evenly distributed unitaries on the structure of unitary designs we clarify the mathematical structure underlying unitary t designs these are sets of unitary matrices evenly distributed in the sense that the average of any tth order polynomial over the design equals the average over the entire unitary group we present a simple necessary and sufficient criterion for deciding if a set of matrices constitutes a design lower bounds for the number of elements of 2 designs are derived we show how to turn mutually unbiased bases into approximate 2 designs whose cardinality is optimal in leading order designs of higher order are discussed and an example of a unitary 5 design is presented we comment on the relation between unitary and spherical designs and outline methods for finding designs numerically or by searching character tables of finite groups further we sketch connections to problems in linear optics and questions regarding typical entanglement

a new class of cutting planes for the symmetric travelling salesman problem a comprehensive class of cutting planes for the symmetric travelling salesman problem tsp is proposed which contains the known comb inequalities the path inequalities and the 3 star constraints as special cases its relation to the clique tree inequalities is discussed the cutting planes are shown to be valid for a relaxed version of the tsp the travelling salesman problem on a road network and under certain conditions to define facets of the polyhedron associated with this problem

traveling salesman path problems in the traveling salesman path problem we are given a set of cities traveling costs between city pairs and fixed source and destination cities the objective is to find a minimum cost path from the source to destination visiting all cities exactly once in this paper we study polyhedral and combinatorial properties of a variant we call the traveling salesman walk problem in which the objective is to find a minimum cost walk from the source to destination visiting all cities at least once we first characterize traveling salesman walk perfect graphs graphs for which the convex hull of incidence vectors of traveling salesman walks can be described by linear inequalities we show these graphs have a description by way of forbidden minors and also characterize them constructively we also address the asymmetric traveling salesman path problem atspp and give a factor o sqrt n approximation algorithm for this problem

kerdock codes determine unitary 2 designs the non linear binary kerdock codes are known to be gray images of certain extended cyclic codes of length n 2 m over mathbb z 4 we show that exponentiating these mathbb z 4 valued codewords by i triangleq sqrt 1 produces stabilizer states that are quantum states obtained using only clifford unitaries these states are also the common eigenvectors of commuting hermitian matrices forming maximal commutative subgroups mcs of the pauli group we use this quantum description to simplify the derivation of the classical weight distribution of kerdock codes next we organize the stabilizer states to form n 1 mutually unbiased bases and prove that automorphisms of the kerdock code permute their corresponding mcs thereby forming a subgroup of the clifford group when represented as symplectic matrices this subgroup is isomorphic to the projective special linear group psl 2 n we show that this automorphism group acts transitively on the pauli matrices which implies that the ensemble is pauli mixing and hence forms a unitary 2 design the kerdock design described here was originally discovered by cleve et al 2016 but the connection to classical codes is new which simplifies its description and translation to circuits significantly sampling from the design is straightforward the translation to circuits uses only clifford gates and the process does not require ancillary qubits finally we also develop algorithms for optimizing the synthesis of unitary 2 designs on encoded qubits i e to construct logical unitary 2 designs software implementations are available at https github com nrenga symplectic arxiv18a which we use to provide empirical gate complexities for up to 16 qubits

the symmetric traveling salesman polytope and its graphical relaxation composition of valid inequalities the graphical relaxation of the traveling salesman problem is the relaxation obtained by requiring that the salesman visit each city at least once instead of exactly once this relaxation has already led to a better understanding of the traveling salesman polytope in cornuejols fonlupt and naddef 1985 we show here how one can compose facet inducing inequalities for the graphical traveling salesman polyhedron and obtain other facet inducing inequalities this leads to new valid inequalities for the symmetric traveling salesman polytope this paper is the first of a series of three papers on the symmetric traveling salesman polytope the next one studies the strong relationship between that polytope and its graphical relaxation and the last one applies all the theoretical developments of the two first papers to prove some new facet inducing results

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

averaging sets a generalization of mean values and spherical designs 0 1 r one cannot of course expect there to be a single point x at which every one assumes its mean value but there is an easy generalization in terms of weighted averages there exist at most m points xi and positive weights oi summing to 1 so that aifj xi i fi dt for j 1 2 m

quantum data hiding we expand on our work on quantum data hiding hiding classical data among parties who are restricted to performing only local quantum operations and classical communication locc we review our scheme that hides one bit between two parties using bell 1964 states and we derive upper and lower bounds on the secrecy of the hiding scheme we provide an explicit bound showing that multiple bits can be hidden bitwise with our scheme we give a preparation of the hiding states as an efficient quantum computation that uses at most one ebit of entanglement a candidate data hiding scheme that does not use entanglement is presented we show how our scheme for quantum data hiding can be used in a conditionally secure quantum bit commitment scheme

small travelling salesman polytopes we introduce a new class of valid inequalities for the polytope of the symmetric travelling salesman problem we also gave complete characterizations of the polytope for 6 and 7 cities for the latter case the new inequalities are needed these results are related to work of r z norman in the 1950 s

the binested inequalities for the symmetric traveling salesman polytope in this paper we define a family of valid inequalities for the symmetric travelling salesman polytope which are defined on two nested sets of vertices of the graph these inequalities generalize the comb inequalities of chvital grotschel and padberg the clique tree inequalities of grotschel and pulleyblank the path inequalities of cornuejols fonlupt and naddef and the hyperstar inequalities of fleischman this is the largest known family of valid inequalities known so far facet inducing inequalities for the symmetric travelling salesman polytope contained in this class and in no other one are given proving that this is a proper generalization of the previously mentioned families

well solvable special cases of the traveling salesman problem a survey the traveling salesman problem tsp belongs to the most basic most important and most investigated problems in combinatorial optimization although it is an cal np hard problem many of its special cases can be solved efficiently in polynomial time we survey these special cases with emphasis on the results that have been obtained during the decade 1985 1995 this survey complements an earlier survey from 1985 compiled by gilmore lawler and shmoys the traveling salesman problem a guided tour of combinatorial optimization wiley chichester pp 87 143

symmetric inequalities and their composition for asymmetric travelling salesman polytopes the asymmetric travelling salesmanatspolytope atspv is the convex hull of incidence vectors of all hamiltonian circuits in a complete digraph with node set v this paper studies classes of valid symmetric inequalities ay a0 for ats polytopes with coefficients satisfying aij aji for all pair of cities i and j of particular interest are those inequalities derived from facet defining inequalities for symmetric travelling salesman polytopes stsps we show that known classes of stsp facet defining inequalities such as path wheelbarrow chain and ladder inequalities induce symmetric atsp facet defining inequalities for atspv with v 8 we show that there are precisely four types of stsp facet defining inequalities that do not induce atsp facets we propose a tree composition of symmetric inequalities and use it to generate a large new class of symmetric facet defining inequalities for ats polytopes subsuming all nonspanning clique tree inequalities a hamiltonian path approach is used for most of the polyhedral proofs in the paper

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

the crown inequalities for the symmetric traveling salesman polytope we define a new family of valid inequalities for the symmetric traveling salesman polytope stspn that is the convex hull of the incidence vectors of all hamiltonian cycles of a complete graph with n nodes the smallest value of n for which the inequalities are defined is 8 we call these inequalities crown inequalities and we prove that they are facet inducing for stspn for n 8 we describe some extensions of the crown inequalities and we prove that they induce facets of stspn as well finally we discuss some issues on the possible applications of these inequalities in a polyhedral cutting plane algorithm

the general routing problem polyhedron facets from the rpp and gtsp polyhedra in this paper we study the polyhedron associated with the general routing problem grp this problem first introduced by orloff in 1974 is a generalization of both the rural postman problem rpp and the graphical traveling salesman problem gtsp and thus is np hard we describe a formulation of the problem such that from every non trivial facet inducing inequality for the rpp and gtsp polyhedra we obtain facet inducing inequalities for the grp polyhedron we describe a new family of facet inducing inequalities for the grp the honeycomb constraints which seem to be very useful for solving grp and rpp instances finally new classes of facets obtained by composition of facet inducing inequalities are presented

facet identification for the symmetric traveling salesman polytope several procedures for the identification of facet inducing inequalities for the symmetric traveling salesman polytope are given an identification procedure accepts as input the support graph of a point which does not belong to the polytope and returns as output some of the facet inducing inequalities violated by the point a procedure which always accomplishes this task is calledexact otherwise it is calledheuristic we give exact procedures for the subtour elimination and the 2 matching constraints based on the gomory hu and padberg rao algorithms respectively efficient reduction procedures for the input graph are proposed which accelerate these two algorithms substantially exact and heuristic shrinking conditions for the input graph are also given that yield efficient procedures for the identification of simple and general comb inequalities and of some elementary clique tree inequalities these procedures constitute the core of a polytopal cutting plane algorithm that we have devised and programmed to solve a substantial number of large scale problem instances with sizes up to 2392 nodes to optimality

the traveling salesman problem on a graph and some related integer polyhedra given a graphg n e and a length functionl e the graphical traveling salesman problem is that of finding a minimum length cycle goingat least once through each node ofg this formulation has advantages over the traditional formulation where each node must be visited exactly once we give some facet inducing inequalities of the convex hull of the solutions to that problem in particular the so called comb inequalities of grotschel and padberg are generalized some related integer polyhedra are also investigated finally an efficient algorithm is given wheng is a series parallel graph

the pickup and delivery problem faces and branch and cut algorithm abstract this paper formulates the pickup and delivery problem also known as the dial a ride problem as an integer program its polyhedral structure is explored and four classes of valid inequalities developed the results of a branch and cut algorithm based on these constraints are presented

induced measures in the space of mixed quantum states we analyse several product measures in the space of mixed quantum states in particular we study measures induced by the operation of partial tracing the natural rotationally invariant measure on the set of all pure states of a n k composite system induces a unique measure in the space of n n mixed states or in the space of k k mixed states if the reduction takes place with respect to the first subsystem for k n the induced measure is equal to the hilbert schmidt measure which is shown to coincide with the measure induced by singular values of non hermitian random gaussian matrices pertaining to the ginibre ensemble we compute several averages with respect to this measure and show that the mean entanglement of n n pure states behaves as lnn 1 2

the domino inequalities facets for the symmetric traveling salesman polytope adam letchford defines in 4 the domino parity inequalities for the symmetric traveling salesman polytope stsp and gives a polynomial algorithm for the separation of such constraints when the support graph is planar generalizing a result of fleischer and tardos 2 for maximally violated comb inequalities naddef in 5 gives a set of necessary conditions for such inequalities to be facet defining for the stsp these conditions lead to the domino inequalities and it is shown in 5 that one does not lose any facet inducing inequality restricting the domino parity inequalities to domino inequalities except maybe for some very particular case we prove here that all the domino inequalities are facet inducing for the stsp settling the conjecture given in 5 as a by product we will also have a complete proof that the comb inequalities are facet inducing

on squares of representations of compact lie algebras we study how tensor products of representations decompose when restricted from a compact lie algebra to one of its subalgebras in particular we are interested in tensor squares which are tensor products of a representation with itself we show in a classification free manner that the sum of multiplicities and the sum of squares of multiplicities in the corresponding decomposition of a tensor square into irreducible representations has to strictly grow when restricted from a compact semisimple lie algebra to a proper subalgebra for this purpose relevant details on tensor products of representations are compiled from the literature since the sum of squares of multiplicities is equal to the dimension of the commutant of the tensor square representation it can be determined by linear algebra computations in a scenario where an a priori unknown lie algebra is given by a set of generators which might not be a linear basis hence our results offer a test to decide if a subalgebra of a compact semisimple lie algebra is a proper one without calculating the relevant lie closures which can be naturally applied in the field of controlled quantum systems

symmetric informationally complete quantum measurements we consider the existence in arbitrary finite dimensions d of a positive operator valued measure povm comprised of d2 rank one operators all of whose operator inner products are equal such a set is called a symmetric informationally complete povm sic povm and is equivalent to a set of d2 equiangular lines in cd sic povms are relevant for quantum state tomography quantum cryptography and foundational issues in quantum mechanics we construct sic povms in dimensions two three and four we further conjecture that a particular kind of group covariant sic povm exists in arbitrary dimensions providing numerical results up to dimension 45 to bolster this claim

the graphical relaxation a new framework for the symmetric traveling salesman polytope a present trend in the study of thesymmetric traveling salesman polytope stsp n is to use as a relaxation of the polytope thegraphical relaxation gtsp n rather than the traditionalmonotone relaxation which seems to have attained its limits in this paper we show the very close relationship between stsp n and gtsp n in particular we prove that every non trivial facet of stsp n is the intersection ofn 1 facets of gtsp n n of which are defined by the degree inequalities this fact permits us to define a standard form for the facet defining inequalities for stsp n that we calltight triangular and to devise a proof technique that can be used to show that many known facet defining inequalities for gtsp n define also facets of stsp n in addition we give conditions that permit to obtain facet defining inequalities by composition of facet defining inequalities for stsp n and general lifting theorems to derive facet defining inequalities for stsp n k from inequalities defining facets of stsp n

a branch and cut algorithm for the resolution of large scale symmetric traveling salesman problems an algorithm is described for solving large scale instances of the symmetric traveling salesman problem stsp to optimality the core of the algorithm is a polyhedral cutting plane procedure that exploits a subset of the system of linear inequalities defining the convex hull of the incidence vectors of the hamiltonian cycles of a complete graph the cuts are generated by several identification procedures that have been described in a companion paper whenever the cutting plane procedure does not terminate with an optimal solution the algorithm uses a tree search strategy that as opposed to branch and bound keeps on producing cuts after branching the algorithm has been implemented in fortran two different linear programming lp packages have been used as the lp solver the implementation of the algorithm and the interface with one of the lp solvers is described in sufficient detail to permit the replication of our experiments computational results are reported with up to 42 stsps with sizes rangin

the symmetric traveling salesman polytope new facets from the graphical relaxation the path the wheelbarrow and the bicycle inequalities have been shown by cornuejols fonlupt and naddef to be facet defining for the graphical relaxation of stsp n the polytope of the symmetric traveling salesman problem on an n node complete graph we show that these inequalities and some generalizations of them define facets also for stsp n in conclusion we characterize a large family of facet defining inequalities for stsp n that include as special cases most of the inequalities currently known to have this property as the comb the clique tree and the chain inequalities most of the results given here come from a strong relationship of stsp n with its graphical relaxation that we have pointed out in another paper where the basic proof techniques are also described

wigner functions and separability for finite systems a discussion of discrete wigner functions in phase space related to mutually unbiased bases is presented this approach requires mathematical assumptions which limits it to systems with density matrices defined on complex hilbert spaces of dimension pn where p is a prime number with this limitation it is possible to define a phase space and wigner functions in close analogy to the continuous case that is we use a phase space that is a direct sum of n two dimensional vector spaces each containing p2 points this is in contrast to the more usual choice of a two dimensional phase space containing p2n points a useful aspect of this approach is that we can relate complete separability of density matrices and their wigner functions in a natural way we discuss this in detail for bipartite systems and present the generalization to arbitrary numbers of subsystems when p is odd special attention is required for two qubits p 2 and our technique fails to establish the separability property for more than two qubits finally we give a brief discussion of hamiltonian dynamics in the language developed in the paper

implementing the dantzig fulkerson johnson algorithm for large traveling salesman problems dantzig fulkerson and johnson 1954 introduced the cutting plane method as a means of attacking the traveling salesman problem this method has been applied to broad classes of problems in combinatorial optimization and integer programming in this paper we discuss an implementation of dantzig et al s method that is suitable for tsp instances having 1 000 000 or more cities our aim is to use the study of the tsp as a step towards understanding the applicability and limits of the general cutting plane method in large scale applications

the circuit polytope facets given an undirected graph g v e and a cost vector c e the weighted girth problem is to find a circuit in g having minimum total cost this problem is in general x1d4a9 x1d4ab hard a promising approach to the solution of hard combinatorial optimization problems is given by the so called cutting plane methods these involve linear programming techniques based on a partial description of the convex hull of the incidence vectors of possible solutions we consider the weighted girth problem in the case where g is the complete graph kn and study the facial structure of the circuit polytope pcn and some related polyhedra in the appendix we give complete characterizations of pcn for n up to 6 we were motivated to study the weighted girth problem by the fact that this problem and variations of it appear as a subproblem namely the pricing problem in a column generation approach to the vehicle routing problem

solution of large scale symmetric travelling salesman problems in this paper we report on a cutting plane procedure with which we solved symmetric travelling salesman problems of up to 1000 cities to optimality our implementation is based on a fast lp solver ibm s mpsx and makes effective use of polyhedral results on the symmetric travelling salesman polytope we describe the important ingredients of our code and give an extensive documentation of its computational performance

worst case comparison of valid inequalities for the tsp we consider most of the known classes of valid inequalities for the graphical travelling salesman polyhedron and compute the worst case improvement resulting from their addition to the subtour polyhedron for example we show that the comb inequalities cannot improve the subtour bound by a factor greater than 10 9 the corresponding factor for the class of clique tree inequalities is 8 7 while it is 4 3 for the path configuration inequalities

separation algorithms for classes of stsp inequalities arising from a new stsp relaxation the problem ofseparating a class of inequalities that are valid for the symmetric traveling salesman problem stsp consists of devising an efficient method that given a fractional pointx for a relaxation of the stsp either finds an inequality in the given class of stsp inequalities that is violated byx or determines that there are no such violated inequalities we can define important classes of stsp inequalities by performing naddef s and rinaldi s 1993 node lifting operation on stsp inequalities defined on small graphs we present efficient methods forexactly separating large classes of stsp inequalities that arise from an stsp relaxation related to node lifting in particular we show how to find efficiently the most violated inequality in the class of stsp inequalities having abackbone set with a constant numberk of vertices and thus separate in polynomial time the class of all stsp inequalities whose backbone set has at mostk vertices

tight informationally complete quantum measurements we introduce a class of informationally complete positive operator valued measures which are in analogy with a tight frame as close as possible to orthonormal bases for the space of quantum states these measures are distinguished by an exceptionally simple state reconstruction formula which allows painless quantum state tomography complete sets of mutually unbiased bases and symmetric informationally complete positive operator valued measures are both members of this class the latter being the unique minimal rank one members recast as ensembles of pure quantum states the rank one members are in fact equivalent to weighted 2 designs in complex projective space these measures are shown to be optimal for quantum cloning and linear quantum state tomography

hamiltonian path and symmetric travelling salesman polytopes we introduce a projective approach for studying symmetric travelling salesman polytopes stsps thesymmetric travelling salesman polytope stsp v resp hamiltonian path polytope hp v is the convex hull of incidence vectors of all hamiltonian cycles resp paths on the complete undirected graph with node setv for any nodeh v hp v is aprojection of stsp v h we show that hp v and stsp v h are isomorphic and hp v is of full dimension minus one by this projective approach we obtain generalclique lifting results all based on simple conditions for deriving large new classes of stsp facets these results apply to all known non trivial stsp facets and generalize clique lifting results of maurras 1975 grotschel and padberg 1979 and naddef and rinaldi 1988

an unconstrained calculus of variations formulation for generalized optimal control problems and for the constrained problem of bolza abstract this paper has two main ideas the first idea is that constrained problems in optimal control theory and the calculus of variations can be associated with unconstrained calculus of variations problems by using multipliers this allows us to obtain a true lagrange multiplier rule where both the original variables and the multipliers can be explicitly and easily determined the second idea is that critical point solutions to the associated problem which include the determination of the multipliers immediately follows from euler lagrange equations for the unconstrained problem this critical point solution is a necessary condition for the original problem we also show that these methods can be combined with previous work by the authors to obtain efficient and accurate numerical solutions to the original problems where no such general numerical methods currently seem to exist this seems to be an important development since the lack of such methods has hindered the usefulness of the theory finally we indicate that a large variety of nonclassical optimization problems can be solved by this association

new inequalities for the general routing problem abstract a large new class of valid inequalities is introduced for the general routing problem which properly contains the class of k c constraints these are also valid for the rural postman problem a separation algorithm is given for a subset of these inequalities which runs in polynomial time

the general routing polyhedron a unifying framework it is shown how to transform the general routing problem grp into a variant of the graphical travelling salesman problem gtsp this transformation yields a projective characterisation of the grp polyhedron using this characterisation it is shown how to convert facets of the gtsp polyhedron into valid inequalities for the grp polyhedron the resulting classes of inequalities subsume several previously published classes

hudson s theorem for finite dimensional quantum systems we show that on a hilbert space of odd dimension the only pure states to possess a non negative wigner function are stabilizer states the clifford group is identified as the set of unitary operations which preserve positivity the result can be seen as a discrete version of hudson s theorem hudson established that for continuous variable systems the wigner function of a pure state has no negative values if and only if the state is gaussian turning to mixed states it might be surmised that only convex combinations of stabilizer states give rise to non negative wigner distributions we refute this conjecture by means of a counterexample further we give an axiomatic characterization which completely fixes the definition of the wigner function and compare two approaches to stabilizer states for hilbert spaces of prime power dimensions in the course of the discussion we derive explicit formulas for the number of stabilizer codes defined on such systems

optimizing quantum process tomography with unitary 2 designs we show that weighted unitary 2 designs define optimal measurements on the system ancilla output state for ancilla assisted process tomography of unital quantum channels examples include complete sets of mutually unbiased unitary operator bases each of these specifies a minimal series of optimal orthogonal measurements general quantum channels are also considered

the symmetric traveling salesman polytope revisited in this paper we present a tour of the symmetric traveling salesman polytope focusing on inequalities that can be defined on sets of nodes most widely known inequalities are of this type many papers have appeared that give increasingly complex valid inequalities for this polytope but little intuition on why these inequalities are valid has been given to help in understanding these inequalities we develop an intuition into their validity by giving a unifying way of defining them through a sequential lifting procedure this procedure is based on lifting the slack variables associated with subtour elimination inequalities defined on sets of nodes called teeth we apply this procedure to some known classes of valid inequalities for the traveling salesman polytope tsp respectively comb brush star path and bipartition inequalities where the lifting coefficients are sequence independent for comb star and bipartition inequalities we provide new and noninductive proofs of validity directly inspired by this lifting procedure we also give an example where a facet defining inequality is derived from the lifting procedure but where the lifting coefficients are sequence dependent we finally study the ladder inequalities and show that they can be generated by an extension of the general procedure where the lifted variables are different from the slack variables of subtour elimination inequalities

efficient separation routines for the symmetric traveling salesman problem i general tools and comb separation this is the first of a series of two papers dedicated to efficient separation heuristics for violated inequalities in the context of solving to optimality instances of the traveling salesman problem via branch and cut in this paper we give the basic ideas behind these heuristics and design heuristics for comb separation the second paper will deal with more complex inequalities such as clique tree star or path and ladder inequalities and give computational results

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

the black and white traveling salesman problem the black and white traveling salesman problem bwtsp is defined on a graph g whose vertex set is partitioned into black and white vertices the aim is to design a shortest hamiltonian tour on g subject to cardinality and length constraints both the number of white vertices as well as the length of the tour between two consecutive black vertices are bounded above the bwtsp has applications in airline scheduling and in telecommunications this paper proposes an integer linear formulation for the undirected bwtsp as well as several classes of valid inequalities an exact branch and cut algorithm is then developed extensive tests show that it can solve exactly instances involving up to 100 vertices the algorithm can also be applied directly to solve unit demand vehicle routing problems of similar sizes

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

symmetric informationally complete positive operator valued measures and the extended clifford group we describe the structure of the extended clifford group defined to be the group consisting of all operators unitary and antiunitary which normalize the generalized pauli group or weyl heisenberg group as it is often called we also obtain a number of results concerning the structure of the clifford group proper i e the group consisting just of the unitary operators which normalize the generalized pauli group we then investigate the action of the extended clifford group operators on symmetric informationally complete positive operator valued measures or sic povms covariant relative to the action of the generalized pauli group we show that each of the fiducial vectors which has been constructed so far including all the vectors constructed numerically by renes et al is an eigenvector of one of a special class of order 3 clifford unitaries this suggests a strengthening of a conjecture of zauner s we give a complete characterization of the orbits and stability groups in dimensions 2 7 fina

unconditionally secure key distribution in higher dimensions by depolarization this paper presents a prepare and measure scheme using n dimensional quantum particles as information carriers where n is a prime power one of the key ingredients used to resist eavesdropping in this scheme is to depolarize all pauli errors introduced to the quantum information carriers using the shor preskill type argument we prove that this scheme is unconditionally secure against all attacks allowed by the laws of quantum physics for n 2n 2 each information carrier can be replaced by n entangled qubits in this case there is a family of eavesdropping attacks on which no unentangled qubit based prepare and measure pm quantum key distribution scheme known to date can generate a provably secure key in contrast under the same family of attacks our entangled qubit based scheme remains secure whenever 2nges4 this demonstrates the advantage of using entangled particles as information carriers and of using depolarization of pauli errors to combat eavesdropping attacks more drastic than those that can be handled by unentangled qubit based prepare and measure schemes