on the complexity of finding chordless paths in bipartite graphs and some interval operators in graphs and hypergraphs in this paper we show that the problem of finding a chordless path between a vertex s and a vertex t containing a vertex v remains np complete in bipartite graphs thereby strengthening the previous results on the same problem we show a relation between this problem and two interval operators the simple path interval operator in hypergraphs and the even chorded path interval operator in graphs we show that the problem of computing the two mentioned intervals is np complete

a short proof of zheludev s theorem we give a short proof of zheludev s theorem that states the existence of precisely one eigenvalue in sufficiently distant spectral gaps of a hill operator subject to certain short range perturbations as a by product we simultaneously recover rofe beketov s result about the finiteness of the number of eigenvalues in essential spectral gaps of the perturbed hill operator our methods are operator theoretic in nature and extend to other one dimensional systems such as perturbed periodic dirac operators and weakly perturbed second order finite difference operators we employ the trick of using a selfadjoint birman schwinger operator even in cases where the perturbation changes sign a method that has already been successfully applied in different contexts and appears to have further potential in the study of point spectra in essential spectral gaps

the asymmetric exclusion process and brownian excursions we consider the totally asymmetric exclusion process tasep in one dimension in its maximal current phase we show by an exact calculation that the non gaussian part of the fluctuations of density can be described in terms of the statistical properties of a brownian excursion numerical simulations indicate that the description in terms of a brownian excursion remains valid for more general one dimensional driven systems in their maximal current phase

lattice gas model in random medium and open boundaries hydrodynamic and relaxation to the steady state we consider a lattice gas interacting by the exclusion rule in the presence of a random field given by i i d bounded random variables in a bounded domain in contact with particles reservoir at different densities we show in dimensions d 3 that the rescaled empirical density field almost surely with respect to the random field converges to the unique weak solution of a quasilinear parabolic equation having the diffusion matrix determined by the statistical properties of the external random field and boundary conditions determined by the density of the reservoir further we show that the rescaled empirical density field in the stationary regime almost surely with respect to the random field converges to the solution of the associated stationary transport equation

some properties of graphs determined by edge zeta functions in 1989 hashimoto introduced an edge zeta function of a finite graph which is a generalization of the ihara zeta function the edge zeta function is the reciprocal of a polynomial in twice as many indeterminants as edges in the graph and can be computed via a determinant expression we look at graph properties which we can determine using the edge zeta function in particular the edge zeta function is enough to deduce the clique number the number of hamiltonian cycles and whether a graph is perfect or chordal finally we present a new example illustrating that the ihara zeta function cannot necessarily do the same

lattice gas models in contact with stochastic reservoirs local equilibrium and relaxation to the steady state extending the results of a previous work we consider a class of discrete lattice gas models in a finite interval whose bulk dynamics consists of stochastic exchanges which conserve the particle number and with stochastic dynamics at the boundaries chosen to model infinite particle reservoirs at fixed chemical potentials we establish here the local equilibrium structure of the stationary measures for these models further we prove as a law of large numbers that the time dependent empirical density field converges to a deterministic limit process which is the solution of the initial boundary value problem for a nonlinear diffusion equation

np hard graph problems and boundary classes of graphs any graph problem which is np hard in general graphs becomes polynomial time solvable when restricted to graphs in special classes when does a difficult problem become easy to answer this question we study the notion of boundary classes in the present paper we define this notion in its most general form describe several approaches to identify boundary classes and apply them to various graph problems

small deviations from local equilibrium for a process which exhibits hydrodynamical behavior i the symmetric simple exclusion process on with sources at l l e is considered the stationary measure l is studied in the limit asl diverges the first order correction to its limit is proven to be of order 1 l and it is explicitly computed the result is in agreement with the analysis of the model from the hydrodynamical point of view initiated in ref 1

reversible and irreversible information networks it is shown that there exist information networks where messages can be sent utilizing network coding more easily in one direction than in the opposite direction this is valid even though each channel is assumed to have the same capacity in both directions it is shown that irreversible information networks only have solutions that use nonlinear network coding this correspondence argues that this result is more surprising than it might appear at first sight and that it follows using ideas resembling the path integral in quantum mechanics

on the toda and kac van moerbeke hierarchies we provide a comprehensive treatment of the single and double commutation method as a tool for constructing soliton solutions of the toda and kac van moerbeke hierarchy on arbitrary background in addition we present a novel construction based on the single commutation method as an illustration we compute the n soliton solution of the toda and kac van moerbeke hierarchy

hydrodynamics of stationary non equilibrium states for some stochastic lattice gas models we consider discrete lattice gas models in a finite interval with stochastic jump dynamics in the interior which conserve the particle number and with stochastic dynamics at the boundaries chosen to model infinite particle reservoirs at fixed chemical potentials the unique stationary measures of these processes support a steady particle current from the reservoir of higher chemical potential into the lower and are non reversible we study the structure of the stationary measure in the hydrodynamic limit as the microscopic lattice size goes to infinity in particular we prove as a law of large numbers that the empirical density field converges to a deterministic limit which is the solution of the stationary transport equation and the empirical current converges to the deterministic limit given by fick s law

a combinatorial approach to nonlocality and contextuality so far most of the literature on quantum contextuality and the kochen specker theorem seems either to concern particular examples of contextuality or be considered as quantum logic here we develop a general formalism for contextuality scenarios based on the combinatorics of hypergraphs which significantly refines a similar recent approach by cabello severini and winter csw in contrast to csw we explicitly include the normalization of probabilities which gives us a much finer control over the various sets of probabilistic models like classical quantum and generalized probabilistic in particular our framework specializes to quantum nonlocality in the case of bell scenarios which arise very naturally from a certain product of contextuality scenarios due to foulis and randall in the spirit of csw we find close relationships to several graph invariants the recently proposed local orthogonality principle turns out to be a special case of a general principle for contextuality scenarios related to the shannon capacity of graphs our results imply that it is strictly dominated by a low level of the navascues pironio acin hierarchy of semidefinite programs which we also apply to contextuality scenarios

hybrid tractability of valued constraint problems the constraint satisfaction problem csp is a central generic problem in computer science and artificial intelligence it provides a common framework for many theoretical problems as well as for many real life applications valued constraint problems are a generalisation of the csp which allow the user to model optimisation problems considerable effort has been made in identifying properties which ensure tractability in such problems in this work we initiate the study of hybrid tractability of valued constraint problems that is properties which guarantee tractability of the given valued constraint problem but which do not depend only on the underlying structure of the instance such as being tree structured or only on the types of valued constraints in the instance such as submodularity we present several novel hybrid classes of valued constraint problems in which all unary constraints are allowed which include a machine scheduling problem constraint problems of arbitrary arities with no overlapping nogoods and the softalldiff constraint with arbitrary unary valued constraints an important tool in our investigation will be the notion of forbidden substructures

combinatorial symbolic powers abstract symbolic powers are studied in the combinatorial context of monomial ideals when the ideals are generated by quadratic squarefree monomials the generators of the symbolic powers are obstructions to vertex covering in the associated graph and its blowups as a result perfect graphs play an important role in the theory dual to the role played by perfect graphs in the theory of secants of monomial ideals we use grobner degenerations as a tool to reduce questions about symbolic powers of arbitrary ideals to the monomial case among the applications are a new unified approach to the grobner bases of symbolic powers of determinantal and pfaffian ideals

nonreversibility and equivalent constructions of multiple unicast networks we prove that for any finite directed acyclic network there exists a corresponding multiple unicast network such that for every alphabet each network is solvable if and only if the other is solvable and for every finite field alphabet each network is linearly solvable if and only if the other is linearly solvable the proof is constructive and creates an extension of the original network by adding exactly s 5m r 1 new nodes where in the original network m is the number of messages r is the average number of receiver nodes demanding each source message and s is the number of messages emitted by more than one source the construction is then used to create a solvable multiple unicast network which becomes unsolvable over every alphabet size if all of its edge directions are reversed and if the roles of source receiver pairs are reversed

zeros of the wronskian and renormalized oscillation theory for general sturm liouville operators with separated boundary conditions we prove the following if e 1 2 r and if u 1 2 solve the differential equation hu j e ju j j 1 2 and respectively satisfy the boundary condition on the left right then the dimension of the spectral projection p e 1 e 2 h of h equals the number of zeros of the wronskian of u 1 and u 2

the strong perfect graph theorem a graph g is perfect if for every induced subgraph h the chromatic number of h equals the size of the largest complete subgraph of h and g is berge if no induced subgraph of g is an odd cycle of length at least five or the complement of one the strong perfect graph conjecture berge 1961 asserts that a graph is perfect if and only if it is berge a stronger conjecture was made recently by conforti cornu ejols and vuiskovi c that every berge graph either falls into one of a few basic classes or admits one of a few kinds of separation designed so that a minimum counterexample to berge s conjecture cannot have either of these properties in this paper we prove both of these conjectures

spectral deformations of jacobi operators we extend recent work concerning isospectral deformations for one dimensional schrodinger operators to the case of jacobi operators we provide a complete spectral characterization of a new method that constructs isospectral deformations of a given jacobi operator hu n a n u n 1 a n 1 u n 1 b n u n our technique is connected to dirichlet data that is the spectrum of the operator h n0 on 2 n0 n0 with a dirichlet boundary condition at n0 the transformation moves a single eigenvalue of h n0 and perhaps flips which side of n0 the eigenvalue lives on the remainder of the spectrum the transformation is realized by a unitary operator

characterizing and bounding the imperfection ratio for some classes of graphs perfect graphs constitute a well studied graph class with a rich structure which is reflected by many characterizations with respect to different concepts perfect graphs are for instance precisely those graphs g where the stable set polytope stab g equals the fractional stable set polytope qstab g the dilation ratio rm min t rm qstab g subseteq t rm stab g of the two polytopes yields the imperfection ratio of g it is np hard to compute and for most graph classes it is even unknown whether it is bounded for graphs g such that all facets of stab g are rank constraints associated with antiwebs we characterize the imperfection ratio and bound it by 3 2 outgoing from this result we characterize and bound the imperfection ratio for several graph classes including near bipartite graphs and their complements namely quasi line graphs by means of induced antiwebs and webs respectively

packing and covering with linear programming a survey this paper considers the polyhedral results and the min max results on packing and covering problems of the decade since the strong perfect graph theorem published in 2006 the main such results are available for the packing problem however there are still important polyhedral questions that remain open for the covering problem the main questions are still open although there has been important progress we survey some of the main results with emphasis on those where linear programming and graph theory come together they mainly concern the covering of cycles or dicycles in graphs or signed graphs either with vertices or edges this includes the multicut and integral multiflow problems

on the switch markov chain for perfect matchings we study a simple markov chain the switch chain on the set of all perfect matchings in a bipartite graph this markov chain was proposed by diaconis graham and holmes as a possible approach to a sampling problem arising in statistics we ask for which hereditary classes of graphs is the markov chain ergodic and for which is it rapidly mixing we provide a precise answer to the ergodicity question and close bounds on the mixing question we show for the first time that the mixing time of the switch chain is polynomial in the case of monotone graphs a class that includes examples of interest in the statistical setting

semitotal domination new hardness results and a polynomial time algorithm for graphs of bounded mim width abstract a semitotal dominating set of a graph g with no isolated vertex is a dominating set d of g such that every vertex in d is within distance two of another vertex in d the minimum size t 2 g of a semitotal dominating set of g is squeezed between the domination number g and the total domination number t g semitotal dominating set is the problem of finding given a graph g a semitotal dominating set of g of size t 2 g in this paper we continue the systematic study on the computational complexity of this problem when restricted to special graph classes in particular we show that it is solvable in polynomial time for the class of graphs of bounded mim width by a reduction to total dominating set and we provide several approximation lower bounds for subclasses of subcubic graphs moreover we obtain complexity dichotomies in monogenic classes for the decision versions of semitotal dominating set and total dominating set finally we show that it is np complete to recognise the graphs such that t 2 g t g and those such that g t 2 g even if restricted to be planar and with maximum degree at most 4 and we provide forbidden induced subgraph characterisations for the graphs hereditarily satisfying either of these two equalities

list matrix partitions of chordal graphs it is well known that a clique with k 1 vertices is the only minimal obstruction to k colourability of chordal graphs a similar result is known for the existence of a cover by l cliques both of these problems are in fact partition problems restricted to chordal graphs the first seeks partitions into k independent sets and the second is equivalent to finding partitions into l cliques in an earlier paper we proved that a chordal graph can be partitioned into k independent sets and l cliques if and only if it does not contain an induced disjoint union of l 1 cliques of size k 1 a linear time algorithm for finding such partitions can be derived from the proof in this paper we expand our focus and consider more general partitions of chordal graphs for each symmetric matrix m over 0 1 the m partition problem seeks a partition of the input graph into independent sets cliques or arbitrary sets with certain pairs of sets being required to have no edges or to have all edges joining them as encoded in the matrix m moreover the vertices of the input chordal graph can be equipped with lists restricting the parts to which a vertex can be placed such list partitions generalize list colourings and list homomorphisms and arise frequently in the theory of graph perfection we show that many m partition problems that are np complete in general become solvable in polynomial time for chordal graphs even in the presence of lists on the other hand we show that there are m partition problems without lists that remain np complete for chordal graphs it is not known whether or not each list m partition problem is np complete or polynomial but it has been shown that each is np complete or quasi polynomial no log n for chordal graphs even this quasi dichotomy is not known but we do identify large families of matrices m for which dichotomy or at least quasi dichotomy holds we also discuss forbidden subgraph characterizations of graphs admitting an m partition such characterizations have recently been investigated for partitions of perfect graphs and we focus on highlighting the improvements one can obtain for the class of chordal rather than just perfect graphs

bulk diffusion in a system with site disorder we consider a system of random walks in a random environment interacting via exclusion the model is reversible with respect to a family of disordered bernoulli measures assuming some weak mixing conditions it is shown that under diffusive scaling the system has a deterministic hydrodynamic limit which holds for almost every realization of the environment the limit is a nonlinear diffusion equation with diffusion coefficient given by a variational formula the model is nongradient and the method used is the long jump variation of the standard nongradient method which is a type of renormalization the proof is valid in all dimensions

scattering theory for jacobi operators with quasi periodic background we develop direct and inverse scattering theory for jacobi operators which are short range perturbations of quasi periodic finite gap operators we show existence of transformation operators investigate their properties derive the corresponding gel fand levitan marchenko equation and find minimal scattering data which determine the perturbed operator uniquely

an upper bound on the number of eigenvalues of an infinite dimensional jacobi matrix an upper bound is given on the number of eigenvalues of a class of infinite dimensional jacobi matrices the theorem presented is a discrete analog of the celebrated result of v bargmann on the number of bound states of the schrodinger equation

the complexity of recognizing linear systems with certain integrality properties let a be a 0 1 matrix with precisely two 1 s in each column and let 1 be the all one vector we show that the problems of deciding whether the linear system a bf x ge bf 1 bf x ge bf 0 1 defines an integral polyhedron 2 is totally dual integral tdi and 3 box totally dual integral box tdi are all co np complete thereby confirming the conjecture on np hardness of recognizing tdi systems made by edmonds and giles in 1984

graph theoretical constructions for graph entropy and network coding based communications the guessing number of a directed graph digraph equivalent to the entropy of that digraph was introduced as a direct criterion on the solvability of a network coding instance this paper makes two contributions on the guessing number first we introduce an undirected graph on all possible configurations of the digraph referred to as the guessing graph which encapsulates the essence of dependence amongst configurations we prove that the guessing number of a digraph is equal to the logarithm of the independence number of its guessing graph therefore network coding solvability is no more a problem on the operations made by each node but is simplified into a problem on the messages that can transit through the network by studying the guessing graph of a given digraph and how to combine digraphs or alphabets we are thus able to derive bounds on the guessing number of digraphs second we construct specific digraphs with high guessing numbers yielding network coding instances where a large amount of information can transit we first propose a construction of digraphs with finite parameters based on cyclic codes with guessing number equal to the degree of the generator polynomial we then construct an infinite class of digraphs with arbitrary girth for which the ratio between the linear guessing number and the number of vertices tends to one despite these digraphs being arbitrarily sparse these constructions yield solvable network coding instances with a relatively small number of intermediate nodes for which the node operations are known and linear although these instances are sparse and the sources are arbitrarily far from their corresponding sinks

hydrodynamics and hydrostatics for a class of asymmetric particle systems with open boundaries we consider attractive particle systems in mathbb z d with product invariant measures we prove that when particles are restricted to a subset of mathbb z d with birth and death dynamics at the boundaries the hydrodynamic limit is given by the unique entropy solution of a conservation law with boundary conditions in the sense of bardos et al comm part diff equ 4 1017 1034 1979 for the hydrostatic limit between parallel hyperplanes we prove a multidimensional version of the phase diagram conjectured in popkov and schutz europhys lett 48 257 263 1999 and show that it is robust with respect to perturbations of the boundaries

hydrodynamic limit of a disordered lattice gas we consider a model of lattice gas dynamics in d in the presence of disorder if the particle interaction is only mutual exclusion and if the disorder field is given by i i d bounded random variables we prove the almost sure existence of the hydrodynamical limit in dimension d 3 the limit equation is a non linear diffusion equation with diffusion matrix characterized by a variational principle

parameterized complexity of vertex deletion into perfect graph classes vertex deletion problems are at the heart of parameterized complexity for a graph class f the f deletion problem takes as input a graph g and an integer k the question is whether it is possible to delete at most k vertices from g such that the resulting graph belongs to f whether perfect deletion is fixed parameter tractable and whether chordal deletion admits a polynomial kernel when parameterized by k have been stated as open questions in previous work we show that perfect deletion and weakly chordal deletion are w 2 hard when parameterized by k in search of positive results we study a restricted variant of the f deletion problem in this restricted variant the deleted vertices must be taken from a specified set x and we parameterize by x we show that for perfect deletion and weakly chordal deletion although this restriction immediately ensures fixed parameter tractability it is not enough to yield polynomial kernels unless np conp poly on the positive side for chordal deletion the restriction enables us to obtain a kernel with o x 4 vertices

nonlinear diffusion limit for a system with nearest neighbor interactions we consider a system of interacting diffusions the variables are to be thought of as charges at sites indexed by a periodic one dimensional lattice the diffusion preserves the total charge and the interaction is of nearest neighbor type with the appropriate scaling of lattice spacing and time a nonlinear diffusion equation is derived for the time evolution of the macroscopic charge density

oscillation theorems for discrete symplectic systems with nonlinear dependence in spectral parameter abstract in this paper we open a new direction in the study of discrete symplectic systems and sturm liouville difference equations by introducing nonlinear dependence in the spectral parameter we develop the notions of finite eigenvalues and finite eigenfunctions and their multiplicities and prove the corresponding oscillation theorem for dirichlet boundary conditions the present theory generalizes several known results for discrete symplectic systems which depend linearly on the spectral parameter our results are new even for special discrete symplectic systems namely for sturm liouville difference equations symmetric three term recurrence equations and linear hamiltonian difference systems

progress on perfect graphs a graph is perfect if for every induced subgraph the chromatic number is equal to the maximum size of a complete subgraph the class of perfect graphs is important for several reasons for instance many problems of interest in practice but intractable in general can be solved efficiently when restricted to the class of perfect graphs also the question of when a certain class of linear programs always have an integer solution can be answered in terms of perfection of an associated graph

riccati type transformations for second order linear difference equations abstract oscillation and comparison theorems for a linear homogeneous second order difference equation are proved by employing various equivalent non linear equations obtained by means of transformations analogous to the riccati transformation for ordinary differential equations

alternatives for testing total dual integrality in this paper we provide characterizing properties of totally dual integral tdi systems among others the following a system of linear inequalities is tdi if and only if its coefficient vectors form a hilbert basis and there exists a test set for the system s dual integer programs where all test vectors have positive entries equal to 1 reformulations of this provide relations between computational algebra and integer programming and they contain applegate cook and mccormick s sufficient condition for the tdi property and sturmfels theorem relating toric initial ideals generated by square free monomials to unimodular triangulations we also study the theoretical and practical efficiency and limits of the characterizations of the tdi property presented here in the particular case of set packing polyhedra our results correspond to endowing the weak perfect graph theorem with an additional computationally interesting geometric feature the normal fan of the stable set polytope of a perfect graph can be refined into a regular triangulation consisting only of unimodular cones

perfectness is an elusive graph property a graph property is called elusive or evasive if every algorithm for testing this property has to read in the worst case n choose 2 entries of the adjacency matrix of the given graph several graph properties have been shown to be elusive e g planarity or k colorability a famous conjecture of karp says that every nontrivial monotone graph property is elusive we prove that a nonmonotone but hereditary graph property is elusive perfectness

fluctuations in the weakly asymmetric exclusion process with open boundary conditions we investigate the fluctuations around the average density profile in the weakly asymmetric exclusion process with open boundaries in the steady state we show that these fluctuations are given in the macroscopic limit by a centered gaussian field and we compute explicitly its covariance function we use two approaches the first method is dynamical and based on fluctuations around the hydrodynamic limit we prove that the density fluctuations evolve macroscopically according to an autonomous stochastic equation and we search for the stationary distribution of this evolution the second approach which is based on a representation of the steady state as a sum over paths allows one to write the density fluctuations in the steady state as a sum over two independent processes one of which is the derivative of a brownian motion the other one being related to a random path in a potential

on the complexity of 4 coloring graphs without long induced paths we show that deciding if a graph without induced paths on nine vertices can be colored with 4 colors is an np complete problem improving a previous np completeness result proved by woeginger and sgall in 2001 the complexity of 4 coloring graphs without induced paths on five vertices remains open we show that deciding if a graph without induced paths or cycles on five vertices can be colored with 4 colors can be done in polynomial time a step in our algorithm uses the well known and deep fact due to grotschel lovasz and schrijver stating that perfect graphs can be optimally colored in polynomial time

regular inference as vertex coloring abstract this paper is concerned with the problem of supervised learning of deterministic finite state automata in the technical sense of identification in the limit from complete data by finding a minimal dfa consistent with a sample we solve this problem by translating it in its entirety to a vertex coloring problem essentially such a problem consists of two types of constraints that restrict the hypothesis space inequivalence and equivalence constraints the inequivalence constraints translate to the vertex coloring problem in a very natural way the equivalence constraints however greatly complicate the translation to vertex coloring in previous coloring based translations these were therefore encoded either dynamically by modifying the vertex coloring instance on the fly or by encoding them as satisfiability problems we provide the first translation that encodes both types of constraints together in a pure vertex coloring instance and prove the correctness of the construction the coloring approach offers many opportunities for applying insights from combinatorial optimization and graph theory to regular inference we immediately obtain new complexity bounds as well as a family of new learning algorithms which can be used to obtain exact hypotheses as well as fast approximations we also demonstrate that satisfying all inequivalence constraints in regular inference and vertex coloring are in some sense equally hard this immediately implies that regular inference is not approximable within s 1 s being the cardinality of the sample for any 1 2 unless p np it also indicates that solving only for inequivalence constraints is easier than the full regular inference problem

boundary properties of graphs for algorithmic graph problems the notion of a boundary graph property was recently introduced as a relaxation of that of a minimal property and was applied to several problems of both algorithmic and combinatorial nature in the present paper we first survey recent results related to this notion and then apply it to two algorithmic graph problems hamiltonian cycle and vertexk colorability in particular we discover the first two boundary classes for the hamiltonian cycle problem and prove that for any k 3 there is a continuum of boundary classes for vertexk colorability

trace formulas and inverse spectral theory for jacobi operators based on high energy expansions and herglotz properties of green and weyl m functions we develop a self contained theory of trace formulas for jacobi operators in addition we consider connections with inverse spectral theory in particular uniqueness results as an application we work out a new approach to the inverse spectral problem of a class of reflectionless operators producing explicit formulas for the coefficients in terms of minimal spectral data finally trace formulas are applied to scattering theory with periodic backgrounds

the linear guessing number of undirected graphs abstract this paper investigates linear guessing number introduced by riis in connection with the network coding theory this paper establishes properties of linear guessing number on undirected graphs in particular lower and upper bounds for linear guessing number are established finally linear guessing numbers of product graphs are investigated

discrete sampling a graph theoretic approach to orthogonal interpolation we study the problem of finding unitary submatrices of the n times n discrete fourier transform matrix in the context of interpolating a discrete bandlimited signal using an orthogonal basis this problem is related to a diverse set of questions on idempotents on mathbb z n and tiling mathbb z n in this work we establish a graph theoretic approach and connections to the problem of finding maximum cliques we identify the key properties of these graphs that make the interpolation problem tractable when n is a prime power and we identify the challenges in generalizing to arbitrary n finally we investigate some connections between graph properties and the spectral tile direction of the fuglede conjecture

structural investigation of piecewise linearized network flow problems in this work we study polyhedra in the context of network flow problems in which the flow value on each arc lies in one of several predefined intervals this is motivated by nonlinear problems on transportation networks where nonlinearities are handled by piecewise linear approximation or relaxation a common and established approach in many applications several methods for modeling piecewise linear functions which provide a complete description for a single network arc are known however in general this property is lost when considering multiple arcs we show how to strengthen the formulation for specific substructures consisting of multiple arcs by linear inequalities for the case of paths of degree two nodes we give a complete description of the polyhedron projected to the integer variables our model is based on but not limited to the connection between binary variables and flow variables that is used in the multiple choice method or the convex combination method we also show how to transfe

on the inapproximability of independent domination in 2p3 free perfect graphs we consider the complexity of approximation for the independent dominating set problem in 2p3 free graphs i e graphs that do not contain two disjoint copies of the chordless path on three vertices as an induced subgraph we show that if p np the problem cannot be approximated for 2p3 free graphs in polynomial time within a factor of n1 e for any constant e 0 where n is the number of vertices in the graph moreover we show that the result holds even if the 2p3 free graph is restricted to being weakly chordal and thereby perfect

cycle transversals in perfect graphs and cographs a cycle transversal or feedback vertex set of a graph g is a subset t v g such that t v c 0 for every cycle c of g this work considers the problem of finding special cycle transversals in perfect graphs and cographs we prove that finding a minimum cycle transversal t in a perfect graph g is np hard even for bipartite graphs with maximum degree four since g t is acyclic this result implies that finding a maximum acyclic induced subgraph of a perfect graph is also np hard other special properties of t are considered a clique stable respectively cycle transversal or simply cct sct respectively is a cycle transversal which is a clique stable set respectively recognizing graphs which admit a cct can be done in polynomial time however no structural characterization of such graphs is known even for perfect graphs we characterize cographs with cct in terms of forbidden induced subgraphs and describe their structure this leads to linear time recognition of cographs with cct we also prove that deciding whether a perfect graph admits an sct is np complete we characterize cographs with sct in terms of forbidden induced subgraphs this characterization also leads to linear time recognition

critical and subcritical jacobi operators defined as friedrichs extensions motivated by a treatment of the toda lattice and kac van moerbeke equations we study critical and subcritical 2nd order finite difference jacobi operators t on z in the course of our analysis we also derive an explicit characterization of the friedrichs extension tf of t in terms of principal solutions u of tu 0 near

renormalized oscillation theory for hamiltonian systems abstract we extend a result on renormalized oscillation theory originally derived for sturm liouville and dirac type operators on arbitrary intervals in the context of scalar coefficients to the case of general hamiltonian systems with block matrix coefficients in particular this contains the cases of general sturm liouville and dirac type operators with block matrix valued coefficients as special cases the principal feature of these renormalized oscillation theory results consists in the fact that by replacing solutions by appropriate wronskians of solutions oscillation theory now applies to intervals in essential spectral gaps where traditional oscillation theory typically fails

fixed parameter algorithms for cochromatic number and disjoint rectangle stabbing via iterative localization given a permutation p of 1 n and a positive integer k can p be partitioned into at most k subsequences each of which is either increasing or decreasing we give an algorithm with running time 2 o k 2 l o g k n o 1 that solves this problem thereby showing that it is fixed parameter tractable this np complete problem is equivalent to deciding whether the cochromatic number of a given permutation graph on n vertices is at most k our algorithm solves in fact a more general problem within the mentioned running time it decides whether the cochromatic number of a given perfect graph on n vertices is at most k to obtain our result we use a combination of two well known techniques within parameterized algorithms iterative compression and greedy localization consequently we name this combination iterative localization we further demonstrate the power of this combination by giving an algorithm with running time 2 o k 2 l o g k nlogn that decides whether a given set of n non overlapping axis parallel rectangles can be stabbed by at most k of a given set of horizontal and vertical lines

reconfiguration of colorable sets in classes of perfect graphs abstract a set of vertices in a graph is c colorable if the subgraph induced by the set has a proper c coloring in this paper we study the problem of finding a step by step transformation called a reconfiguration sequence between two c colorable sets in the same graph this problem generalizes the well studied independent set reconfiguration problem as the first step toward a systematic understanding of the complexity of this general problem we study the problem on classes of perfect graphs we first focus on interval graphs and give a combinatorial characterization of the distance between two c colorable sets this gives a linear time algorithm for finding an actual shortest reconfiguration sequence for interval graphs since interval graphs are exactly the graphs that are simultaneously chordal and co comparability we then complement the positive result by showing that even deciding reachability is pspace complete for chordal graphs and for co comparability graphs the hardness for chordal graphs holds even for split graphs we also consider the case where c is a fixed constant and show that in such a case the reachability problem is polynomial time solvable for split graphs but still pspace complete for co comparability graphs the complexity of this case for chordal graphs remains unsettled as by products our positive results give the first polynomial time solvable cases split graphs and interval graphs for feedback vertex set reconfiguration

reduction and fixed points of boolean networks and linear network coding solvability linear network coding transmits data through networks by letting the intermediate nodes combine the messages they receive and forward the combinations toward their destinations the solvability problem asks whether the demands of all the destinations can be simultaneously satisfied by using linear network coding the guessing number approach converts this problem into determining the number of fixed points of coding functions f a n to a n over a finite alphabet a usually referred to as boolean networks if a 0 1 with a given interaction graph that describes which local functions depend on which variables in this paper we generalize the so called reduction of coding functions in order to eliminate variables we then determine the maximum number of fixed points of a fully reduced coding function whose interaction graph has a loop on every vertex since the reduction preserves the number of fixed points we then apply these ideas and results to obtain four main results on the linear network coding solvability problem first we prove that non decreasing coding functions cannot solve any more instances than routing already does second we show that the triangle free undirected graphs are linearly solvable if and only if they are solvable by routing this is the first classification result for the linear network coding solvability problem third we exhibit a new class of non linearly solvable graphs fourth we determine large classes of strictly linearly solvable graphs

error correction for index coding with side information a problem of index coding with side information was first considered by birk and kol in 1998 in this study a generalization of index coding scheme where transmitted symbols are subject to errors is studied error correcting methods for such a scheme and their parameters are investigated in particular the following question is discussed given the side information hypergraph of index coding scheme and the maximal number of erroneous symbols what is the shortest length of a linear index code such that every receiver is able to recover the required information this question turns out to be a generalization of the problem of finding a shortest length error correcting code with a prescribed error correcting capability in the classical coding theory the singleton bound and two other bounds referred to as the bound and the bound for the optimal length of a linear error correcting index code ecic are established for large alphabets a construction based on concatenation of an optimal index code with a maximum distance separable classical code is shown to attain the singleton bound for smaller alphabets however this construction may not be optimal a random construction is also analyzed it yields another inexplicit bound on the length of an optimal linear ecic further the problem of error correcting decoding by a linear ecic is studied it is shown that in order to decode correctly the desired symbol the decoder is required to find one of the vectors belonging to an affine space containing the actual error vector the syndrome decoding is shown to produce the correct output if the weight of the error pattern is less or equal to the error correcting capability of the corresponding ecic finally the notion of static ecic which is suitable for use with a family of instances of an index coding problem is introduced several bounds on the length of static ecics are derived and constructions for static ecics are discussed connections of these codes to weakly resilient boolean functions are established

relative oscillation theory weighted zeros of the wronskian and the spectral shift function we develop an analog of classical oscillation theory for sturm liouville operators which rather than measuring the spectrum of one single operator measures the difference between the spectra of two different operators

lovasz schrijver sdp operator near perfect graphs and near bipartite graphs we study the lovasz schrijver lift and project operator mathrm text ls ls based on the cone of symmetric positive semidefinite matrices applied to the fractional stable set polytope of graphs the problem of obtaining a combinatorial characterization of graphs for which the mathrm text ls ls operator generates the stable set polytope in one step has been open since 1990 we call these graphs mathrm text ls ls perfect in the current contribution we pursue a full combinatorial characterization of mathrm text ls ls perfect graphs and make progress towards such a characterization by establishing a new close relationship among mathrm text ls ls perfect graphs near bipartite graphs and a newly introduced concept of full support perfect graphs

index coding with side information motivated by a problem of transmitting supplemental data over broadcast channels birk and kol infocom 1998 we study the following coding problem a sender communicates with n receivers r1 rn he holds an input x 0 01l n and wishes to broadcast a single message so that each receiver ri can recover the bit xi each ri has prior side information about x induced by a directed graph grain nodes ri knows the bits of a in the positions j i j is an edge of g g is known to the sender and to the receivers we call encoding schemes that achieve this goal indexcodes for 0 1 n with side information graph g in this paper we identify a measure on graphs the minrank which exactly characterizes the minimum length of linear and certain types of nonlinear index codes we show that for natural classes of side information graphs including directed acyclic graphs perfect graphs odd holes and odd anti holes minrank is the optimal length of arbitrary index codes for arbitrary index codes and arbitrary graphs we obtain a lower bound in terms of the size of the maximum acyclic induced subgraph this bound holds even for randomized codes but has been shown not to be tight

almost all webs are not rank perfect graphs with circular symmetry called webs are crucial for describing the stable set polytopes of two larger graph classes quasi line graphs 8 12 and claw free graphs 7 8 providing a complete linear description of the stable set polytopes of claw free graphs is a long standing problem 9 ben rebea conjectured a description for quasi line graphs see 12 chudnovsky and seymour 2 verified this conjecture recently for quasi line graphs not belonging to the subclass of fuzzy circular interval graphs and showed that rank facets are required in this case only fuzzy circular interval graphs contain all webs and even the problem of finding all facets of their stable set polytopes is open so far it is only known that stable set polytopes of webs with clique number 3 have rank facets only 5 17 while there are examples with clique number 4 having non rank facets 10 12 15

contraction and deletion blockers for perfect graphs and h free graphs abstract we study the following problem for given integers d k and graph g can we reduce some fixed graph parameter of g by at least d via at most k graph operations from some fixed set s as parameters we take the chromatic number clique number and independence number and as operations we choose edge contraction ec and vertex deletion vd we determine the complexity of this problem for s ec and s vd and for a number of subclasses of perfect graphs we use these results to determine the complexity of the problem for s ec and s vd and restricted to h free graphs

oscillation theory and renormalized oscillation theory for jacobi operators we provide a comprehensive treatment of oscillation theory for jacobi operators with separated boundary conditions our main results are as follows ifusolves the jacobi equation hu n a n u n 1 a n 1 u n 1 b n u n u n r in the weak sense on an arbitrary interval and satisfies the boundary condition on the left or right then the dimension of the spectral projectionp h ofhequals the number of nodes i e sign flips ifa n 0 ofu moreover we present a reformulation of oscillation theory in terms of wronskians of solutions thereby extending the range of applicability for this theory if 1 2 r and ifu1 2solve the jacobi equationhuj juj j 1 2 and respectively satisfy the boundary condition on the left right then the dimension of the spectral projectionp 1 2 h equals the number of nodes of the wronskian ofu1andu2 furthermore these results are applied to establish the finiteness of the number of eigenvalues in essential spectral gaps of perturbed periodic jacobi operators

colorings of hypergraphs perfect graphs and associated primes of powers of monomial ideals abstract there is a natural one to one correspondence between squarefree monomial ideals and finite simple hypergraphs via the cover ideal construction let h be a finite simple hypergraph and let j j h be its cover ideal in a polynomial ring r we give an explicit description of all associated primes of r j s for any power j s of j in terms of the coloring properties of hypergraphs arising from h we also give an algebraic method for determining the chromatic number of h proving that it is equivalent to a monomial ideal membership problem involving powers of j our work yields two new purely algebraic characterizations of perfect graphs independent of the strong perfect graph theorem the first characterization is in terms of the sets ass r j s while the second characterization is in terms of the saturated chain condition for associated primes

generalized prufer variables for perturbations of jacobi and cmv matrices abstract prufer variables are a standard tool in spectral theory developed originally for perturbations of the free schrodinger operator we adapt the generalized prufer variables introduced by kiselev remling simon for perturbations of an arbitrary schrodinger operator to the setting of jacobi and szeg recursions we present an application to random l 2 perturbations of jacobi and cmv matrices and an application to decaying oscillatory perturbations of periodic jacobi and cmv matrices

symmetric simple exclusion process regularity of the self diffusion coefficient we prove that the self diffusion coefficient of a tagged particle in the symmetric exclusion process in zd which is in equilibrium at density is of class c as a function of in the closed interval 0 1 the proof provides also a recursive method to compute the taylor expansion at the boundaries

permanents pfaffian orientations and even directed circuits given a 0 1 square matrix a when can some of the 1 s be changed toi1 s in such a way that the permanent of a equals the determinant of the modifled matrix when does a real square matrix have the property that every real matrix with the same sign pattern that is the corresponding entries either have the same sign or are both zero is nonsingular when is a hypergraph with n vertices and n hyperedges minimally nonbipartite when does a bipartite graph have a pfa an orientation given a digraph does it have no directed circuit of even length given a digraph does it have a subdivision with no even directed circuit it is known that all of the above problems are equivalent we prove a structural characterization of the feasible instances which implies a polynomial time algorithm to solve all of the above problems the structural characterization says roughly speaking that a bipartite graph has a pfa an orientation if and only if it can be obtained by piecing together in a specifled way planar bipartite graphs and one sporadic nonplanar bipartite graph

on algorithms for p 5 gem free graphs a graph is p5 gem free when it does not contain p5 an induced path with five vertices or a gem a graph formed by making an universal vertex adjacent to each of the four vertices of the induced path p4 as an induced subgraph we present o n2 time recognition algorithms for chordal gem free graphs and for p5 gem free graphs using a characterization of p5 gem free graphs by their prime graphs with respect to modular decomposition and their modular decomposition trees a brandstadt d kratsch on the structure of p5 gem free graphs discrete appl math 145 2005 155 166 we give linear time algorithms for the following np complete problems on p5 gem free graphs minimum coloring maximum weight stable set maximum weight clique and minimum clique cover