random capacities and their distributions we formalize the notion of an increasing and outer continuous random process indexed by a class of compact sets that maps the empty set on zero existence and convergence theorems for distributions of such processes are proved these results generalize or are similar to those known in the special cases of random measures random closed sets and random upper semicontinuous functions for the latter processes infinite divisibility under the maximum is introduced and characterized our result generalizes known characterizations of infinite divisibility for random sets and max infinite divisibility for random vectors also discussed is the convergence in distribution of the row vise maxima of a null array of random semicontinuous functions

determining the chromatic number of triangle free 2p3 free graphs in polynomial time let 2p 3 denote the disjoint union of two paths on three vertices a graph g that has no subgraph isomorphic to a graph h is called h free the vertex coloring problem is the problem to determine the chromatic number of a graph its computational complexity for triangle free h free graphs has been classified for every fixed graph h on at most 6 vertices except for the case h 2p 3 this remaining case is posed as an open problem by dabrowski lozin raman and ries we solve their open problem by showing polynomial time solvability

predictable dynamics of opinion forming for networks with antagonistic interactions for communities of agents which are not necessarily cooperating distributed processes of opinion forming are naturally represented by signed graphs with positive edges representing friendly and cooperative interactions and negative edges the corresponding antagonistic counterpart unlike for nonnegative graphs the outcome of a dynamical system evolving on a signed graph is not obvious and it is in general difficult to characterize even when the dynamics are linear in this paper we identify a significant class of signed graphs for which the linear dynamics are however predictable and show many analogies with positive dynamical systems these cases correspond to adjacency matrices that are eventually positive for which the perron frobenius property still holds and implies the existence of an invariant cone contained inside the positive orthant as examples of applications we determine cases in which it is possible to anticipate or impose unanimity of opinion in decision voting processes even in presence of stubborn agents and show how it is possible to extend the pagerank algorithm to include negative links

property testing and its connection to learning and approximation in this paper we consider the question of determining whether a function f has property p or is e far from any function with property p a property testing algorithm is given a sample of the value of f on instances drawn according to some distribution in some cases it is also allowed to query f on instances of its choice we study this question for different properties and establish some connections to problems in learning theory and approximation in particular we focus our attention on testing graph properties given access to a graph g in the form of being able to query whether an edge exists or not between a pair of vertices we devise algorithms to test whether the underlying graph has properties such as being bipartite k colorable or having a p clique clique of density p with respect to the vertex set our graph property testing algorithms are probabilistic and make assertions that are correct with high probability while making a number of queries that is independent of the size of the graph moreover the property testing algorithms can be used to efficiently i e in time linear in the number of vertices construct partitions of the graph that correspond to the property being tested if it holds for the input graph

bipartite and cooperative output synchronizations of linear heterogeneous agents this paper investigates cooperative output synchronization and bipartite output synchronization of a group of linear heterogeneous agents in a unified framework for a structurally balanced signed graph we prove that the bipartite output synchronization is equivalent to the cooperative output synchronization over an unsigned graph whose adjacency matrix is obtained by taking the absolute value of each entry in the adjacency matrix of the signed graph we obtain a new h criterion which is sufficient for both cooperative output synchronization and bipartite output synchronization

parallel algorithms in graph theory planarity testing we present efficient o log 2 n parallel algorithms for two classical graph problems planarity testing and finding triconnected components the algorithms use only a polynomial number of processors previous algorithms used omega n operations regardless of the number of available processors

updating the complexity status of coloring graphs without a fixed induced linear forest a graph is h free if it does not contain an induced subgraph isomorphic to the graph h the graph p k denotes a path on k vertices the coloring problem is the problem to decide whether a graph can be colored with at most colors such that adjacent vertices receive different colors we show that 4 coloring is np complete for p 8 free graphs this improves a result of le randerath and schiermeyer who showed that 4 coloring is np complete for p 9 free graphs and a result of woeginger and sgall who showed that 5 coloring is np complete for p 8 free graphs additionally we prove that the precoloring extension version of 4 coloring is np complete for p 7 free graphs but that the precoloring extension version of 3 coloring can be solved in polynomial time for p 2 p 4 free graphs a subclass of p 7 free graphs here p 2 p 4 denotes the disjoint union of a p 2 and a p 4 we denote the disjoint union of s copies of a p 3 by sp 3 and involve ramsey numbers to prove that the precoloring extension version of 3 coloring can be solved in polynomial time for sp 3 free graphs for any fixed s combining our last two results with known results yields a complete complexity classification of precoloring extension of 3 coloring for h free graphs when h is a fixed graph on at most 6 vertices the problem is polynomial time solvable if h is a linear forest otherwise it is np complete

counting h colorings of partial k trees the problem of counting all h colorings of a graph g with n vertices is considered while the problem is in general p complete we give linear time algorithms that solve the main variants of this problem when the input graph g is a k tree or in the case where g is directed when the underlying graph of g is a k tree our algorithms remain polynomial even in the case where k o logn or in the case where the size of h is o n our results are easy to implement and imply the existence of polynomial time algorithms for a series of problems on partial k trees such as core checking and chromatic polynomial computation

supremum self decomposable random vectors an intrinsic definition of sup self decomposable random vectors is given it is proved that they are precisely the limits in distribution of certain normalized partial maxima of sequences of independent random vectors the main further result is a representation of sup self decomposable random vectors as functions of poisson processes which is the analogue of wolfe s 1982 representation of additively self decomposable random variables

the complexity of colouring problems on dense graphs abstract we consider the problem of deciding whether or not a graph has a vertex k colouring restricted to the class of graphs in which each vertex has degree at least n where is a fixed positive constant and n is the number of vertices of the graph we show that there is a deterministic polynomial time algorithm for this problem when k 3 k 2 and that this is best possible we also consider the corresponding enumeration problems

output regulation of linear heterogeneous multi agent systems via output and state feedback in this paper we propose output feedback and state feedback controllers for output regulation of linear heterogeneous multi agent systems we use a general model for the output regulation problem of linear multi agent systems which accounts for a general directed and fixed communication graph heterogeneity of the agents an arbitrary number of inputs and outputs and both state and output feedback first we derive global and local sufficient conditions for the existence of local output feedback and local state feedback controllers we show that both controllers have an h criterion as a part of their sufficient conditions second we design these controllers locally to insure those sufficient conditions third for the case of output feedback we show that the design procedure requires a feasible solution to the set of two lmis and one non convex algebraic constraint we provide an existing algorithm to find a feasible solution to this problem fourth for the case of state feedback we prove that the h criterion is satisfied if and only if there exists a feasible solution to an lmi moreover we define an output regulation region such that selecting the state feedback coupling gain from this region ensures that the sufficient condition is always satisfied

on the consensus and bipartite consensus in high order multi agent dynamical systems with antagonistic interactions abstract the aim of this paper is to address consensus and bipartite consensus for a group of homogeneous agents under the assumption that their mutual interactions can be described by a weighted signed connected and structurally balanced communication graph this amounts to assuming that the agents can be split into two antagonistic groups such that interactions between agents belonging to the same group are cooperative and hence represented by nonnegative weights while interactions between agents belonging to opposite groups are antagonistic and hence represented by nonpositive weights in this framework bipartite consensus can always be reached under the stabilizability assumption on the state space model describing the dynamics of each agent on the other hand nontrivial standard consensus may be achieved only under very demanding requirements both on the laplacian associated with the communication graph and on the agents description in particular consensus may be achieved only if there is a sort of equilibrium between the two groups both in terms of cardinality and in terms of the weights of the conflicting interactions amongst agents

cooperative output regulation of heterogeneous multi agent systems an h_ infty criterion in this note the cooperative output regulation problem of heterogeneous linear multi agent systems is investigated our approach is based on internal model and dynamic error feedback among the agents under some standard assumptions the problem is solved by an internal model and a stabilizing h controller we showed that the exo signal free system is stabilized if each controlled agent is robust in the sense of h norm to resist the disturbance measured by the second largest eigenvalue of the graph adjacency matrix then the synthesis problem is solved using algebraic riccati equation furthermore we showed that this condition is a generalization of existing results

a taxonomy of problems with fast parallel algorithms the class nc consists of problems solvable very fast in time polynomial in log n in parallel with a feasible polynomial number of processors many natural problems in nc are known in this paper an attempt is made to identify important subclasses of nc and give interesting examples in each subclass the notion of nc 1 reducibility is introduced and used throughout problem r is nc 1 reducible to problem s if r can be solved with uniform log depth circuits using oracles for s problems complete with respect to this reducibility are given for many of the subclasses of nc a general technique the parallel greedy algorithm is identified and used to show that finding a minimum spanning forest of a graph is reducible to the graph accessibility problem and hence is in nc 2 solvable by uniform boolean circuits of depth o log 2 n and polynomial size the class logcfl is given a new characterization in terms of circuit families the class det of problems reducible to integer determinants is defined and many examples given a new problem complete for deterministic polynomial time is given namely finding the lexicographically first maximal clique in a graph this paper is a revised version of s a cook 1983 in proceedings 1983 intl found comut sci conf lecture notes in computer science vol 158 pp 78 93 springer verlag berlin new york

constructing the highest degree subgraph for dense graphs in ncal abstract in this paper we first show that the highest degree subgraph problem remains p complete for dense graphs i e when m n 2 poly log n this hardness result gives a clear motivation in studying the approximability of the highest degree problem even for this restricted case we then provide an n b approximation scheme computing approximate solutions for dense graphs thus proving that in this case the problem belongs to the n b a s class

the maximum flow problem is log space complete for p the space complexity of the maximum flow problem is investigated it is shown that the problem is log space complete for deterministic polynomial time thus the maximum flow problem probably has no algorithm which needs only o logk n storage space for any constant k another consequence is that there is probably no fast parallel algorithm for the maximum flow problem

parallel construction of perfect matchings and hamiltonian cycles on dense graphs the intensive study of fast parallel and distributed algorithms for various routing and communications problems on graphs with good expanding properties ref pu1 ref pu2 ref ss is being carried out recently the parallel solutions for expanders that already exist ref pu 2 required an extensive randomization and an application of a randomized subroutine for the maximum matching problem in this paper we attack the problem of fast parallel algorithms for constructing perfect matchings and hamiltonian cycles on dense graph networks undirected graphs of minimal degree v over 2 somewhat surprisingly we design fast deterministic parallel algorithms for constructing both the perfect matchings and the hamiltonian cycles on the dense graphs the algorithm for constructing perfect matchings on dense graphs works in o log 2 n parallel time and o n 8 processors or o log 4 n parallel time and o n 4 processors on a crew pram the algorithm for constructing hamiltonian cycles on dense graphs works in o log 5 n parallel time and o n 4 processors on a crew pram our method of the parallel solution involves a development of new dense graph combinatorics suitable for fast parallelization

parallel matrix and graph algorithms matrix multiplication algorithms for cube connected and perfect shuffle computers are presented it is shown that in both these models two n times n matrices can be multiplied in o n m log m time when n 2 m 1 leqq m leqq n processing elements pes are available when only m 2 1 leqq m leqq n pes are available two n times n matrices can be multiplied in o n 2 m m n m 2 61 time it is shown that many graph problems can be solved efficiently using the matrix multiplication algorithms

consensus problems on networks with antagonistic interactions in a consensus protocol an agreement among agents is achieved thanks to the collaborative efforts of all agents expresses by a communication graph with nonnegative weights the question we ask in this paper is the following is it possible to achieve a form of agreement also in presence of antagonistic interactions modeled as negative weights on the communication graph the answer to this question is affirmative on signed networks all agents can converge to a consensus value which is the same for all agents except for the sign necessary and sufficient conditions are obtained to describe cases in which this is possible these conditions have strong analogies with the theory of monotone systems linear and nonlinear laplacian feedback designs are proposed