constrained likelihood for reconstructing a directed acyclic gaussian graph directed acyclic graphs are widely used to describe directional pairwise relations such relations are estimated by reconstructing a directed acyclic graph s structure which is challenging when the ordering of nodes of the graph is unknown in such a situation existing methods such as the neighbourhood and search and score methods have high estimation errors or computational complexities especially when a local or sequential approach is used to enumerate edge directions by testing or optimizing a criterion locally as a local method may break down even for moderately sized graphs we propose a novel approach to simultaneously identifying all estimable directed edges and model parameters using constrained maximum likelihood with nonconvex constraints we develop a constraint reduction method that constructs a set of active constraints from super exponentially many constraints this coupled with an alternating direction method of multipliers and a difference convex method permits efficient computation for large graph learning we show that the proposed method consistently reconstructs identifiable directions of the true graph and achieves the optimal performance in terms of parameter estimation numerically the method compares favourably with competitors a protein network is analysed to demonstrate that the proposed method can make a difference in identifying the network s structure

a note on clique web facets for multicut polytopes in this note we provide a previously undiscovered necessary condition for the facet defining property of clique web inequalities for the multicut polytope this condition imposes a minimum cardinality requirement on the node set of the clique thus implying in general that clique web inequalities associated with relatively small cliques are not facet defining for multicut polytopes

aggregation and mixed integer rounding to solve mips in this paper we discuss the use of mixed integer rounding mir inequalities to solve mixed integer programs mir inequalities are essentially gomory mixed integer cuts however as we wish to use problem structure we insist that mir inequalities be generated from constraints or simple aggregations of constraints of the original problem this idea is motivated by the observation that several strong valid inequalities based on specific problem structure can be derived as mir inequalities here we present and test a separation routine for such mir inequalities that includes a heuristic row aggregation procedure to generate a single knapsack plus continuous variables constraint complementation of variables and finally the generation of an mir inequality inserted in a branch and cut system the results suggest that such a routine is a useful additional tool for tackling a variety of mixed integer programming problems

facets and algorithms for capacitated lot sizing the dynamic economic lot sizing model which lies at the core of numerous production planning applications is one of the most highly studied models in all of operations research and yet capacitated multi item versions of this problem remain computationally elusive we study the polyhedral structure of an integer programming formulation of a single item capacitated version of this problem and use these results to develop solution methods for multi item applications in particular we introduce a set of valid inequalities for the problem and show that they define facets of the underlying integer programming polyhedron computational results on several single and multiple product examples show that these inequalities can be used quite effectively to develop an efficient cutting plane branch and bound procedure moreover our results show that in many instances adding certain of these inequalities a priori to the problem formulation and avoiding the generation of cutting planes can be equally effective

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

0 1 2 chva tal gomory cuts given the integer polyhedronp t conv x n ax b wherea m n andb m achvatal gomory cg cut is a valid inequality forp 1 of the type ax b for some such that a n in this paper we study 0 1 2 cg cuts arising for 0 1 2 m we show that the associated separation problem 0 1 2 sep is equivalent to finding a minimum weight member of a binary clutter this implies that 0 1 2 sep is np complete in the general case but polynomially solvable whena is related to the edge path incidence matrix of a tree we show that 0 1 2 sep can be solved in polynomial time for a convenient relaxation of the systemax b this leads to an efficient separation algorithm for a subclass of 0 1 2 cg cuts which often contains wide families of strong inequalities forp 1 applications to the clique partitioning asymmetric traveling salesman plant location acyclic subgraph and linear ordering polytopes are briefly discussed

on the monotonization of polyhedra in polyhedral combinatorics one often has to analyze the facial structure of less than full dimensional polyhedra the presence of implicit or explicit equations in the linear system defining such a polyhedron leads to technical difficulties when analyzing its facial structure it is therefore customary to approach the study of such a polytopep through the study of one of its full dimensional relaxations monotonizations known as the submissive and the dominant ofp finding sufficient conditions for an inequality that induces a facet of the submissive or the dominant of a polyhedron to also induce a facet of the polyhedron itself has been posed in the literature as an important research problem our paper goes a long way towards solving this problem we address the problem in the framework of a generalized monotonization of a polyhedronp g mon p that subsumes both the submissive and the dominant and give a sufficient condition for an inequality that defines a facet of g mon p to define a facet ofp for the important cases of the traveling salesman ts polytope in both its symmetric and asymmetric variants and of the linear ordering polytope we give sufficient conditions trivially easy to verify for a facet of the monotone completion to define a facet of the original polytope itself

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

on the exact solution of a facility layout problem we consider the layout problem of arranging a number of departments on a line this problem is known as single row facility layout the problem is very difficult to be solved in fact it has as a particular case the linear ordering problem which is strongly np hard in this paper a new mixed integer linear programming model is proposed for the problem theoretical arguments as well as computational results are given which demonstrate the efficiency of the new model relatively to previous mixed integer linear programming models proposed for the problem

clustering qualitative data based on binary equivalence relations neighborhood search heuristics for the clique partitioning problem the clique partitioning problem cpp requires the establishment of an equivalence relation for the vertices of a graph such that the sum of the edge costs associated with the relation is minimized the cpp has important applications for the social sciences because it provides a framework for clustering objects measured on a collection of nominal or ordinal attributes in such instances the cpp incorporates edge costs obtained from an aggregation of binary equivalence relations among the attributes we review existing theory and methods for the cpp and propose two versions of a new neighborhood search algorithm for efficient solution the first version ns r uses a relocation algorithm in the search for improved solutions whereas the second ns ts uses an embedded tabu search routine the new algorithms are compared to simulated annealing sa and tabu search ts algorithms from the cpp literature although the heuristics yielded comparable results for some test problems the neighborhood search algorithms generally yielded the best performances for large and difficult instances of the cpp

median linear orders heuristics and a branch and bound algorithm abstract after recalling some facts about the median linear orders problem two heuristics are studied to solve it then a branch and bound method is designed which computes all the solutions

new bounds for the max k cut and chromatic number of a graph we consider several semidefinite programming relaxations for the max k cut problem with increasing complexity the optimal solution of the weakest presented semidefinite programming relaxation has a closed form expression that includes the largest laplacian eigenvalue of the graph under consideration this is the first known eigenvalue bound for the max k cut when k 2 that is applicable to any graph this bound is exploited to derive a new eigenvalue bound on the chromatic number of a graph for regular graphs the new bound on the chromatic number is the same as the well known hoffman bound however the two bounds are incomparable in general we prove that the eigenvalue bound for the max k cut is tight for several classes of graphs we investigate the presented bounds for specific classes of graphs such as walk regular graphs strongly regular graphs and graphs from the hamming association scheme

clustering data that are graph connected abstract a new combinatorial model for clustering is proposed for all applications in which individual and relational data are available individual data refer to the intrinsic features of units they are stored in a matrix d and are the typical input of all clustering algorithms proposed so far relational data refer to the observed links between units representing social ties such as friendship joint participation to social events and so on relational data are stored in the graph g v e and the data available for clustering are the triplet g v e d called attributed graph known clustering algorithms can take advantage of the relational structure of g to redefine and refine the units membership for example uncertain membership of units to groups can be resolved using the sociological principle that ties are more likely to form between similar units the model proposed here shows how to take into account the graph information combining the clique partitioning objective function a known clustering methodology with connectivity as the structural constraint of the resulting clusters the model can be formulated and solved using integer linear programming and a new family of cutting planes moderate size problems are solved and heuristic procedures are developed for instances in which the optimal solution can only be approximated finally tests conducted on simulated data show that the clusters quality is greatly improved through this methodology

new results on the windy postman problem in this paper we study the windy postman problem wpp this is a well known arc routing problem that contains the mixed chinese postman problem mcpp as a special case we extend to arbitrary dimension some new inequalities that complete the description of the polyhedron associated with the windy postman problem over graphs with up to four vertices and ten edges we introduce two new families of facet inducing inequalities and prove that these inequalities along with the already known odd zigzag inequalities are chvatal gomory inequalities of rank at most 2 moreover a branch and cut algorithm that incorporates two new separation algorithms for all the previously mentioned inequalities and a new heuristic procedure to obtain upper bounds are presented finally the performance of a branch and cut algorithm over several sets of large wpp and mcpp instances with up to 3 000 nodes and 9 000 edges and arcs in the mcpp case shows that to our knowledge this is the best algorithm to date for the exact resolution of the wpp and the mcpp

polynomial time separation of a superclass of simple comb inequalities the comb inequalities are a well known class of facet inducing inequalities for the traveling salesman problem defined in terms of certain vertex sets called the handle and the teeth we say that a comb inequality is simple if the following holds for each tooth either the intersection of the tooth with the handle has cardinality one or the part of the tooth outside the handle has cardinality one or both the simple comb inequalities generalize the classical 2 matching inequalities of edmonds edmonds j 1965 maximum matching and a polyhedron with 0 1 vertices j res nat bur standards69b 125 130 and also the so called chvatal comb inequalities in 1982 padberg and rao padberg m w m r rao 1982 odd minimum cut sets and b matchings math oper res 7 67 80 gave a polynomial time separation algorithm for the 2 matching inequalities i e an algorithm for testing if a given fractional solution to an lp relaxation violates a 2 matching inequality we extend this significantly by giving a polynomial time separation algorithm for a class of valid inequalities which includes all simple comb inequalities

on linear characterizations of combinatorial optimization problems we show that there can be no computationally tractable description by linear inequalities of the polyhedron associated with any np complete combinatorial optimization problem unless np co np a very unlikely event we also apply the ellipsoid method for linear programming to show that a combinatorial optimization problem is solvable in polynomial time if and only if it admits a small generator of violated inequalities

facets of the linear ordering polytope letd n be the complete digraph onn nodes and letp lo denote the convex hull of all incidence vectors of arc sets of linear orderings of the nodes ofd n i e these are exactly the acyclic tournaments ofd n we show that various classes of inequalities define facets ofp lo e g the 3 dicycle inequalities the simplek fence inequalities and various mobius ladder inequalities and we discuss the use of these inequalities in cutting plane approaches to the triangulation problem of input output matrices i e the solution of permutation resp linear ordering problems

set packing relaxations of some integer programs this paper is about set packing relaxations of combinatorial optimization problems associated with acyclic digraphs and linear orderings cuts and multicuts and set packings themselves families of inequalities that are valid for such a relaxation as well as the associated separation routines carry over to the problems under investigation

odd minimum cut sets and b matchings we show that the determination of a minimum cut set of odd cardinality in a graph with even and odd vertices can be dealt with by a minor modification of the polynomially bounded algorithm of gomory and hu for multi terminal networks we connect this problem to the problem of identifying a matching or blossom constraint that chops off a point which is not contained in the convex hull of matchings or proving that no such inequality exists both the b matching problems without and with upper bounds are considered we discuss how the results of this paper can be used in conjunction with commercial lp packages lo solve b matching problems

transitive packing a unifying concept in combinatorial optimization this paper attempts to provide a better understanding of the facial structure of polyhedra previously investigated separately it introduces the notion of transitive packing and the transitive packing polytope polytopes that turn out to be special cases of the transitive packing polytope include the node packing acyclic subdigraph bipartite subgraph planar subgraph clique partitioning partition transitive acyclic subdigraph interval order and relatively transitive subgraph polytopes we give cutting plane proofs for several rich classes of valid inequalities of the transitive packing polytope thereby introducing generalized cycle generalized clique generalized antihole generalized antiweb and odd partition inequalities on the one hand these classes subsume several known classes of valid inequalities for several special cases on the other hand they yield many new inequalities for several other special cases for some of the classes we also prove a lower bound on their gomory chvatal rank finally we relate the concept of transitive packing to generalized set packing and covering as well as to balanced and ideal matrices

facets of the clique partitioning polytope a subseta of the edge set of a graphg v e is called a clique partitioning ofg is there is a partition of the node setv into disjoint setsw 1 w k such that eachw i induces a clique i e a complete but not necessarily maximal subgraph ofg and such thata 1 uv u v w i u v given weightsw e for alle e the clique partitioning problem is to find a clique partitioninga ofg such that e a w e is as small as possible this problem known to be hard see wakabayashi 1986 comes up for instance in data analysis and here the underlying graphg is typically a complete graph in this paper we study the clique partitioning polytope of the complete graphk n i e is the convex hull of the incidence vectors of the clique partitionings ofk n we show that triangles 2 chorded odd cycles 2 chorded even wheels and other subgraphs ofk n induce facets of the theoretical results described here have been used to design an empirically efficient cutting plane algorithm with which large real world instances of the clique partitioning problem could be solved these computational results can be found in grotschel and wakabayashi 1989

efficient semidefinite branch and cut for map mrf inference we propose a branch and cut b c method for solving general map mrf inference problems the core of our method is a very efficient bounding procedure which combines scalable semidefinite programming sdp and a cutting plane method for seeking violated constraints in order to further speed up the computation several strategies have been exploited including model reduction warm start and removal of inactive constraints we analyze the performance of the proposed method under different settings and demonstrate that our method either outperforms or performs on par with state of the art approaches especially when the connectivities are dense or when the relative magnitudes of the unary costs are low we achieve the best reported results experiments show that the proposed algorithm achieves better approximation than the state of the art methods within a variety of time budgets on challenging non submodular map mrf inference problems

an inexact algorithm for the sequential ordering problem abstract given the directed g n a and the penalty matrix c the sequential ordering problem hereafter sop consists of finding the permutation of the nodes from the set n such that it minimizes a c based function and does not violate the precedence relationships given by the set a strong sufficient conditions for the infeasibility of a sop s instance are embedded in a procedure for the sop s pre processing note that it is one of the key steps in any algorithm that attempts to solve sop by dropping the constraints related to the precedence relationships sop can be converted in the classical asymmetric traveling salesman problem hereafter astp the algorithm obtains hopefully satisfactory solutions by modifying the optimal solution to the related assignment problem hereafter ap if it is not a feasible sequential ordering hereafter fso the new solution patches the subtours if any giving preference to the patches with zero reduced cost in the linking arcs the ap based lower bound on the optimal solution to atsp is tightened by using some of the procedures given in 1 in any case a local search for improving the initial fso is performed it uses 3 and 4 changed based procedures computational results on a broad set of cases are reported

the partition problem in this paper we describe several forms of thek partition problem and give integer programming formulations of each case the dimension of the associated polytopes and some basic facets are identified we also give several valid and facet defining inequalities for each of the polytopes

the equipartition polytope i formulations dimension and basic facets the following basic clustering problem arises in different domains ranging from physics statistics and boolean function minimization given a graphg v e and edge weightsce partition the setv into two sets of 1 2 v and 1 2 v nodes in such a way that the sum of the weights of edges not having both endnodes in the same set is maximized or minimized anequicut is a feasible solution of the above problem and theequicut polytope q g is the convex hull of the incidence vectors of equicuts ing in this paper we give some integer programming formulations of the equicut problem study the dimension of the equicut polytope and describe some basic classes of facet inducing inequalities forq g

a branch and bound algorithm for the linear ordering problem with cumulative costs abstract the linear ordering problem with cumulative costs is an np hard combinatorial optimization problem arising from an application in umts mobile phone communication systems this paper presents a polynomially computable lower bound that is particularly effective when embedded in a branch and bound algorithm the same idea can be further exploited to sort the children nodes at each node of the search tree in order to find the optimal solution earlier a suitable truncation of the resulting branch and bound algorithm results in a fast constructive heuristic

the mod 2 cohomology of fixed point sets of anti symplectic involutions abstract let m be a compact connected symplectic manifold with a hamiltonian action of a compact n dimensional torus g tn suppose that is an anti symplectic involution compatible with the g action the real locus of m is x the fixed point set of duistermaat uses morse theory to give a description of the ordinary cohomology of x in terms of the cohomology of m there is a residual g r z 2 z n action on x and we can use duistermaat s result as well as some general facts about equivariant cohomology to prove an equivariant analogue to duistermaat s theorem in some cases we can also extend theorems of goresky kottwitz macpherson and goldin holm to the real locus

column generation algorithms for exact modularity maximization in networks finding modules or clusters in networks currently attracts much attention in several domains the most studied criterion for doing so due to newman and girvan phys rev e 69 026113 2004 is modularity maximization many heuristics have been proposed for maximizing modularity and yield rapidly near optimal solution or sometimes optimal ones but without a guarantee of optimality there are few exact algorithms prominent among which is a paper by xu et al eur phys j b 60 231 2007 modularity maximization can also be expressed as a clique partitioning problem and the row generation algorithm of grotschel and wakabayashi math program 45 59 1989 applied we propose to extend both of these algorithms using the powerful column generation methods for linear and non linear integer programming performance of the four resulting algorithms is compared on problems from the literature instances with up to 512 entities are solved exactly moreover the computing time of previously solved problems are reduced substantially

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

on some exact sequences in the theory of cyclotomic fields abstract the purpose of this article is to show the existence of some exact sequences which relate the ideal class groups of cyclotomic fields to gauss sums these exact sequences imply the results of hachimori ichimura and beliaeva furthermore we study the relations between the etale cohomology groups and some conjectures on cyclotomic fields

clique web facets for multicut polytopes let g v e be a graph an edge set uv e u si v sj i j where s1 sk is a partition of v is called a multicut with k shores we investigate the polytopes mck n and mck n that are defined as the convex hulls of the incidence vectors of all multicuts with at most k shores and at least k shores respectively of the complete graph kn we introduce a large class of inequalities called clique web inequalities valid for these polytopes and provide a quite complete characterization of those clique web inequalities that define facets of mck n and mck n using general facet manipulation techniques like collapsing and node splitting we construct further new classes of facets for these multicut polytopes we also exhibit a class of clique web facets for which the separation problem can be solved in polynomial time

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

the equipartition polytope ii valid inequalities and facets theequipartition problem is defined as follows given a graphg v e and edge weightsce partition the setv into two sets of 1 2 v and 1 2 v nodes in such a way that the sum of the weights of edges not having both endnodes in the same set is maximized or minimized anequicut is a feasible solution of the above problem and theequicut polytope q g is the convex hull of the incidence vectors of equicuts ing in this paper we describe some facet inducing inequalities of this polytope

cluster analysis and mathematical programming given a set of entities cluster analysis aims at finding subsets called clusters which are homogeneous and or well separated as many types of clustering and criteria for homogeneity or separation are of interest this is a vast field a survey is given from a mathematical programming viewpoint steps of a clustering study types of clustering and criteria are discussed then algorithms for hierarchical partitioning sequential and additive clustering are studied emphasis is on solution methods i e dynamic programming graph theoretical algorithms branch and bound cutting planes column generation and heuristics

on the separation of split cuts and related inequalities the split cuts of cook kannan and schrijver are general purpose valid inequalities for integer programming which include a variety of other well known cuts as special cases to detect violated split cuts one has to solve the associated separation problem the complexity of split cut separation was recently cited as an open problem by cornuejols li cl01 in this paper we settle this question by proving strong completeness of separation for split cuts as a by product we also show completeness of separation for several other classes of inequalities including the mir inequalities of nemhauser and wolsey and some new inequalities which we call balanced split cuts and binary split cuts we also strengthen completeness results of caprara fischetti cf96 for cuts and eisenbrand e99 for chvatal gomory cuts to compensate for this bleak picture we also give a positive result for the symmetric travelling salesman problem we show how to separate in polynomial time over a class of split cuts which includes all comb inequalities with a fixed handle

equivariant cohomology and analytic descriptions of ring isomorphisms in this paper we consider a class of connected closed g manifolds with a non empty finite fixed point set each m of which is totally non homologous to zero in mg or g equivariantly formal where g bbb z 2 with the help of the equivariant index we give an explicit description of the equivariant cohomology of such a g manifold in terms of algebra so that we can obtain analytic descriptions of ring isomorphisms among equivariant cohomology rings of such g manifolds and a necessary and sufficient condition that the equivariant cohomology rings of such two g manifolds are isomorphic this also leads us to analyze how many there are equivariant cohomology rings up to isomorphism for such g manifolds in 2 and 3 dimensional cases

a polyhedral approach to the asymmetric traveling salesman problem several branch and bound algorithms for the exact solution of the asymmetric traveling salesman problem atsp based on the assignment problem ap relaxation have been proposed in the literature these algorithms perform very well for some instances e g those with uniformly random integer costs but very poorly for others the aim of this paper is to evaluate the effectiveness of a branch and cut algorithm exploiting atsp specific facet defining cuts to be used to attack hard instances that cannot be solved by the ap based procedures from the literature we present new separation algorithms for some classes of facet defining cuts and a new variable pricing technique for dealing with highly degenerate primal lp problems a branch and cut algorithm based on these new results is designed and evaluated through computational analysis on several classes of both random and real world instances the outcome of the research is that on hard instances the branch and cut algorithm clearly outperforms the best ap based algorithms from the literature

an extended formulation of the convex recoloring problem on a tree we introduce a strong extended formulation of the convex recoloring problem on a tree which has an application in analyzing phylogenetic trees the extended formulation has only a polynomial number of constraints but dominates the conventional formulation and the exponentially many valid inequalities introduced by campelo et al math progr 156 303 330 2016 we show that all valid inequalities introduced by campelo et al can be derived from the extended formulation we also show that the natural restriction of the extended formulation provides a complete inequality description of the polytope of subtrees of a tree the solution time using the extended formulation is much smaller than that with the conventional formulation moreover the extended formulation solves all the problem instances attempted in campelo et al 2016 and larger sized instances at the root node of the branch and bound tree without branching

a lagrangian relaxation approach to the edge weighted clique problem the b clique polytope cp n b is the convex hull of the node and edge incidence vectors of all subcliques of size at most b of a complete graph on n nodes including the boolean quadric polytope qp n as a special case and being closely related to the quadratic knapsack polytope it has received considerable attention in the literature in particular the max cut problem is equivalent with optimizing a linear function over qp n n the problem of optimizing linear functions over cp n b has so far been approached via heuristic combinatorial algorithms and cutting plane methods we study the structure of cp n b in further detail and present a new computational approach to the linear optimization problem based on lucena s suggestion of integrating cutting planes into a lagrangian relaxation of an integer programming problem in particular we show that the separation problem for tree inequalities becomes polynomial in our lagrangian framework finally computational results are presented

the boolean quadric polytope some characteristics facets and relatives we study unconstrained quadratic zero one programming problems havingn variables from a polyhedral point of view by considering the boolean quadric polytope qpn inn n 1 2 dimensions that results from the linearization of the quadratic form we show that qpn has a diameter of one descriptively identify three families of facets of qpn and show that qpn is symmetric in the sense that all facets of qpn can be obtained from those that contain the origin by way of a mapping the naive linear programming relaxation qpnlp of qpn is shown to possess the trubin property and its extreme points are shown to be 0 1 2 1 valued furthermore o n3 facet defining inequalities of qpn suffice to cut off all fractional vertices of qpnlp whereas the family of facets described by us has at least o 3n members the problem is also studied for sparse quadratic forms i e when several or many coefficients are zero and conditions are given for the previous results to carry over to this case polynomially solvable problem instances are discussed and it is shown that the known polynomiality result for the maximization of nonnegative quadratic forms can be obtained by simply rounding the solution to the linear programming relaxation in the case that the graph induced by the nonzero coefficients of the quadratic form is series parallel a complete linear description of the associated boolean quadric polytope is given the relationship of the boolean quadric polytope associated to sparse quadratic forms with the vertex packing polytope is discussed as well

formulations and valid inequalities for the node capacitated graph partitioning problem we investigate the problem of partitioning the nodes of a graph under capacity restriction on the sum of the node weights in each subset of the partition the objective is to minimize the sum of the costs of the edges between the subsets of the partition this problem has a variety of applications for instance in the design of electronic circuits and devices we present alternative integer programming formulations for this problem and discuss the links between these formulations having chosen to work in the space of edges of the multicut we investigate the convex hull of incidence vectors of feasible multicuts in particular several classes of inequalities are introduced and their strength and robustness are analyzed as various problem parameters change

generalizations of cliques odd cycles and anticycles and their relation to independence system polyhedra padberg padberg m w 1973 on the facial structure of set packing polyhedra math programming5 199 215 has shown that if a graph g with vertex set v constitutes a clique odd cycle odd anticycle then the inequality x vxv 1 x vxv v 1 x vxv 2 defines a facet of pistg the convex hull of the incidence vectors of stable sets in g we generalize the concepts of cliques odd cycles and anticycles to arbitrary independence systems e i and show that the corresponding inequalities define facets of pi the convex hull of the incidence vectors of independent sets

facets for the cut cone i we study facets of the cut conec n i e the cone of dimension 1 2n n 1 generated by the cuts of the complete graph onn vertices actually the study of the facets of the cut cone is equivalent in some sense to the study of the facets of the cut polytope we present several operations on facets and in particular a lifting procedure for constructing facets ofc n 1 from given facets of the lower dimensional conec n after reviewing hypermetric valid inequalities we describe the new class of cycle inequalities and prove the facet property for several subclasses the new class of parachute facets is developed and other known facets and valid inequalities are presented

least cost influence propagation in social networks influence maximization problems aim to identify key players in social networks and are typically motivated from viral marketing in this work we introduce and study the generalized least cost influence problem glcip that generalizes many previously considered problem variants and allows to overcome some of their limitations a formulation that is based on the concept of activation functions is proposed together with strengthening inequalities exact and heuristic solution methods are developed and compared for the new problem our computational results also show that our approaches outperform the state of the art on relevant special cases of the glcip

lagrangian relaxation versus genetic algorithm based matheuristic for a large partitioning problem abstract this paper is concerned with a partitioning problem one of the applications and the motivation for this research is the problem of class formation for training and retraining sessions at large electricity distributors two different approaches are developed one is based on the quadratic multiple knapsack formulation and lagrangian relaxation the other is a matheuristic developed as an amalgamation of genetic algorithms and integer programming the approaches are tested by means of computational experiments both heuristics outperformed the direct application of quadratic programming with the lagrangian relaxation based approach performing the best on average and the genetic algorithm based approach performing the best on the larger test cases

the clique partitioning problem abstract we study the problem of clique partitioning a graph we prove a new general upper bound result on the number of clique partitions this upper bound is the best possible given information of just the vertices and the number of edges next we show that there exists an optimal partition in which one of the cliques is a maximal clique finally we present two new efficient methods to clique partition a graph since the clique partitioning of a graph is equivalent to the coloring of the complement of the graph any coloring algorithm can also be used to clique partition a graph we present detailed statistics comparing the performance of our algorithms against two of the best known coloring algorithms and against a recently published clique partitioning algorithm both functional and timing comparisons are given and we show that our algorithms compare very favorably on both counts

a branch and bound algorithm for the acyclic subgraph problem abstract a branch and bound algorithm for the acyclic subgraph problem feedback are set problem is described the branching scheme lexicographically enumerates all permutations skipping initial segments known by some easy tests not to have any optimal completion a lower bound for the number of feedback arcs is given by the size of any collection of disjoint cycles we propose a heuristic algorithm to find a large collection the size of the problems our branch and bound algorithm can solve varies from 25 to 34 nodes depending on the nature of the problem

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

the linear ordering problem with cumulative costs abstract the optimization problem of finding a permutation of a given set of items that minimizes a certain cost function is naturally modeled by introducing a complete digraph g whose vertices correspond to the items to be sorted depending on the cost function to be used different optimization problems can be defined on g the most familiar one is the min cost hamiltonian path problem or its closed path version the traveling salesman problem arising when the cost of a given permutation only depends on consecutive node pairs a more complex situation arises when a given cost has to be paid whenever an item is ranked before another one in the final permutation in this case a feasible solution is associated with an acyclic tournament the transitive closure of an hamiltonian path and the resulting problem is known as the linear ordering problem lop in this paper we introduce and study a relevant case of lop arising when the overall permutation cost can be expressed as the sum of terms u associated with each item u each defined as a linear combination of the values v of all items v that follow u in the permutation this setting implies a cumulative nonlinear propagation of the value of variables v along the node permutation hence the name linear ordering problem with cumulative costs we illustrate the practical application in wireless telecommunication system that motivated the present study we prove complexity results and propose a mixed integer linear programming model as well as an ad hoc enumerative algorithm for the exact solution of the problem a dynamic programming heuristic is also described extensive computational results on large sets of instances are presented showing that the proposed techniques are capable of solving in reasonable computing times all the instances coming from our application

on the partial order polytope of a digraph we introduce the partial order polytope of a digraphd defined as the convex hull of the incidence vectors of all transitive acyclic arc sets ofd for this polytope we prove some classes of inequalities to be facet defining and show that there is a polynomial separation algorithm for each of these classes the results imply a polynomial separation algorithm for a class of valid inequalities of the clique partitioning polytope that includes the two chorded odd cycle inequalities

single commodity robust network design with finite and hose demand sets we study a single commodity robust network design problem srnd defined on an undirected graph our goal is to determine minimum cost capacities such that any traffic demand from a given uncertainty set can be satisfied by a feasible single commodity flow we consider two ways of representing the uncertainty set either as a finite list of scenarios or as a polytope we propose a branch and cut algorithm to derive optimal solutions to srnd built on a capacity based integer linear programming formulation it is strengthened with valid inequalities derived as 0 frac 1 2 0 12 chvatal gomory cuts since the formulation contains exponentially many constraints we provide practical separation algorithms extensive computational experiments show that our approach is effective in comparison to existing approaches from the literature as well as to solving a flow based formulation by a general purpose solver

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

combinatorial benders cuts for mixed integer linear programming mixed integer programs mips involving logical implications modeled through big m coefficients are notoriously among the hardest to solve in this paper we propose and analyze computationally an automatic problem reformulation of quite general applicability aimed at removing the model dependency on the big m coefficients our solution scheme defines a master integer linear problem ilp with no continuous variables which contains combinatorial information on the feasible integer variable combinations that can be distilled from the original mip model the master solutions are sent to a slave linear program lp which validates them and possibly returns combinatorial inequalities to be added to the current master ilp the inequalities are associated to minimal or irreducible infeasible subsystems of a certain linear system and can be separated efficiently in case the master solution is integer the overall solution mechanism closely resembles the benders one but the cuts we produce are purely co

binary positive semidefinite matrices and associated integer polytopes we consider the positive semidefinite psd matrices with binary entries along with the corresponding integer polytopes we begin by establishing some basic properties of these matrices and polytopes then we show that several families of integer polytopes in the literature the cut boolean quadric multicut and clique partitioning polytopes are faces of binary psd polytopes finally we present some implications of these polyhedral relationships in particular we answer an open question in the literature on the max cut problem by showing that the rounded psd inequalities define a polytope

polyhedral structure and properties of a model for layout design abstract in flexible manufacturing systems the type of materials handling system strongly influences the machine layout this paper studies the problem of determining a machine layout for uni directional loop networks which minimizes the maximum number of loop traversals for a family of parts which we show to be np complete we also show that the polyhedral structure of its integer programming formulation is closely related to that of the maximum weight acyclic subgraph problem we introduce inequalities that define facets and faces of the convex hull of solutions for the layout problem which also define corresponding facets and faces for the acyclic subgraph polytope preliminary computational results are also presented

min cut clustering we describe a decomposition framework and a column generation scheme for solving a min cut clustering problem the subproblem to generate additional columns is itself an np hard mixed integer programming problem we discuss strong valid inequalities for the subproblem and describe some efficient solution strategies computational results on compiler construction problems are reported

the confined primal integral a measure to benchmark heuristic minlp solvers against global minlp solvers it is a challenging task to fairly compare local solvers and heuristics against each other and against global solvers how does one weigh a faster termination time against a better quality of the found solution in this paper we introduce the confined primal integral a new performance measure that rewards a balance of speed and solution quality it emphasizes the early part of the solution process by using an exponential decay thereby it avoids that the order of solvers can be inverted by choosing an arbitrarily large time limit we provide a closed analytic formula to compute the confined primal integral a posteriori and an incremental update formula to compute it during the run of an algorithm for the latter we show that we can drop one of the main assumptions of the primal integral namely the knowledge of a fixed reference solution to compare against furthermore we prove that the confined primal integral is a transitive measure when comparing local solves with different final solution values finally we present a computational experiment where we compare a local minlp solver that uses certain classes of cutting planes against a solver that does not both versions show very different tendencies w r t average running time and solution quality and we use the confined primal integral to argue which of the two is the preferred setting

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

facets and lifting procedures for the set covering polytope given a family of subsets of an arbitrary groundsete acover of is any setc e having non empty intersection with every subset in in this paper we deal with thecovering polytope i e the convex hull of the incidence vectors of all the covers of in section 2 we review all the known properties of the covering polytope in sections 3 and 4 we introduce two new classes of non boolean facets of such a polytope in sections 5 and 6 we describe some non sequential lifting procedures in section 7 a generalization of the notion ofweb introduced by l e trotter is presented together with the facets of the covering polytope produced by such a structure moreover the strong connections between several combinatorial problems and the covering problem are pointed out and exploiting those connections some examples are presented of new facets for the knapsack and acyclic subdigraph polytopes

a cutting plane algorithm for a clustering problem in this paper we consider a clustering problem that arises in qualitative data analysis this problem can be transformed to a combinatorial optimization problem the clique partitioning problem we have studied the latter problem from a polyhedral point of view and determined large classes of facets of the associated polytope these theoretical results are utilized in this paper we describe a cutting plane algorithm that is based on the simplex method and uses exact and heuristic separation routines for some of the classes of facets mentioned before we discuss some details of the implementation of our code and present our computational results we mention applications from e g zoology economics and the political sciences

separating a superclass of comb inequalities in planar graphs many classes of valid and facet inducing inequalities are known for the family of polytopes associated with thesymmetric travelling salesman problem stsp includingsubtour elimination 2 matching andcomb inequalities for a given class of inequalities anexact separation algorithm is a procedure which given an lp relaxation vectorx finds one or more inequalities in the class which are violated byx or proves that none exist such algorithms are at the core of the highly successfulbranch and cut algorithms for the stsp however whereas polynomial time exact separation algorithms are known for subtour elimination and 2 matching inequalities the complexity of comb separation is unknown apartial answer to the comb problem is provided in this paper we define ageneralization of comb inequalities and show that the associated separation problem can be solved efficiently when the subgraph induced by the edges withx e 0 is planar the separation algorithm runs in o n3 time wheren is the number of vertices in the graph

integer linear programming for the bayesian network structure learning problem abstract bayesian networks are a commonly used method of representing conditional probability relationships between a set of variables in the form of a directed acyclic graph dag determination of the dag which best explains observed data is an np hard problem 1 this problem can be stated as a constrained optimisation problem using integer linear programming ilp this paper explores how the performance of ilp based bayesian network learning can be improved through ilp techniques and in particular through the addition of non essential implied constraints there are exponentially many such constraints that can be added to the problem this paper explores how these constraints may best be generated and added as needed the results show that using these constraints in the best discovered configuration can lead to a significant improvement in performance and show significant improvement in speed using a state of the art bayesian network structure learner

a cutting plane algorithm for the linear ordering problem the linear ordering problem is an np hard combinatorial optimization problem with a large number of applications including triangulation of input output matrices archaeological senation minimizing total weighted completion time in one machine scheduling and aggregation of individual preferences in a former paper we have investigated the facet structure of the 0 1 polytope associated with the linear ordering problem here we report on a new algorithm that is based on these theoretical results the main part of the algorithm is a cutting plane procedure using facet defining inequalities this procedure is combined with various heuristics and branch and bound techniques our computational results compare favorably with the results of existing codes in particular we could triangulate all input output matrices of size up to 60 60 available to us within acceptable time bounds

computational results of an interior point algorithm for large scale linear programming this paper gives computational results for an efficient implementation of a variant of dual projective algorithm for linear programming the implementation uses the preconditioned conjugate gradient method for computing projections our computational experience reported in this paper indicates that this algorithm has potential as an alternative for solving very large lps in which the direct methods fail due to memory and cpu time requirements the conjugate gradient algorithm was able to find very accurate directions even when the system was ill conditioned the paper also discusses a new mathematical technique called the reciprocal estimates for estimating the primal variables we have conducted extensive computational experiments on problems representative of large classes of applications of current interest we have also chosen instances of the problems of future potential interest which could not be solved in the past due to the weakness of the prior solution methods but which represent a large class of new applications the hypergraph model is such an example comparison of our implementation with minos 5 1 shows that our implementation is orders of magnitude faster than minos 5 1 for these problems

a graph theoretic heuristic for designing loop layout manufacturing systems abstract in flexible manufacturing systems the type of materials handling system strongly influence the machine layout for layouts where the machines are arranged in a loop and the material handling system is uni directional a measure of congestion for a part is the number of times the part traverses the loop before its processing is completed this paper investigates the problem of determining a machine layout for uni directional loop networks which minimizes the maximum number of loop traversals for a family of parts based on graph theoretic arguments a heuristic is developed which constructs a layout from a solution to the linear programming relaxation of the problem preliminary computational results are also presented

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

hierarchical image segmentation using correlation clustering in this paper we apply efficient implementations of integer linear programming to the problem of image segmentation the image is first grouped into superpixels and then local information is extracted for each pair of spatially adjacent superpixels given local scores on a map of several hundred superpixels we use correlation clustering to find the global segmentation that is most consistent with the local evidence we show that although correlation clustering is known to be np hard finding the exact global solution is still feasible by breaking the segmentation problem down into subproblems each such sub problem can be viewed as an automatically detected image part we can further accelerate the process by using the cutting plane method which provides a hierarchical structure of the segmentations the efficiency and improved performance of the proposed method is compared to several state of the art methods and demonstrated on several standard segmentation data sets