Algorithms for Singleton Attractor Detection in Planar and Nonplanar AND/OR Boolean Networks

Singleton attractor (also called fixed point) detection is known to be NP-hard even for AND/OR Boolean networks (AND/OR BNs in short, i.e., BNs consisting of AND/OR nodes), where BN is a mathematical model of genetic networks and singleton attractors correspond to steady states. In our recent paper, we developed an O(1.787ⁿ) time algorithm for detecting a singleton attractor of a given AND/OR BN where n is the number of nodes. In this paper, we present an O(1.757ⁿ) time algorithm with which we succeeded in improving the above algorithm. We also show that this problem can be solved in time, which is less than O((1 + ∈)ⁿ) for any positive constant ∈, when a BN is planar. A preliminary version of this paper has appeared in Proc. 3rd International Conference on Algebraic Biology (AB2008) [27].     

Constructing Ramsey graphs from Boolean function representations

Explicit construction of Ramsey graphs or graphs with no large clique or independent set, has remained a challenging open problem for a long time. While Erdös’ probabilistic argument shows the existence of graphs on 2n vertices with no clique or independent set of size 2 ⁿ , the best explicit constructions achieve a far weaker bound. There is a connection between Ramsey graph constructions and polynomial representations of Boolean functions due to Grolmusz; a low degree representation for the OR function can be used to explicitly construct Ramsey graphs [17,18]. We generalize the above relation by proposing a new framework. We propose a new definition of OR representations: a pair of polynomials represent the OR function if the union of their zero sets contains all points in {0, 1} ⁿ except the origin. We give a simple construction of a Ramsey graph using such polynomials. Furthermore, we show that all the known algebraic constructions, ones to due to Frankl-Wilson [12], Grolmusz [18] and Alon [2] are captured by this framework; they can all be derived from various OR representations of degree O(√n) based on symmetric polynomials. Thus the barrier to better Ramsey constructions through such algebraic methods appears to be the construction of lower degree representations. Using new algebraic techniques, we show that better bounds cannot be obtained using symmetric polynomials.     

Generation of assembly graphs by systematic analysis of assembly structures

In assembly line balancing problems, parallel execution of assembly operations is often advocated because of its enhanced flexibility and minimum lead-time. Although the theoretical maximum number of possible assembly sequences combinatorially explodes with the number of components in a product, graphical representations can depict these sequences in a surveyable way. The AND/OR graph representation is an appropriate basis for optimum sequence selection, which can be achieved via heuristic, metaheuristic, and exact methods. The exact method, based on binary linear programming, is described. To arrive at the appropriate model, a novel approach for AND/OR graph generation, based on subassembly detection, is presented. The method is demonstrated with simple cases and next extended to increasingly complex products. A modification of the optimization method is applied, which enables a search for sequences with maximum parallelism.     

On the importance of behavioral operational research: The case of understanding and communicating about dynamic systems

We point out the need for Behavioral Operational Research (BOR) in advancing the practice of OR. So far, in OR behavioral phenomena have been acknowledged only in behavioral decision theory but behavioral issues are always present when supporting human problem solving by modeling. Behavioral effects can relate to the group interaction and communication when facilitating with OR models as well as to the possibility of procedural mistakes and cognitive biases. As an illustrative example we use well known system dynamics studies related to the understanding of accumulation. We show that one gets completely opposite results depending on the way the phenomenon is described and how the questions are phrased and graphs used. The results suggest that OR processes are highly sensitive to various behavioral effects. As a result, we need to pay attention to the way we communicate about models as they are being increasingly used in addressing important problems like climate change.     

Petri net approach to disassembly process planning for products with complex AND/OR precedence relationships

We present a Petri net (PN)-based approach to automatically generate disassembly process plans (DPPs) for product recycling or remanufacturing. We define an algorithm to generate a geometrically-based disassembly precedence matrix (DPM) from a CAD drawing of the product. We then define an algorithm to automatically generate a disassembly Petri net (DPN) from the DPM; the DPN is live, bounded, and reversible. The resulting DPN can be analyzed using the reachability tree method to generate feasible DPPs, and cost functions can be used to determine the optimal DPP. Since reachability tree generation is NP-complete, we develop a heuristic to dynamically explore the v likeliest lowest cost branches of the tree, to identify optimal or near-optimal DPPs. The cost function incorporates tool changes, changes in direction of movement, and individual part characteristics (e.g., hazardous). An example is used to illustrate the procedure. This approach can be used for products containing AND, OR, and complex AND/OR disassembly precedence relationships.     

ContributionsUpper domination and upper irredundance perfect graphs

Let β(G), Γ(G) and IR(G) be the independence number, the upper domination number and the upper irredundance number, respectively. A graph G is calledΓ-perfect if β(H) = Γ(H), for every induced subgraph H of G. A graph G is called IR-perfect if Γ(H) = IR(H), for every induced subgraph H of G. In this paper, we present a characterization of Γ-perfect graphs in terms of a family of forbidden induced subgraphs, and show that the class of Γ-perfect graphs is a subclass of IR-perfect graphs and that the class of absorbantly perfect graphs is a subclass of Γ-perfect graphs. These results imply a number of known theorems on Γ-perfect graphs and IR-perfect graphs. Moreover, we prove a sufficient condition for a graph to be Γ-perfect and IR-perfect which improves a known analogous result.     

Recognizing graphs without asteroidal triples

We consider the problem of recognizing AT-free graphs. Although there is a simple O(n³) algorithm, no faster method for solving this problem had been known. Here we give three different algorithms which have a better time complexity for graphs which are sparse or have a sparse complement; in particular we give algorithms which recognize AT-free graphs in , , and O(n^2.82+nm). In addition we give a new characterization of graphs with bounded asteroidal number by the help of the knotting graph, a combinatorial structure which was introduced by Gallai for considering comparability graphs.     

The reduction of graph families closed under contraction

Let be a family of graphs. Suppose there is a nontrivial graph H such that for any supergraph G of H, G is in if and only if the contraction G/H is in . Examples of such an : graphs with a spanning closed trail; graphs with at least k edge-disjoint spanning trees; and k-edge-connected graphs (k fixed). We give a reduction method using contractions to find when a given graph is in and to study its structure if it is not in . This reduction method generalizes known special cases.     

Planar graphs with no 6-wheel minor

Tutte's wheels theorem states that the k-spoked wheel graphs, W_k, are the basic building blocks for the collection of simple, 3-connected graphs. Therefore it is of interest to examine the structure of the graphs that do not have a minor isomorphic to W_k for small values of k. Dirac determined that the graphs having no W₃-minor are the series-parallel networks. An easy consequence of Tutte's wheels theorem is that W₃ is the only simple, 3-connected graph that has a W₃-minor and no W₄-minor. Oxley characterized the graphs that have a W₄-minor and no W₅-minor. This paper characterizes the planar graphs that have a W₅-minor and no W₆-minor. A best-possible upper bound on the number of edges of such a graph is also determined.     

Enumeration of labelled chain graphs and labelled essential directed acyclic graphs

A chain graph is a digraph whose strong components are undirected graphs and a directed acyclic graph (ADG or DAG) G is essential if the Markov equivalence class of G consists of only one element. We provide recurrence relations for counting labelled chain graphs by the number of chain components and vertices; labelled essential DAGs by the number of vertices. The second one is a lower bound for the number of labelled essential graphs. The formula for labelled chain graphs can be extended in such a way, that allows us to count digraphs with two additional properties, which essential graphs have.     

Star partitions and the graph isomorphism problem

A canonical basis of Rⁿ associated with a graph G on n vertices has been defined in [15] in connection with eigenspaces and star partitions of G. The canonical star basis together with eigenvalues of G determines G to an isomorphism. We study algorithms for finding the canonical basis and some of its variations. The emphasis is on the following three special cases; graphs with distinct eigenvalues, graphs with bounded eigenvalue multiplicities and strongly regular graphs. We show that the procedure is reduced in some parts to special cases of some well known combinatorial optimization problems, such as the maximal matching problem. the minimal cut problem, the maximal clique problem etc. This technique provides another proof of a result of L. Babai et al. [2] that isomorphism testing for graphs with bounded eigenvalue multiplicities can be performend in a polynomial time. We show that the canonical basis in strongly regular graphs is related to the graph decomposition into two strongly regular induced subgraphs. Examples of distinguishing between cospectral strongly regular graphs by means of the canonical basis are provided. The behaviour of star partitions of regular graphs under operations of complementation and switching is studied.     

The strong perfect graph conjecture holds for diamonded odd cycle-free graphs

In this paper, we show that the strong perfect graph conjecture holds for a new class of graphs which we call diamonded odd cycle-free graphs. The class of diamonded odd cycle-free graphs contains the classes of threshold graphs and K₄\e-free graphs.     

Using Discrete Distributions to Approximate General Service Time Distributions in Queueing Models

In solving discrete time queueing models by numerical techniques, the computational requirements (computer memory and time) are a practical limitation and are particularly dependent on the number of discrete time intervals required in the discrete distribution chosen to match the general service distribution. This paper shows that the minimum number of points required for matching to the first two moments depends on the size of the discrete interval relative to the mean and also on the coefficient of variation. Equations and graphs are provided that will enable the OR practitioner to select the discrete distribution to be used as an approximation. Additionally, it is concluded that discrete time modelling, using these approximations to model service time, now provides a practical means to model both steady-state measures and transient behaviour of M/G/c, M(t)/G/c and M(t)/G/c(t) queueing systems on a personal computer.     

Polynomial algorithms for (integral) maximum two-flows in vertex edge-capacitated planar graphs

In this paper we study the maximum two-flow problem in vertex- and edge-capacitated undirected ST₂-planar graphs, that is, planar graphs where the vertices of each terminal pair are on the same face. For such graphs we provide an O(n) algorithm for finding a minimum two-cut and an O(n log n) algorithm for determining a maximum two-flow and show that the value of a maximum two-flow equals the value of a minimum two-cut. We further show that the flow obtained is half-integral and provide a characterization of edge and vertex capacitated ST₂-planar graphs that guarantees a maximum two-flow that is integral. By a simple variation of our maximum two-flow algorithm we then develop, for ST₂-planar graphs with vertex and edge capacities, an O(n log n) algorithm for determining an integral maximum two-flow of value not less than the value of a maximum two-flow minus one.     

A note on the approximability of the toughness of graphs

We show that, if NP≠ZPP, for any >0, the toughness of a graph with n vertices is not approximable in polynomial time within a factor of . We give a 4-approximation for graphs with toughness bounded by and we show that this result cannot be generalized to graphs with a bounded toughness. More exactly we prove that there is no constant approximation for graphs with bounded toughness, unless P=NP.     

Completeness for intersection classes

An intersection representation of a graph is a function gf mapping vertices to sets such that for any u ≠ v, u is adjacent to v iff gf(u) ∩ gf(v) ≠ . The intersection class defined by a set of sets ∑ is the set of all graphs having an intersection representation using sets from ∑. Interval graphs and chordal graphs are well-studied examples of intersection classes.

This paper introduces the notion of completeness for intersection classes. That is, determining precisely what restrictions can be made on the allowable sets without losing the power to represent any graph in the intersection class. The solution to this problem is presented for the chordal graphs.     

Algorithmic aspects of clique-transversal and clique-independent sets

A minimum clique-transversal set MCT(G) of a graph G=(V,E) is a set SV of minimum cardinality that meets all maximal cliques in G. A maximum clique-independent set MCI(G) of G is a set of maximum number of pairwise vertex-disjoint maximal cliques. We prove that the problem of finding an MCT(G) and an MCI(G) is NP-hard for cocomparability, planar, line and total graphs. As an interesting corollary we obtain that the problem of finding a minimum number of elements of a poset to meet all maximal antichains of the poset remains NP-hard even if the poset has height two, thereby generalizing a result of Duffas et al. (J. Combin. Theory Ser. A 58 (1991) 158–164). We present a polynomial algorithm for the above problems on Helly circular-arc graphs which is the first such algorithm for a class of graphs that is not clique-perfect. We also present polynomial algorithms for the weighted version of the clique-transversal problem on strongly chordal graphs, chordal graphs of bounded clique size, and cographs. The algorithms presented run in linear time for strongly chordal graphs and cographs. These mark the first attempts at the weighted version of the problem.     

Matching extension and minimum degree

Let G be a simple connected graph on 2n vertices with a perfect matching. For a given positive integer k, 1 k n − 1, G is k-extendable if for every matching M of size k in G, there exists a perfect matching in G containing all the edges of M. The problem that arises is that of characterizing k-extendable graphs. In this paper, we establish a necessary condition, in terms of minimum degree, for k-extendable graphs. Further, we determine the set of realizable values for minimum degree of k-extendable graphs. In addition, we establish some results on bipartite graphs including a sufficient condition for a bipartite graph to be k-extendable.     

Tough spiders

Spider graphs are the intersection graphs of subtrees of subdivisions of stars. Thus, spider graphs are chordal graphs that form a common superclass of interval and split graphs. Motivated by previous results on the existence of Hamilton cycles in interval, split and chordal graphs, we show that every 3/2‐tough spider graph is hamiltonian. The obtained bound is best possible since there are (3/2 – ε)‐tough spider graphs that do not contain a Hamilton cycle. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 23–40, 2007     

Weighted domination of cocomparability graphs

It is shown in this paper that the weighted domination problem and its three variants, the weighted connected domination, total domination, and dominating clique problems are NP-complete on cobipartite graphs when arbitrary integer vertex weights are allowed and all of them can be solved in polynomial time on cocomparability graphs if vertex weights are integers and less than or equal to a constant c. The results are interesting because cocomparability graphs properly contain cobipartite graphs and the cardinality cases of the above problems are trivial on cobipartite graphs. On the other hand, an O(¦V¦²) algorithm is given for the weighted independent perfect domination problem of a cocomparability graph G = (V.E).