polyhedral properties for the intersection of two knapsacks we address the question to what extent polyhedral knowledge about individual knapsack constraints suffices or lacks to describe the convex hull of the binary solutions to their intersection it turns out that the sign patterns of the weight vectors are responsible for the types of combinatorial valid inequalities appearing in the description of the convex hull of the intersection in particular we introduce the notion of an incomplete set inequality which is based on a combinatorial principle for the intersection of two knapsacks we outline schemes to compute nontrivial bounds for the strength of such inequalities w r t the intersection of the convex hulls of the initial knapsacks an extension of the inequalities to the mixed case is also given this opens up the possibility to use the inequalities in an arbitrary simplex tableau

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

separation algorithms for 0 1 knapsack polytopes valid inequalities for 0 1 knapsack polytopes often prove useful when tackling hard 0 1 linear programming problems to generate such inequalities one needs separation algorithms for them i e routines for detecting when they are violated we present new exact and heuristic separation algorithms for several classes of inequalities namely lifted cover extended cover weight and lifted pack inequalities moreover we show how to improve a recent separation algorithm for the 0 1 knapsack polytope itself extensive computational results on miplib and or library instances show the strengths and limitations of the inequalities and algorithms considered

the merchant subtour problem we consider the problem of a travelling merchant who makes money by buying commodities where they are cheap and selling them in other places where he can make a profit the merchant ships commodities of his own choice in a van of fixed capacity given the prices of all the commodities in all of the places and the cost of driving from one place to another the problem the merchant faces each day is to select a subset of the cities that he can visit in a day and to determine the order in which the cities are visited such that the total profit is maximised we call this problem the merchant subtour problem the msp models the pricing problem of a rather complex pickup and delivery problem that was given to us by the dutch logistics company van gend loos we show that a special case of the msp has a totally unimodular constraint matrix this knowledge enables us to develop a tabu search algorithm for finding good feasible solutions to the msp and a branch and price and cut algorithm for solving the msp to optimality the relaxations solved in each node of the branch and bound tree are strengthened by lifted knapsack inequalities lifted cycle inequalities and mod k cuts we present computational results on data sets derived from our main instance of the van gend loos pickup and delivery problem

lifted inequalities for 0 1 mixed integer programming superlinear lifting we study the mixed 0 1 knapsack polytope which is defined by a single knapsack constraint that contains 0 1 and bounded continuous variables through the lifting of continuous variables fixed at their upper bounds we introduce the concept of a superlinear inequality and show that in this case lifting is significantly simpler than for general inequalities we use the superlinearity theory together with the traditional lifting of 0 1 variables to describe families of facets of the mixed 0 1 knapsack polytope finally we show that superlinearity results can be extended to nonsuperlinear inequalities when the coefficients of the variables fixed at their upper bounds are large

solving airline crew scheduling problems by branch and cut the crew scheduling problem is one that has been studied almost continually for the past 40 years but all prior approaches have always approximated the problem of finding an optimal schedule for even the smallest of an airline s fleets the problem is especially important today since costs for flying personnel of major u s carriers have grown and now often exceed 1 3 billion a year and are the second largest item next to fuel cost of the total operating cost of major u s carriers thus even small percentage savings amount to substantial dollar amounts we present a branch and cut approach to solving to proven optimality large set partitioning problems arising within the airline industry we first provide some background related to this important application and then describe the approach for solving representative problems in this problem class the branch and cut solver generates cutting planes based on the underlying structure of the polytope defined by the convex hull of the feasible integer points and incorporates these cuts into a tree search algorithm that uses automatic reformulation procedures heuristics and linear programming technology to assist in the solution numerical experiments are reported for a sample of 68 large scale real world crew scheduling problems these problems include both pure set partitioning problems and set partitioning problems with side constraints these base constraints represent contractual labor requirements and have heretofore not been represented explicitly in the construction of crew schedules thus making it impossible to provide any measure of how far the obtained solution was from optimality an interesting result of obtaining less costly schedules is that the crews themselves are happier with the schedules because they spend more of their duty time flying than waiting on the ground

easily computable facets of the knapsack polytope it is known that facets and valid inequalities for the knapsack polytope p can be obtained by lifting a simple inequality derived from each minimal cover we study the computational complexity of such lifting in particular we show that the task of computing a lifted facet can be accomplished in o ns where s n is the cardinality of the minimal cover also for a lifted inequality with integer coefficients we show that the dual tasks of recognizing whether the inequality is valid for p or is a facet of p can be done within the same time bound

the 0 1 knapsack problem with a single continuous variable constraints arising in practice often contain many 0 1 variables and one or a small number of continuous variables existing knapsack separation routines cannot be used on such constraints here we study such constraint sets and derive valid inequalities that can be used as cats for such sets as well for more general mixed 0 1 constraints specifically we investigate the polyhedral structure of the knapsack problem with a single continuous variable called the mixed 0 1 knapsack problem first different classes of facet defining inequalities are derived based on restriction and lifting the order of lifting particularly of the continuous variable plays an important role secondly we show that the flow cover inequalities derived for the single node flow set consisting of are flows into and out of a single node with binary variable lower and upper bounds on each are can be obtained from valid inequalities for the mixed 0 1 knapsack problem thus the separation heuristic we derive for mixed knapsack sets can also be used to derive cuts for more general mixed 0 1 constraints initial computational results on a variety of problems are presented

on splittable and unsplittable flow capacitated network design arc set polyhedra we study the polyhedra of splittable and unsplittable single arc set relaxations of multicommodity flow capacitated network design problems we investigate the optimization problems over these sets and the separation and lifting problems of valid inequalities for them in particular we give a linear time separation algorithm for the residual capacity inequalities 19 and show that the separation problem of c strong in equalities 7 is np hard but can be solved over the subspace of fractional variables only we introduce two classes of inequalities for the unsplittable flow problems we present a summary of computational experiments with a branch and cut algorithm for multicommodity flow capacitated network design problems to test the effectiveness of the results presented here empirically

spectral asymptotics for arithmetic quotients of sl n r so n in this article we study the asymptotic distribution of the cuspidal spectrum of arithmetic quotients of the symmetric space sl n r so n in particular we obtain weyl s law with an estimation on the remainder term this extends some of the main results of duistermaat kolk and varadarajan dkv1 to this setting

on the facets of the mixed integer knapsack polyhedron we study the mixed integer knapsack polyhedron that is the convex hull of the mixed integer set defined by an arbitrary linear inequality and the bounds on the variables we describe facet defining inequalities of this polyhedron that can be obtained through sequential lifting of inequalities containing a single integer variable these inequalities strengthen and or generalize known inequalities for several special cases we report computational results on using the inequalities as cutting planes for mixed integer programming

hilbert bases and the facets of special knapsack polytopes we present a linear description of the 0 1 knapsack polytope for the special case when the number of different weights of the items is two the inequalities needed for this description involve elements of the hilbert basis of the cone of exchange vectors

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

contravariantly finite resolving subcategories over commutative rings contravariantly finite resolving subcategories of the category of finitely generated modules have been playing an important role in the representation theory of algebras in this paper we study contravariantly finite resolving subcategories over commutative rings the main purpose of this paper is to classify contravariantly finite resolving subcategories over a henselian gorenstein local ring in fact there exist only three such subcategories our method to obtain this classification also recovers as a by product the theorem of christensen piepmeyer striuli and takahashi concerning the relationship between the contravariant finiteness of the full subcategory of totally reflexive modules and the gorenstein property of the base ring

solving a truck dispatching scheduling problem using branch and cut a branch and cut ip solver is developed for a class of structured 0 1 integer programs arising from a truck dispatching scheduling problem this problem involves a special class of knapsack equality constraints families of facets for the polytopes associated with individual knapsack constraints are identified in addition a notion of conflict graph is utilized to obtain an approximating node packing polytope for the convex hull of all 0 1 solutions the branch and cut solver generates cuts based on both the knapsack equality constraints and the approximating node packing polytope and incorporates these cuts into a tree search algorithm that uses problem reformulation and linear programming based heuristics at each node in the search tree to assist in the solution process numerical experiments are performed on large scale real instances supplied by texaco trading transportation inc the optimal schedules correspond to cost savings for the company and greater job satisfaction for drivers due to more balanced work schedules and income distribution

a cut and solve based algorithm for the single source capacitated facility location problem abstract in this paper we present a cut and solve cs based exact algorithm for the single source capacitated facility location problem sscflp at each level of cs s branching tree it has only two nodes corresponding to the sparse problem sp and the dense problem dp respectively the sp whose solution space is relatively small with the values of some variables fixed to zero is solved to optimality by using a commercial mip solver and its solution if it exists provides an upper bound to the sscflp meanwhile the resolution of the lp of dp provides a lower bound for the sscflp a cutting plane method which combines the lifted cover inequalities and fenchel cutting planes to separate the 0 1 knapsack polytopes is applied to strengthen the lower bound of sscflp and that of dp these lower bounds are further tightened with a partial integrality strategy numerical tests on benchmark instances demonstrate the effectiveness of the proposed cutting plane algorithm and the partial integrality strategy in reducing integrality gap and the effectiveness of the cs approach in searching an optimal solution in a reasonable time computational results on large sized instances are also presented

solving multiple knapsack problems by cutting planes in this paper we consider the multiple knapsack problem which is defined as follows given a set n of items with weights f i i in n a set m of knapsacks with capacities f k k in m and a profit function c ik i in n k in m find an assignment of a subset of the set of items to the set of knapsacks that yields minimum cost we consider the multiple knapsack problem from a polyhedral point of view the inequalities that we describe here serve as the theoretical basis for a cutting plane algorithm we present some of the implementation details of this algorithm including a discussion and evaluation of different separation and primal heuristics our algorithm is applied to practical problem instances arising in the design of main frame computers in the layout of electronic circuits and in sugar cane alcohol production

the selberg trace formula for sl 3 ℤ sl 3 ℝ so 3 ℝ in this paper we compute the trace formula for sl 3 z in detail and refine it to a greater extent than has previously been done we show that massive cancellation occurs in the parabolic terms leading to a far simpler formula than had been thought possible

routing through virtual paths in layered telecommunication networks we study a network configuration problem in telecommunications where one wants to set up paths in a capacitated network to accommodate given point to point traffic demand the problem is formulated as an integer linear programming model where 0 1 variables represent different paths an associated integral polytope is studied and different classes of facets are described these results are used in a cutting plane algorithm computational results for some realistic problems are reported

solving mixed integer programming problems using automatic reformulation in this paper we describe computational experience in solving mixed 0 1 programming problems using strong valid inequalities as cutting planes in particular we report on the solution to optimality of 18 medium to large size problems including production planning problems with setup costs and capacity constraints multilevel distribution planning problems drainage and heating system design problems and electricity generator scheduling problems the solution approach uses the theory of strong valid inequalities that we developed in a series of earlier papers here we report specifically on the implementation of an experimental system mpsarx which consists of the sciconic mathematical programming system and an automatic reformulation executor arx that use this theory and on the results obtained with this system

cutting planes for integer programs with general integer variables we investigate the use of cutting planes for integer programs with general integer variables we show how cutting planes arising from knapsack inequalities can be generated and lifted as in the case of 0 1 variables we also explore the use of gomory s mixed integer cuts we address both theoretical and computational issues and show how to embed these cutting planes in a branch and bound framework we compare results obtained by using our cut generation routines in two existing systems with a commercially available branch and bound code on a range of test problems arising from practical applications 1998 the mathematical programming society inc published by elsevier science b v

the sequential knapsack polytope in this paper we describe the convex hull of all solutions of the integer bounded knapsack problem in the special case when the weights of the items are divisible the corresponding inequalities are defined via an inductive scheme that can also be used in a more general setting

on the 0 1 knapsack polytope this paper deals with the 0 1 knapsack polytope in particular we introduce the class ofweight inequalities this class of inequalities is needed to describe the knapsack polyhedron when the weights of the items lie in certain intervals a generalization of weight inequalities yields the so called weight reduction principle and the class of extended weight inequalities the latter class of inequalities includes minimal cover and l k configuration inequalities the properties of lifted minimal cover inequalities are extended to this general class of inequalities

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

density results for automorphic forms on hilbert modular groups ii we obtain an asymptotic formula for a weighted sum over cuspidal eigenvalues in a specific region for sl2 over a totally real number field f with a discrete subgroup of hecke type 0 i for a non zero ideal i in the ring of integers of f the weights are products of fourier coefficients this implies in particular the existence of infinitely many cuspidal automorphic representations with multi eigenvalues in various regions growing to infinity for instance in the quadratic case the regions include floating boxes floating balls sectors slanted strips see 1 2 4 1 2 13 and products of prescribed small intervals for all but one of the infinite places of f the main tool in the derivation is a sum formula of kuznetsov type sum formula for sl2 over a totally real number field theorem 2 1