subset algebra lift operators for 0 1 integer programming we extend the sherali adams lovasz schrijver balas ceria cornuejols and lasserre lift and project methods for 0 1 optimization by considering liftings to subset algebras our methods yield polynomial time algorithms for solving a relaxation of a set covering problem at least as strong as that given by the set of all valid inequalities with small coefficients and more generally all valid inequalities where the right hand side is not very large relative to the positive coefficients in the left hand side applied to generalizations of vertex packing problems our methods yield in polynomial time relaxations that have unbounded rank using for example the n operator

additive one dimensional cellular automata are chaotic according to devaney s definition of chaos abstract we study the chaotic behavior of a particular class of dynamical systems cellular automata we specialize the definition of chaos given by devaney for general dynamical systems to the case of cellular automata a dynamical system x f is chaotic according to devaney s definition of chaos if its transition map f is sensitive to the initial conditions topologically transitive and has dense periodic orbits on x our main result is the proof that all the additive one dimensional cellular automata defined on a finite alphabet of prime cardinality are chaotic in the sense of devaney

the stable set problem and the lift and project ranks of graphs we study the lift and project procedures for solving combinatorial optimization problems as described by lovasz and schrijver in the context of the stable set problem on graphs we investigate how the procedures performances change as we apply fundamental graph operations we show that the odd subdivision of an edge and the subdivision of a star operations as well as their common generalization the stretching of a vertex operation cannot decrease the n 0 n or n rank of the graph we also provide graph classes which contain the complete graphs where these operations do not increase the n 0 or the n rank hence we obtain the ranks for these graphs and we also present some graph minor like characterizations for them despite these properties we give examples showing that in general most of these operations can increase these ranks finally we provide improved bounds for n ranks of graphs in terms of the number of nodes in the graph and prove that the subdivision of an edge or cloning a vertex can increase the n rank of a graph

solving lift and project relaxations of binary integer programs we propose a method for optimizing the lift and project relaxations of binary integer programs introduced by lovasz and schrijver in particular we study both linear and semidefinite relaxations the key idea is a restructuring of the relaxations which isolates the complicating constraints and allows for a lagrangian approach we detail an enhanced subgradient method and discuss its efficient implementation computational results illustrate that our algorithm produces tight bounds more quickly than state of the art linear and semidefinite solvers

transition phenomena in cellular automata rule space abstract we define several qualitative classes of cellular automata ca behavior based on various statistical measures and describe how the space of all cellular automata is organized as a cellular automaton is changed by varying entries in its rule table abrupt changes in qualitative behavior may occur these abrupt changes have the character of bifurcations in smooth dynamical systems or of phase transitions in statistical mechanical systems the most complex ca rules exhibit many of the characteristics of second order transitions suggesting an association between computation complexity and critical phenomena

on some variational approximations in two dimensional classical lattice systems a new derivation is presented of some variational approximations for classical lattice systems that belong to the class of cluster variation methods among them the well known bethe peierls and kramers wannier approximations the limiting behavior of a hierarchical sequence of cluster variation approximations the so calledc hierarchy is discussed it is shown that this hierarchy provides a monotonically decreasing sequence of upper boundsfn on the free energy per lattice sitef and thatfn f asn our results are based on extension theorems for states given on subsets of the lattice which might be of some independent interest and on an application of transfer matrix concepts to the variational characterization of translation invariant equilibrium states

evolving cellular automata to perform computations mechanisms and impediments abstract we present results from experiments in which a genetic algorithm ga was used to evolve cellular automata cas to perform a particular computational task one dimensional density classification we look in detail at the evolutionary mechanisms producing the ga s behavior on this task and the impediments faced by the ga in particular we identify four epochs of innovation in which new ca strategies for solving the problem are discovered by the ga describe how these strategies are implemented in ca rule tables and identify the ga mechanisms underlying their discovery the epochs are characterized by a breaking of the task s symmetries on the part of the ga the symmetry breaking results in a short term fitness gain but ultimately prevents the discovery of the most highly fit strategies we discuss the extent to which symmetry breaking and other impediments are general phenomena in any ga search

the probable value of the lovasz schrijver relaxations for maximum independent set lov asz and schrijver siam j optim 1 1991 pp 166 190 devised a lift and project method that produces a sequence of convex relaxations for the problem of finding in a graph an independent set or a clique of maximum size each relaxation in the sequence is tighter than the one before it while the first relaxation is already at least as strong as the lov asz theta function ieee trans inform theory 25 1979 pp 1 7 we show that on a random graph gn 1 2 the value of the rth relaxation in the sequence is roughly rule 0pt 7pt smash sqrt rule 0pt 7pt smash n 2 r almost surely it follows that for those relaxations known to be efficiently computable namely for r o 1 the value of the relaxation is comparable to the theta function furthermore a perfectly tight relaxation is almost surely obtained only at the r theta log n relaxation in the sequence

sum of squares hierarchy lower bounds for symmetric formulations we introduce a method for proving sum of squares sos lasserre hierarchy lower bounds when the initial problem formulation exhibits a high degree of symmetry our main technical theorem allows us to reduce the study of the positive semidefiniteness to the analysis of well behaved univariate polynomial inequalities we illustrate the technique on two problems one unconstrained and the other with constraints more precisely we give a short elementary proof of grigoriev laurent lower bound for finding the integer cut polytope of the complete graph we also show that the sos hierarchy requires a non constant number of rounds to improve the initial integrality gap of 2 for the min knapsack linear program strengthened with cover inequalities

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

universality and complexity in cellular automata abstract cellular automata are discrete dynamical systems with simple construction but complex self organizing behaviour evidence is presented that all one dimensional cellular automata fall into four distinct universality classes characterizations of the structures generated in these classes are discussed three classes exhibit behaviour analogous to limit points limit cycles and chaotic attractors the fourth class is probably capable of universal computation so that properties of its infinite time behaviour are undecidable

lower bound for the number of iterations in semidefinite hierarchies for the cut polytope hierarchies of semidefinite relaxations for 0 1 polytopes have been constructed by lasserre 2001a and by lovaisz and schrijver 1991 the cut polytope of a graph onn nodes can be expressed as a projection of such a semidefinite relaxation after at mostn steps we show that n 2 iterations are needed for finding the cut polytope of the complete graphk n

aperiodicity in one dimensional cellular automata abstract cellular automata are a class of mathematical systems characterized by discreteness in space time and state values determinism and local interaction a certain class of one dimensional binary site valued nearest neighbor automata is shown to generate infinitely many aperiodic temporal sequences from arbitrary finite initial conditions on an infinite lattice the class of automaton rules that generate aperiodic temporal sequences are characterized by a particular form of injectivity in their interaction rules included are the nontrivial linear automaton rules that is rules for which the superposition principle holds certain nonlinear automata with injectivity properties similar to those of linear automata and a wider subset of nonlinear automata whose interaction rules satisfy a weaker form of injectivity together with certain symmetry conditions the evolution of this last subset of automata can be viewed as consisting of multiple domains within which behavior exactly mimics that of a linear automaton a linearization technique is used to establish that these domains coalesce in finite time to produce at most two domains and thereby to assert the aperiodicity of their temporal sequences

criticality in cellular automata abstract using recent results obtained for the transition to turbulence via spatiotemporal intermittency in extended dynamical systems critical cellular automata rules are built thanks to a systematic procedure the continuous phase transition observed in a coupled map lattice is translated into a sequence of cellular automata rules for which the critical properties of the original system are reproduced as precisely as desired it is shown that criticality as understood from the point of view of statistical mechanics is intimately related to the various characteristics of wolfram s class iv rules this suggests in turn the picture of a critical surface in the space of rules and provides the basis for a discussion of the problem of classification schemes we review recent cellular automata studies where questions linked to criticality arise and argue that they are unified in light of our results leaving the relationship between computational and statistical characterizations of critical rules as the central problem for future studies on this subject

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

semidefinite programming in combinatorial optimization we discuss the use of semidefinite programming for combinatorial optimization problems the main topics covered include i the lovasz theta function and its applications to stable sets perfect graphs and coding theory ii the automatic generation of strong valid inequalities iii the maximum cut problem and related problems and iv the embedding of finite metric spaces and its relationship to the sparsest cut problem

on lovasz schrijver lift and project procedures on the dantzig fulkerson johnson relaxation of the tsp we study the lovasz schrijver lift and project procedure n on the linear relaxation of the dantzig fulkerson johnson formulation of the traveling salesman problem tsp a long standing conjecture states that the integrality gap of this relaxation is frac43 in the case of metric costs in this paper we show that the n rank of 2 matching inequalities relative to this relaxation can be arbitrarily high and obtain as a corollary that even after applying n to the relaxation a fixed number of times the integrality gap of the resulting relaxation is at least frac43

lift project systems performing on the partial vertex cover polytope abstract we study integrality gap ig lower bounds on strong lp and sdp relaxations derived by the sherali adams sa lovasz schrijver sdp ls sherali adams sdp sa and lasserre sdp la lift and project l p systems for the t partial vertex cover t pvc problem a variation of the classic vertex cover problem in which only t edges need to be covered t pvc admits a 2 approximation using various algorithmic techniques all relying on a natural lp relaxation with starting point this lp relaxation our main results assert that for every 0 level n lps or sdps derived by all known l p systems that have been used for positive algorithmic results but the lasserre hierarchy have igs at least 1 n t where n is the number of vertices of the input graph our lower bounds are nearly tight in that level n relaxations even of the weakest systems have integrality gap 1 additionally we give a o n integrality gap for the level 1 lasserre system and a superconstant general integrality gap for all level n lasserre derived sdps as lift and project systems have given the best algorithms known for numerous combinatorial optimization problems our results show that restricted yet powerful models of computation derived by many l p systems fail to witness c approximate solutions to t pvc for any constant c and for t o n as further motivation for our results we show that the sdp that has given the best algorithm known for t pvc has integrality gap n t on instances that can be solved by the level 1 lp relaxation derived by the ls system this constitutes another rare phenomenon where even in specific instances a static lp outperforms an sdp that has been used for the best approximation guarantee for the problem at hand finally we believe our results are of independent interest as they are among the very few known integrality gap lower bounds for lp and sdp 0 1 relaxations in which not all variables possess the same semantics in the underlying combinatorial optimization problem to achieve our results we utilize a common methodology of constructing solutions to lp relaxations that almost trivially satisfy constraints derived by all sdp l p systems known to be useful for algorithmic positive results except the la system the latter sheds some light as to why la tightenings seem strictly stronger than ls or sa tightenings

a hierarchical classification of cellular automata abstract a central issue in the theory of cellular automata is classification a classification is a structure imposed on the space of automata which groups together cellular automata with related properties in this paper a classification based on the action of cellular automata on n step markov measures is presented this classification is hierarchical and parametric an algorithm is described which efficiently constructs all rules in a given class at a given level of the hierarchy the utility of the classification is explored in the study of the invariant statistical properties of next nearest neighbor cellular automata on two states per cell

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

when does the positive semidefiniteness constraint help in lifting procedures we study the lift and project procedures of lovaisz and schrijver for 0 1 integer programming problems we prove that the procedure using the positive semidefiniteness constraint is not better than the one without it in the worst case various examples are considered we also provide geometric conditions characterizing when the positive semidefiniteness constraint does not help

the free boundary euler equations with large surface tension abstract we study the free boundary euler equations with surface tension in three spatial dimensions showing that the equations are well posed if the coefficient of surface tension is positive then we prove that under natural assumptions the solutions of the free boundary motion converge to solutions of the euler equations in a domain with fixed boundary when the coefficient of surface tension tends to infinity

cones of matrices and set functions and 0 1 optimization it has been recognized recently that to represent a polyhedron as the projection of a higher dimensional but simpler polyhedron is a powerful tool in polyhedral combinatorics a general method is developed to construct higher dimensional polyhedra or in some cases convex sets whose projection approximates the convex hull of 0 1 valued solutions of a system of linear inequalities an important feature of these approximations is that one can optimize any linear objective function over them in polynomial time in the special case of the vertex packing polytope a sequence of systems of inequalities is obtained such that the first system already includes clique odd hole odd antihole wheel and orthogonality constraints in particular for perfect and many other graphs this first system gives the vertex packing polytope for various classes of graphs including t perfect graphs it follows that the stable set polytope is the projection of a polytope with a polynomial number of facets an extension o

on a representation of the matching polytope via semidefinite liftings we consider the relaxation of the matching polytope defined by the non negativity and degree constraints we prove that given an undirected graph on n nodes and the corresponding relaxation of the matching polytope n 2 iterations of the lovasz schrijver semidefinite lifting procedure are needed to obtain the matching polytope in the worst case we show that n 2 iterations of the procedure always suffice

a new lift and project operator abstract in this paper we analyze the strength of split cuts in a lift and project framework we first observe that the lovasz schrijver and sherali adams lift and project operator hierarchies can be viewed as applying specific 0 1 split cuts to an appropriate extended formulation and demonstrate how to strengthen these hierarchies using additional split cuts more precisely we define a new operator that adds all 0 1 split cuts to the extended formulation for 0 1 mixed integer sets with k binary variables this new operator is guaranteed to obtain the integer hull in k 2 steps compared to k steps for the lovasz schrijver or the sherali adams operator we also present computational results on the stable set problem with our new operator

is there a sharp phase transition for deterministic cellular automata abstract previous work has suggested that there is a kind of phase transition between deterministic automata exhibiting periodic behavior and those exhibiting chaotic behavior however unlike the usual phase transitions of physics this transition takes place over a range of values of the parameter rather than at a specific value the present paper asks whether the transition can be made sharp either by taking the limit of an infinitely large rule table or by changing the parameter in terms of which the space of automata is explored we find strong evidence that for the class of automata we consider the transition does become sharp in the limit of an infinite number of symbols the size of the neighborhood being held fixed our work also suggests an alternative parameter in terms of which it is likely that the transition will become fairly sharp even if one does not increase the number of symbols in the course of our analysis we find that mean field theory which is our main tool gives surprisingly good predictions of the statistical properties of the class of automata we consider

approximation limits of linear programs beyond hierarchies we develop a framework for proving approximation limits of polynomial size linear programs lps from lower bounds on the nonnegative ranks of suitably defined matrices this framework yields unconditional impossibility results that are applicable to any lp as opposed to only programs generated by hierarchies using our framework we prove that o n1 2 approximations for clique require lps of size 2n this lower bound applies to lps using a certain encoding of clique as a linear optimization problem moreover we establish a similar result for approximations of semidefinite programs by lps our main technical ingredient is a quantitative improvement of razborov s 38 rectangle corruption lemma for the high error regime which gives strong lower bounds on the nonnegative rank of shifts of the unique disjointness matrix

tighter linear and semidefinite relaxations for max cut based on the lovasz schrijver lift and project procedure we study how the lift and project method introduced by lovasz and schrijver siam j optim 1 1991 pp 166 190 applies to the cut polytope we show that the cut polytope of a graph can be found in k iterations if there exist k edges whose contraction produces a graph with no k5 minor therefore for a graph g with n ge 4 nodes with stability number alpha g n 4 iterations suffice instead of the m number of edges iterations required in general and under some assumption n alpha g 3 iterations suffice the exact number of needed iterations is determined for small n le 7 by a detailed analysis of the new relaxations if positive semidefiniteness is added to the construction then one finds in one iteration a relaxation of the cut polytope which is tighter than its basic semidefinite relaxation and than another one introduced recently by anjos and wolkowicz discrete appl math to appear we also show how the lovasz schrijver relaxations for the stable set polytope of g can be strengthened using the corresponding relaxations for the cut polytope of the graph g nabla obtained from g by adding a node adjacent to all nodes of g

applications of ergodic theory and sofic systems to cellular automata abstract we indicate a mathematical framework for analysing the evolution of cellular automata those automata obeying an additive rule are shown to be the same as endomorphisms of a compact abelian group and therefore their statistical and dynamical behavior can be told exactly by using fourier analysis and ergodic theory those obeying certain nonlinear rules are closely tied to a finitely described object called a sofic system but the underlying statistics appear to be more complicated and interesting we conclude by formulating several conjectures about one such system

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

local structure theory for cellular automata abstract the mean field theory for cellular automata wolfram and schulman and seiden is generalized to the local structure theory the local structure theory is a sequence of finitely parameterized models of the statistical features of a cellular automaton s evolution the n th model in the sequence takes into account correlations in terms of the probability of blocks of n states a class of measures the n block measures is introduced the local structure operator of order n maps n block measures to n block measures in a manner which reflects the cellular automaton map on blocks of states the fixed points of the map on measures approximate the invariant measures of the cellular automaton the ability of the local structure theory to model evolution from uncorrelated initial distributions is studied the theory gives exact results in simple cases in more complex cases monte carlo numerical experiments suggest that an accurate statistical portrait of cellular automaton evolution is obtained the invariant measures of a cellular automaton and the stability of these measures may be obtained from the local structure theory the local structure theory appears to be a powerful method for characterization and classification of cellular automata nearest neighbor cellular automata with two states per cell are studied using this method

on the matrix cut rank of polyhedra lovasz and schrijver 1991 described a semidefinite operator for generating strong valid inequalities for the 0 1 vectors in a prescribed polyhedron among their results they showed that n iterations of the operator are sufficient to generate the convex hull of 0 1 vectors contained in a polyhedron in n space we give a simple example having chvatal rank 1 that meets this worst case bound of n we describe another example requiring n iterations even when combining the semidefinite and gomory chvatal operators this second example is used to show that the standard linear programming relaxation of a k city traveling salesman problem requires at least k 8 iterations of the combined operator this bound is best possible up to a constant factor as k 1 iterations suffice

approximate fixed rank closures of covering problems consider a 0 1 integer program min ctx ax b x 0 1 n where a is nonnegative we show that if the number of minimal covers of ax b is polynomially bounded then for any e 0 and any fixed q there is a polynomially large lift and project relaxation whose value is at least 1 e times the value of the rank q relaxation a special case of this result is that given by set covering problems or generally problems where the coefficients in a and b are bounded

computation of the lasserre ranks of some polytopes over the years various lift and project methods have been proposed to construct hierarchies of successive linear or semidefinite relaxations of a 0 1 polytope p rn that converge to p in n steps many such methods have been shown to require n steps in the worst case in this paper we show that the method of lasserre also requires n steps in the worst case

towards the classification of all boolean cellular automata abstract a classification scheme for all boolean cellular automata models involving an arbitrary number of inputs is proposed the scheme is based upon counting the number of stable points for a given rule i e points whose time development is stable with respect to small perturbations when applied to all cellular automata with 2 3 4 or 5 boolean inputs the latter involving more than 4000 million cases it yields in each case a nontrivial classification i e there is no single class in which nearly all the rules lie in addition it is shown how rules satisfying a given boolean differential equation can easily be obtained

exponential lower bounds on the lengths of some classes of branch and cut proofs we examine the complexity of branch and cut proofs in the context of 0 1 integer programs we establish an exponential lower bound on the length of branch and cut proofs that use 0 1 branching and lift and project cuts called simple disjunctive cuts by some authors gomory chvatal cuts and cuts arising from the n0 matrix cut operator of lovasz and schrijver a consequence of the lower bound result in this paper is that branch and cut methods of the type described above have exponential running time in the worst case

cellular automata in fuzzy backgrounds abstract the main purpose of this work is to understand some limitations introduced by the classical definitions of cellular automata ca to this end we have defined a new model of cas fuzzy ca s which allows the observation of interesting chaotic properties of elementary cas to date neither a formal nor a precise definition of chaos in cas exists we believe that the proposed model provides a sharper tool to detect which properties can be associated to a chaotic behavior we also define a measure rule entropy which gives information about the ca s dynamics solely on the basis of the rule table and provides theoretical explanations to some of the empirical observations

tightening simple mixed integer sets with guaranteed bounds in this paper we study how to reformulate knapsack sets and simple mixed integer sets in order to obtain provably tight polynomially large formulations

theoretical challenges towards cutting plane selection while many classes of cutting planes are at the disposal of integer programming solvers our scientific understanding is far from complete with regards to cutting plane selection i e the task of selecting a portfolio of cutting planes to be added to the lp relaxation at a given node of the branch and bound tree in this paper we review the different classes of cutting planes available known theoretical results about their relative strength important issues pertaining to cut selection and discuss some possible new directions to be pursued in order to accomplish cutting plane selection in a more principled manner finally we review some lines of work that we undertook to provide a preliminary theoretical underpinning for some of the issues related to cut selection