aggregation based cutting planes for packing and covering integer programs in this paper we study the strength of chvatal gomory cg cuts and more generally aggregation cuts for packing and covering integer programs ips aggregation cuts are obtained as follows given an ip formulation we first generate a single implied inequality using aggregation of the original constraints then obtain the integer hull of the set defined by this single inequality with variable bounds and finally use the inequalities describing the integer hull as cutting planes our first main result is to show that for packing and covering ips the cg and aggregation closures can be 2 approximated by simply generating the respective closures for each of the original formulation constraints without using any aggregations on the other hand we use computational experiments to show that aggregation cuts can be arbitrarily stronger than cuts from individual constraints for general ips the proof of the above stated results for the case of covering ips with bounds require the development of some new structural results which may be of independent interest finally we examine the strength of cuts based on k different aggregation inequalities simultaneously the so called multi row cuts and show that every packing or covering ip with a large integrality gap also has a large k aggregation closure rank in particular this rank is always at least of the order of the logarithm of the integrality gap

on the rational polytopes with chvatal rank 1 abstract in this paper we study the following problem given a polytope p with chvatal rank 1 does p contain an integer point boyd and pulleyblank observed that this problem is in thecomplexity class np co np an indication that it is probably not np complete we solvethis problem in polynomial time for polytopes arising from the satisfiability problem of a givenformula with at least three literals per clause for simplices whose integer hull can be obtainedby adding at most a constant number of chvatal inequalities and for rounded polytopes weprove that any closed convex set whose chvatal closure is empty has an integer width of atmost n and we give an example showing that this bound is tight within an additive constantof 1 the promise that a polytope has chvatal rank 1 seems hard to verify though we provethat deciding emptiness of the chvatal closure of a given rational polytope p is np complete even when p is contained in the unit hypercube or is a rational simplex and even when p doesnot contain any integer point this has two implications i it is np hard to decide whethera given rational polytope p has chvatal rank 1 even when p is contained in the unit cube oris a rational simplex ii the optimization and separation problems over the chvatal closureof a given rational polytope contained in the unit hypercube or of a given rational simplex arenp hard these results improve earlier complexity results of cornuejols and li and eisenbrand finally we prove that for any positive integer k it is np hard to decide whether adding atmost k chvatal inequalities is sufficient to describe the integer hull of a given rational polytope

on the set covering polytope i all the facets with coefficients in 0 1 2 while the set packing polytope through its connection with vertex packing has lent itself to fruitful investigations little is known about the set covering polytope we characterize the class of valid inequalities for the set covering polytope with coefficients equal to 0 1 or 2 and give necessary and sufficient conditions for such an inequality to be minimal and to be facet defining we show that all inequalities in the above class are contained in the elementary closure of the constraint set and that 2 is the largest value ofk such that all valid inequalities for the set covering polytope with integer coefficients no greater thank are contained in the elementary closure we point out a connection between minimal inequalities in the class investigated and certain circulant submatrices of the coefficient matrix finally we discuss conditions for an inequality to cut off a fractional solution to the linear programming relaxation of the set covering problem and to improve the lower bound given by a feasible solution to the dual of the linear programming relaxation

critical cutsets of graphs and canonical facets of set packing polytopes a cutset in a graph g with node set n is the set of edges joining the nodes in a subset of n to those not in the subset we call a cutset critical if its removal produces a graph whose independence number is greater than that of g we use the concept of a critical cutset to give first a necessary then a sufficient condition for an inequality with 0 1 coefficients to be a facet of the set packing polytope defined by g

equilibrium states for factor maps between subshifts abstract let x y be a factor map where x x and y y are subshifts over finite alphabets assume that x satisfies weak specification let a a 1 a 2 r 2 with a 1 0 and a 2 0 let f be a continuous function on x with sufficient regularity holder continuity for instance we show that there is a unique shift invariant measure on x that maximizes f d a 1 h x a 2 h 1 y in particular taking f 0 we see that there is a unique invariant measure on x that maximizes the weighted entropy a 1 h x a 2 h 1 y which answers an open question raised by gatzouras and peres 1996 in 15 an extension is given to high dimensional cases as an application we show that for each compact invariant set k on the k torus under a diagonal endomorphism if the symbolic coding of k satisfies weak specification then there is a unique invariant measure supported on k so that dim h dim h k

on the cut polytope the cut polytopep c g of a graphg v e is the convex hull of the incidence vectors of all edge sets of cuts ofg we show some classes of facet defining inequalities ofp c g we describe three methods with which new facet defining inequalities ofp c g can be constructed from known ones in particular we show that inequalities associated with chordless cycles define facets of this polytope moreover for these inequalities a polynomial algorithm to solve the separation problem is presented we characterize the facet defining inequalities ofp c g ifg is not contractible tok 5 we give a simple characterization of adjacency inp c g and prove that for complete graphs this polytope has diameter one and thatp c g has the hirsch property a relationship betweenp c g and the convex hull of incidence vectors of balancing edge sets of a signed graph is studied

valid inequalities for mixed integer linear programs this tutorial presents a theory of valid inequalities for mixed integer linear sets it introduces the necessary tools from polyhedral theory and gives a geometric understanding of several classical families of valid inequalities such as lift and project cuts gomory mixed integer cuts mixed integer rounding cuts split cuts and intersection cuts and it reveals the relationships between these families the tutorial also discusses computational aspects of generating the cuts and their strength

a comparison of the sherali adams lov 341sz schrijver and lasserre relaxations for 0 1 programming sherali and adams sa90 lov asz and schrijver ls91 and recently lasserre las01b have proposed lift and project methods for constructing hierarchies of successive linear or semidefinite relaxations of a 0 1 polytope p subseteq or n converging to p in n steps lasserre s approach uses results about representations of positive polynomials as sums of squares and the dual theory of moments we present the three methods in a common elementary framework and show that the lasserre construction provides the tightest relaxations of p as an application this gives a direct simple proof for the convergence of the lasserre s hierarchy we describe applications to the stable set polytope and to the cut polytope

sensitivity theorems in integer linear programming we consider integer linear programming problems with a fixed coefficient matrix and varying objective function and right hand side vector among our results we show that for any optimal solution to a linear program max wx ax b the distance to the nearest optimal solution to the corresponding integer program is at most the dimension of the problem multiplied by the largest subdeterminant of the integral matrixa using this we strengthen several integer programming proximity results of blair and jeroslow graver and wolsey we also show that the chvatal rank of a polyhedron x ax b can be bounded above by a function of the matrixa independent of the vectorb a result which as blair observed is equivalent to blair and jeroslow s theorem that each integer programming value function is a gomory function

sofic systems and encoding data techniques of symbolic dynamics are applied to prove the existence of codes suitable for certain input restricted channels this generalizes the earlier work of adler coppersmith and hassner on the same problem

on the facial structure of the set covering polytope dimensional linear programming given a bipartite graphg v u e a cover ofg is a subset d subseteq v with the property that each nodeu u is adjacent to at least one nodev d if a positive weightcv is associated with each nodev v the covering problem cp is to find a cover ofg having minimum total weight

relative entropy functions for factor maps between subshifts let x s and y t be topological dynamical systems and x y a factor map a function f c x is a compensation function for if p f o p for all c v for a factor code between subshifts of finite type we analyze the associated relative entropy function and give a necessary condition for the existence of saturated compensation functions necessary and sufficient conditions for a map to be a saturated compensation function will be provided

design and verify a new scheme for generating cutting planes a cutting plane procedure for integer programming ip problems usually involves invoking a black box procedure such as the gomory chvatal procedure to compute a cutting plane in this paper we describe an alternative paradigm of using the same cutting plane black box this involves two steps in the first step we design an inequality cx le d where c and d are integral independent of the cutting plane black box in the second step we verify that the designed inequality is a valid inequality by verifying that the set p cap x in mathbb r n mid cx ge d 1 cap mathbb z n is empty using cutting planes from the black box here p is the feasible region of the linear programming relaxation of the ip we refer to the closure of all cutting planes that can be verified to be valid using a specific cutting plane black box as the verification closure of the considered cutting plane black box this paper undertakes a systematic study of properties of verification closures of various cutting plane black box procedures

entropies and factorizations of topological markov shifts on presente une caracterisation des valeurs possibles pour l entropie topologique de decalages de markov si est un nombre de penon alors il y a une matrice non negative aperiodique entiere dont le rayon spectral est

lower bounds for lovasz schrijver systems and beyond follow from multiparty communication complexity we prove that an omega log 4 n lower bound for the three party number on the forehead nof communication complexity of the set disjointness function implies an n omega 1 size lower bound for treelike lovasz schrijver systems that refute unsatisfiable formulas in conjunctive normal form cnfs more generally we prove that an n omega 1 lower bound for the k 1 party nof communication complexity of set disjointness implies a 2 n omega 1 size lower bound for all treelike proof systems whose formulas are degree k polynomial inequalities

lattice closures of polyhedra given p subset mathbb r n a mixed integer set p i p cap mathbb z t times mathbb r n t and a k tuple of n dimensional integral vectors pi 1 ldots pi k where the last n t entries of each vector is zero we consider the relaxation of p i obtained by taking the convex hull of points x in p for which pi 1 tx ldots pi t kx are integral we then define the k dimensional lattice closure of p i to be the intersection of all such relaxations obtained from k tuples of n dimensional vectors when p is a rational polyhedron we show that given any collection of such k tuples there is a finite subcollection that gives the same closure more generally we show that any k tuple is dominated by another k tuple coming from the finite subcollection the k dimensional lattice closure contains the convex hull of p i and is equal to the split closure when k 1 therefore a result of cook et al math program 47 155 174 1990 implies that when p is a rational polyhedron the k dimensional lattice closure is a polyhedron for k 1 and our finiteness result extends this to all k ge 2 we also construct a polyhedral mixed integer set with n integer variables and one continuous variable such that for any k n finitely many iterations of the k dimensional lattice closure do not give the convex hull of the set our result implies that t branch split cuts cannot give the convex hull of the set nor can valid inequalities from unbounded full dimensional convex lattice free sets

binary clutter inequalities for integer programs we introduce a new class of valid inequalities for general integer linear programs called binary clutter bc inequalities they include the 0 frac 1 2 cuts of caprara and fischetti as a special case and have some interesting connections to binary matroids binary clutters and gomory corner polyhedra we show that the separation problem for bc cuts is strongly hard in general but polynomially solvable in certain special cases as a by product we also obtain new conditions under which 0 frac 1 2 cuts can be separated in polynomial time these ideas are then illustrated using the traveling salesman problem tsp as an example this leads to an interesting link between the tsp and two apparently unrelated problems the t join and max cut problems

on some polytopes contained in the 0 1 hypercube that have a small chvatal rank in this paper we consider polytopes p that are contained in the unit hypercube we provide conditions on the set of 0 1 vectors not contained in p that guarantee that p has a small chvatal rank our conditions are in terms of the subgraph induced by these infeasible 0 1 vertices in the skeleton graph of the unit hypercube in particular we show that when this subgraph contains no 4 cycle the chvatal rank is at most 3 and when it has tree width 2 the chvatal rank is at most 4 we also give polyhedral decomposition theorems when this graph has a vertex cutset of size one or two

nonparametric density estimation the l 1 view differentiation of integrals consistency lower bounds for rates of convergence rates of convergence in l1 the automatic kernel estimate l1 and pointwise convergence estimates related to the kernel estimate and the histogram estimate simulation inequalities and random variate generation the transformed kernel estimate applications in discrimination operations on density estimates estimators based on orthogonal series index

infinite to one codes and markov measures we study the structure of infinite to one continuous codes between subshifts of finite type and the behaviour of markov measures under such codes we show that if an infinite to one code lifts one markov measure to a markov measure then it lifts each markov measure to uncountably many markov measures and the fibre over each markov measure is isomorphic to any other fibre calling such a code markovian we characterize markovian codes through pressure we show that a simple condition on periodic points necessary for the existence of a code between two subshifts of finite type is sufficient to construct a markovian code several classes of markovian codes are studied in the process of proving illustrating and providing contrast to the main results a number of examples and counterexamples are given in particular we give a continuous code between two bernoulli shifts such that the defining vector of the image is not a clustering of the defining vector of the domain

factorization dynamics and coxeter toda lattices it is shown that the factorization relation on simple lie groups with standard poisson lie structure restricted to coxeter symplectic leaves gives an integrable dynamical system this system can be regarded as a discretization of the toda flow in case of sl n the integrals of the factorization dynamics are integrals of the relativistic toda system a substantial part of the paper is devoted to the description of symplectic leaves in simple complex lie groups its borel subgroups and their doubles

facets of the bipartite subgraph polytope the bipartite subgraph polytope pbg of a graph g v e is the convex hull of the incidence vectors of all edge sets of bipartite subgraphs of g we show that all complete subgraphs of g of odd order and all so called odd bicycle wheels contained in g induce facets of pbg moreover we describe several methods with which new facet defining inequalities of pbg can be constructed from known ones examples of these methods are contraction of node sets in odd complete subgraphs odd subdivision of edges certain splittings of nodes and subdivision of all edges of a cut using these methods we can construct facet defining inequalities of pbg having coefficients of order v 2

computing degree and class degree abstract let be a factor code from a one dimensional shift of finite type x onto an irreducible sofic shift y if is finite to one then the number of preimages of a typical point in y is an invariant called the degree of in this paper we present an algorithm to compute this invariant the generalized notion of the degree when is not limited to finite to one factor codes is called the class degree of the class degree of a code is defined to be the number of transition classes over a typical point of y and is invariant under topological conjugacy we show that the class degree is computable

0 1 polytopes with quadratic chvatal rank for a polytope p the chvatal closure p p is obtained by simultaneously strengthening all feasible inequalities cx with integral c to cx the number of iterations of this procedure that are needed until the integral hull of p is reached is called the chvatal rank if p 0 1 n then it is known that o n2 log n iterations always suffice and at least 1 1 e o 1 n iterations are sometimes needed leaving a huge gap between lower and upper bounds we prove that there is a polytope contained in the 0 1 cube that has chvatal rank n2 closing the gap up to a logarithmic factor in fact even a superlinear lower bound was mentioned as an open problem by several authors our choice of p is the convex hull of a semi random knapsack polytope and a single fractional vertex the main technical ingredient is linking the chvatal rank to simultaneous diophantine approximations w r t the 1 norm of the normal vector defining p

projection of markov measures may be gibbsian we study the induced measure obtained from a 1 step markov measure supported by a topological markov chain after the mapping of the original alphabet onto another one we give sufficient conditions for the induced measure to be a gibbs measure in the sense of bowen when the factor system is again a topological markov chain this amounts to constructing when it does exist the induced potential and proving its holder continuity this is achieved through a matrix method we provide examples and counterexamples to illustrate our results

on cutting plane proofs in combinatorial optimization abstract gomory s cutting plane technique can be viewed as a recursive procedure for proving the validity of linear inequalities over the set of all integer vectors in a prescribed polyhedron the number of rounds of cutting planes needed to obtain all valid linear inequalities is known as the rank of the polyhedron we prove that polyhedra featured in popular formulations of the stable set problem the set covering problem the set partitioning problem the knapsack problem the bipartite subgraph problem the maximum cut problem the acyclic subdigraph problem the asymmetric traveling salesman problem and the traveling salesman problem have arbitrarily high rank in particular we prove conjectures of barahona grotschel and mahjoub chvatal grotschel and pulleyblank and junger

lifted cycle inequalities for the asymmetric traveling salesman problem we investigate the family of facet defining inequalities for the asymmetric traveling salesman ats polytope obtainable by lifting the cycle inequalities we establish several properties of this family that earmark it as arguably the most important among the asymmetric inequalities for the ats polytope known to date in particular we give several results that characterize the pattern of lifting coefficients and thereby facilitate the identification of facet defining inequalities this characterization is used to investigate three new large classes of lifted cycle inequalities that we call shell fork and curtain inequalities these latter inequalities have unbounded chvatal rank furthermore the pattern of lifting coefficients that we identify makes it easy to develop efficient separation routines finally each member of the family is shown to have a counterpart for the symmetric ts sts polytope that is often new and is obtainable by mapping the inequality for the ats polytope into a certain face of the sts polytope and then lifting the resulting inequality into one for the sts polytope itself

transience bounds for long walks let g v e be a directed graph with rewards re for e e e we show that if there exists an s t walk in g of length k v 2 then there exists a maximum reward s t walk of length k with at least k v 2 arcs from a cycle with maximal mean reward if a is the reward incidence matrix of g then the maximum rewards of s t walks of length k can be determined by computing the k fold matrix product a k where the operations plus and times are replaced by max and plus denoted by and max algebra dynamic systems y k y k 1 a have been used to model certain transportation and automated manufacturing systems and can be interpreted as deterministic dynamic programming recursions if g is strongly connected then y k da y k da 1 for all k ka where is the maximum cycle mean in g and ka and da are the max algebra transient and period cyclicity of a for given y 0 the transient and period of the system may satisfy ka and da we derive a tight bound on ka and give polynomial algorithms for computing ka and and for solving deterministic dynamic programming problems we also give analogous results for walks of maximum discounted reward

class degree and relative maximal entropy given a factor code from a shift of finite type x onto an irreducible sofic shift y and a fully supported ergodic measure on y we give an explicit upper bound on the number of ergodic measures on x which project to and have maximal entropy among all measures in the fiber 1 this bound is invariant under conjugacy we relate this to an important construction for finite to one symbolic factor maps

on the existence of open and bi continuing codes given an irreducible sofic shift x we show that an irreducible shift of finite type y of lower entropy is a factor of x if and only if it is a factor of x by an open bi continuing code if these equivalent conditions hold and y is mixing then any code from a proper subshift of x to y can be extended to an open bi continuing code on x these results are still valid when x is assumed to be only an almost specified shift i e a subshift satisfying an irreducible version of the specification property

chva tal closures for mixed integer programming problems chvatal introduced the idea of viewing cutting planes as a system for proving that every integral solution of a given set of linear inequalities satisfies another given linear inequality this viewpoint has proven to be very useful in many studies of combinatorial and integer programming problems the basic ingredient in these cutting plane proofs is that for a polyhedronp and integral vectorw if max wx x p wx integer t thenwx t is valid for all integral vectors inp we consider the variant of this step where the requirement thatwx be integer may be replaced by the requirement that bar wx be integer for some other integral vector bar w the cutting plane proofs thus obtained may be seen either as an abstraction of gomory s mixed integer cutting plane technique or as a proof version of a simple class of the disjunctive cutting planes studied by balas and jeroslow our main result is that for a given polyhedronp the set of vectors that satisfy every cutting plane forp with respect to a specified subset of integer variables is again a polyhedron this allows us to obtain a finite recursive procedure for generating the mixed integer hull of a polyhedron analogous to the process of repeatedly taking chvatal closures in the integer programming case these results are illustrated with a number of examples from combinatorial optimization our work can be seen as a continuation of that of nemhauser and wolsey on mixed integer cutting planes

clique tree inequalities and the symmetric travelling salesman problem the linear programming cutting plane approach for solving the travelling salesman problem has recently proven to be highly successful cf crowder and padberg crowder h p m w padberg 1980 solving large scale symmetric travelling salesman problems to optimality management sci 26 495 509 grotschel grotschel m 1980a on the symmetric travelling salesman problem solution of a 120 city problem math programming stud 12 61 77 padberg and hong padberg m w s hong 1980 on the symmetric travelling salesman problem a computational study math programming stud 12 78 107 one of the reasons for this success is certainly the fact that instead of ordinary cutting planes gomory cuts etc problem specific cutting planes could be used which define facets of the underlying integer programming polytopes in this paper we shall define a new class of inequalities clique tree inequalities valid for the travelling salesman polytope which properly contains many of the known classes of inequalities like subtour elimination constraints 2 matching constraints comb inequalities and we show that all these new inequalities induce facets of the travelling salesman polytope since the general structure of these new inequalities is quite simple we hope that it will be possible to use the inequalities efficiently in cutting plane procedures for the travelling salesman problem

on the acyclic subgraph polytope the acyclic subgraph problem can be formulated as follows given a digraph with arc weights find a set of arcs containing no directed cycle and having maximum total weight we investigate this problem from a polyhedral point of view and determine several classes of facets for the associated acyclic subgraph polytope we also show that the separation problem for the facet defining dicycle inequalities can be solved in polynomial time this implies that the acyclic subgraph problem can be solved in polynomial time for weakly acyclic digraphs this generalizes a result of lucchesi for planar digraphs

testing the odd bicycle wheel inequalities for the bipartite subgraph polytope in their paper on facets of the bipartite subgraph polytope barahona grotschel and mahjoub pose the question is there a polynomial time algorithm testing the odd bicycle wheel inequalities this note describes such an algorithm

relative pressure relative equilibrium states compensation functions and many to one codes between subshifts soit s x x t y y des applications continues d espaces metrisables compacts et soit x y une surjection continue avec t o s t o on etudie la notion de pression relative et on etudie certaines fonctions de pression relative maximales qui s associent aux etats d equilibres relatifs on relie ces idees a la notion de fonction de compensation et on montre qu une fonction de compensation existe toujours quand s et t sont des sous decalages

on the 0 1 facets of the set covering polytope in this paper we consider inequalities of the form jxj where j equals 0 or 1 and is a positive integer we give necessary and sufficient conditions for such inequalities to define facets of the set covering polytope associated with a 0 1 constraint matrixa these conditions are in terms of critical edges and critical cutsets defined in the bipartite incidence graph ofa and are in the spirit of the work of balas and zemel on the set packing problem where similar notions were defined in the intersection graph ofa furthermore we give a polynomial characterization of a class of 0 1 facets defined from chorded cycles of the bipartite incidence graph this characterization also yields all the 0 1 liftings of odd hole inequalities for the simple plant location polytope