bipartite graphs, and complete graphs with a matching removed. The first family is determined using the Alon and Friedland bound. For the second family, we show that a graph transformation which is known to increase network reliability also increases the number of 1-factors. In fact, more is true: this graph transformation increases the number of k-factors for all k≥1, and “in reverse” also shows that in general, threshold graphs have the fewest k-factors. We are then able to determine precisely which threshold graphs have the fewest 1-factors. We conjecture that the same graphs have the fewest k-factors for all k≥2 as well.     

On 2-factor hamiltonian regular bipartite graphs

A k-regular bipartite graph is said to be 2-factor hamiltonian if each of its 2-factor is hamiltonian. It is well known that if a k-regular bipartite graph is 2-factor hamiltonian, then k?Q3. In this paper, we give a new proof of this fact.     

Periodic decomposition of measurable integer valued functions

We study those functions that can be written as a sum of (almost everywhere) integer valued periodic measurable functions with given periods. We show that being (almost everywhere) integer valued measurable function and having a real valued periodic decomposition with the given periods is not enough. We characterize those periods for which this condition is enough. We also get that the class of bounded measurable (almost everywhere) integer valued functions does not have the so-called decomposition property. We characterize those periods a₁,…,ak for which an almost everywhere integer valued bounded measurable function f has an almost everywhere integer valued bounded measurable (a₁,…,ak)-periodic decomposition if and only if Δa₁?Δakf=0, where Δaf(x)=f(x+a)−f(x).     

Clustering and domination in perfect graphs

A k-cluster in a graph is an induced subgraph on k vertices which maximizes the number of edges. Both the k-cluster problem and the k-dominating set problem are NP-complete for graphs in general. In this paper we investigate the complexity status of these problems on various sub-classes of perfect graphs. In particular, we examine comparability graphs, chordal graphs, bipartite graphs, split graphs, cographs and κ-trees. For example, it is shown that the k-cluster problem is NP-complete for both bipartite and chordal graphs and the independent k-dominating set problem is NP-complete for bipartite graphs. Furthermore, where the k-cluster problem is polynomial we study the weighted and connected versions as well. Similarly we also look at the minimum k-dominating set problem on families which have polynomial k-dominating set algorithms.     

Formulas for approximating pseudo-Boolean random variables

We consider {0,1}ⁿ as a sample space with a probability measure on it, thus making pseudo-Boolean functions into random variables. We then derive explicit formulas for approximating a pseudo-Boolean random variable by a linear function if the measure is permutation-invariant, and by a function of degree at most k if the measure is a product measure. These formulas generalize results due to Hammer-Holzman and Grabisch-Marichal-Roubens. We also derive a formula for the best faithful linear approximation that extends a result due to Charnes-Golany-Keane-Rousseau concerning generalized Shapley values. We show that a theorem of Hammer-Holzman that states that a pseudo-Boolean function and its best approximation of degree at most k have the same derivatives up to order k does not generalize to this setting for arbitrary probability measures, but does generalize if the probability measure is a product measure.     

Relations between threshold and <Emphasis Type="Italic">k</Emphasis>-interval Boolean functions

Every k-interval Boolean function f can be represented by at most k intervals of integers such that vector x is a truepoint of f if and only if the integer represented by x belongs to one of these k (disjoint) intervals. Since the correspondence of Boolean vectors and integers depends on the order of bits an interval representation is also specified with respect to an order of variables of the represented function. Interval representation can be useful as an efficient representation for special classes of Boolean functions which can be represented by a small number of intervals. In this paper we study inclusion relations between the classes of threshold and k-interval Boolean functions. We show that positive 2-interval functions constitute a (proper) subclass of positive threshold functions and that such inclusion does not hold for any k>2. We also prove that threshold functions do not constitute a subclass of k-interval functions, for any k.     

On Perfect k-Matchings

In this paper, we generalize the notions of perfect matchings, perfect 2-matchings to perfect k-matchings and give a necessary and sufficient condition for the existence of perfect k-matchings. We show that a bipartite graph G contains a perfect k-matching if and only if it contains a perfect matching. Moreover, for regular graphs, we provide a sufficient condition for the existence of perfect k-matching in terms of the edge connectivity.     

The basis number of a graph

MacLane proved that a graph is planar if and only if it has a 2-fold basis for its cycle space. We define the basis number of a graph G to be the least integer k such that G has a k-fold basis for its cycle space. We investigate the basis number of the complete graphs, complete bipartite graphs, and the n-cube.     

Very large cliques are easy to detect

It is known that, for every constant k?3, the presence of a k-clique (a complete sub-graph on k vertices) in an n-vertex graph cannot be detected by a monotone boolean circuit using much fewer than n^k gates. We show that, for every constant k, the presence of an (n-k)-clique in an n-vertex graph can be detected by a monotone circuit using only a logarithmic number of fanin-2 OR gates; the total number of gates does not exceed O(n²logn). Moreover, if we allow unbounded fanin, then a logarithmic number of gates is enough.     

Rudin orthogonality problem on the Bergman space

In this paper, we study the Rudin orthogonality problem on the Bergman space, which is to characterize those functions bounded analytic on the unit disk whose powers form an orthogonal set in the Bergman space of the unit disk. We completely solve the problem if those functions are univalent in the unit disk or analytic in a neighborhood of the closed unit disk. As a consequence, it is shown that an analytic multiplication operator on the Bergman space is unitarily equivalent to a weighted unilateral shift of finite multiplicity n if and only if its symbol is a constant multiple of the n-th power of a Möbius transform, which was obtained via the Hardy space theory of the bidisk in Sun et al. (2008) [10].     

Read-Once Functions Revisited and the Readability Number of a Boolean Function

In this paper, we present the first polynomial time algorithm for recognizing and factoring read-once functions. The algorithm is based on algorithms for cograph recognition and a new efficient method for checking normality. Its correctness is based on a classical characterization theorem of Gurvich which states that a positive Boolean function is read-once if and only if it is normal and its co-occurrance graph is P₄-free.We also investigate the problem of factoring certain non-read-once functions. In particular, we show that if the co-occurrence graph of a positive Boolean function f is a tree, then the function is read-twice. We then extend this further proving that if f is normal and its corresponding graph is a partial k-tree, then f is a read 2^k function and a read 2^k formula for F for f can be obtained in polynomial time.     

Rademacher series and isomorphisms of rearrangement invariant spaces on the finite interval and on the semi-axis

Let X be a rearrangement invariant function space on [0,1]. We consider the subspace Radi X of X which consists of all functions of the form , where xk are arbitrary independent functions from X and rk are usual Rademacher functions independent of {xk}. We prove that Radi X is complemented in X if and only if both X and its Köthe dual space X^′ possess the so-called Kruglov property. As a consequence we show that the last conditions guarantee that X is isomorphic to some rearrangement invariant function space on [0,∞). This strengthens earlier results derived in different approach in [W.B. Johnson, B. Maurey, G. Schechtman, L. Tzafriri, Symmetric structures in Banach spaces, Mem. Amer. Math. Soc. 1 (217) (1979)].     

2-resonance of plane bipartite graphs and its applications to boron-nitrogen fullerenes

A set H of disjoint faces of a plane bipartite graph G is a resonant pattern if G has a perfect matching M such that the boundary of each face in H is an M-alternating cycle. An elementary result was obtained [Discrete Appl. Math. 105 (2000) 291-311]: a plane bipartite graph is 1-extendable if and only if every face forms a resonant pattern. In this paper we show that for a 2-extendable plane bipartite graph, any pair of disjoint faces form a resonant pattern, and the converse does not necessarily hold. As an application, we show that all boron-nitrogen (B-N) fullerene graphs are 2-resonant, and construct all the 3-resonant B-N fullerene graphs, which are all k-resonant for any positive integer k. Here a B-N fullerene graph is a plane cubic graph with only square and hexagonal faces, and a B-N fullerene graph is k-resonant if any disjoint faces form a resonant pattern. Finally, the cell polynomials of 3-resonant B-N fullerene graphs are computed.     

Weighted coloring on planar, bipartite and split graphs: Complexity and approximation

We study complexity and approximation of min weighted node coloring in planar, bipartite and split graphs. We show that this problem is NP-hard in planar graphs, even if they are triangle-free and their maximum degree is bounded above by 4. Then, we prove that min weighted node coloring is NP-hard in P₈-free bipartite graphs, but polynomial for P₅-free bipartite graphs. We next focus on approximability in general bipartite graphs and improve earlier approximation results by giving approximation ratios matching inapproximability bounds. We next deal with min weighted edge coloring in bipartite graphs. We show that this problem remains strongly NP-hard, even in the case where the input graph is both cubic and planar. Furthermore, we provide an inapproximability bound of 7/6−ε, for any ε>0 and we give an approximation algorithm with the same ratio. Finally, we show that min weighted node coloring in split graphs can be solved by a polynomial time approximation scheme.     

On teaching sets for 2-threshold functions of two variables

We consider k-threshold functions of n variables, i.e. the functions representable as the conjunction of k threshold functions. For n = 2, k = 2, we give upper bounds for the cardinality of the minimal teaching set depending on the various properties of the function.