faster exact solutions for some np hard problems this paper considers a number of np complete problems and provides faster algorithms for solving them the solutions are based on a recursive partitioning of the problem domain and careful elimination of some of the branches along the search without actually checking them the time complexity of the proposed algorithms is of the form o 2en for constant 0 e 1 where n is the output size of the problem in particular such algorithms are presented for the exact sat and exact hitting set problems with e 0 3212 and for the exact 3sat problem with e 0 2072 both algorithms improve on previous ones proposed in the literature

an o n lg k 2 n 2 time and o k 2 n k space algorithm for certain np complete problems for a subset of np complete problems schroeppel and shamir 1981 described a general purpose algorithm with o 2n2 time and o 2n4 space complexity in this note we try to combine their idea with the selection in x y problem this approach yields a family of algorithms solving a subset of np complete problems in o nlg k 2n2 time and o k 2nk space

optimal multi way number partitioning the np hard number partitioning problem is to separate a multiset s of n positive integers into k subsets such that the largest sum of the integers assigned to any subset is minimized the classic application is scheduling a set of n jobs with different runtimes on k identical machines such that the makespan the elapsed time to complete the schedule is minimized the two way number partitioning decision problem is one of the original 21 problems that richard karp proved np complete it is also one of garey and johnson s six fundamental np complete problems and the only one based on numbers this article explores algorithms for solving multi way number partitioning problems optimally we explore previous algorithms as well as our own algorithms which fall into three categories sequential number partitioning snp a branch and bound algorithm binary search improved bin completion bsibc a bin packing algorithm and cached iterative weakening ciw an iterative weakening algorithm we show experimentally that for large random numbers snp and ciw are state of the art algorithms depending on the values of n and k both algorithms outperform the previous state of the art by up to seven orders of magnitude in terms of runtime

a simplex algorithm whose average number of steps is bounded between two quadratic functions of the smaller dimension it has been a challenge for mathematicians to confirm theoretically the extremely good performance of simplex type algorithms for linear programming in this paper the average number of steps performed by a simplex algorithm the so called self dual method is analyzed the algorithm is not started at the traditional point 1 l t but points of the form 1 e e 2 t with e sufficiently small are used the result is better in two respects than those of the previous analyses first it is shown that the expected number of steps is bounded between two quadratic functions c 1 min m n 2 and c 2 min m n 2 of the smaller dimension of the problem this should be compared with the previous two major results in the field borgwardt proves an upper bound of o n 4 m 1 n 1 under a model that implies that the zero vector satisfies all the constraints and also the algorithm under his consideration solves only problems from that particular subclass smale analyzes the self dual algorithm starting at 1 1 t he shows that for any fixed m there is a constant c m such the expected number of steps is less than c m ln n m m 1 megiddo has shown that under smale s model an upper bound c m exists thus for the first time a polynomial upper bound with no restrictions except for nondegeneracy on the problem is proved and for the first time a nontrivial lower bound of precisely the same order of magnitude is established both borgwardt and smale require the input vectors to be drawn from spherically symmetric distributions in the model in this paper invariance is required only under certain

extremality conditions for generalized channels a generalized channel is a completely positive map that preserves trace on a given subspace we find conditions under which a generalized channel with respect to a positively generated subspace j is an extreme point in the set of all such generalized channels as a special case this yields extremality conditions for quantum protocols in particular we obtain new extremality conditions for quantum 1 testers with 2 outcomes which correspond to yes no measurements on the set of quantum channels

evolutionary topology optimization using design space adjustment based on fixed grid design space optimization for topology based on fixed grid is proposed and its superiority to conventional topology optimization is shown in the conventional topology optimization the design domain is fixed it is however desirable to make the design domain evolve into a better one during optimization process by increasing or decreasing the number of design pixels or variables which we call design space optimization a breakthrough in obtaining sensitivities when design space expands has been made recently with necessary mathematical background but due to coupling effect and others sensitivity results have not been satisfactory three innovative implementations are developed in this paper firstly the proper characteristics of artificial material are defined the second one is to decouple neighbouring elements for exact design space sensitivities the previous design space optimization has been tedious because only one layer can be added so the third innovation is a new expansion strategy with multi layers based on design space sensitivities as a result the proposed evolutionary method can get an optimum much faster than ever before especially for large scale problems it is also conjectured that this gives higher probability of getting the global optimum as confirmed by numerical examples

generalized channels channels for convex subsets of the state space let k be a convex subset of the state space of a finite dimensional c algebra we study the properties of channels on k which are defined as affine maps from k into the state space of another algebra extending to completely positive maps on the subspace generated by k we show that each such map is the restriction of a completely positive map on the whole algebra called a generalized channel we characterize the set of generalized channels and also the equivalence classes of generalized channels having the same value on k moreover if k contains the tracial state the set of generalized channels forms again a convex subset of a multipartite state space this leads to a definition of a generalized supermap which is a generalized channel with respect to this subset we prove a decomposition theorem for generalized supermaps and describe the equivalence classes the set of generalized supermaps having the same value on equivalent generalized channels is also characterized special cases include quantum

on the minimum gap between sums of square roots of small integers let k and n be positive integers n k define r n k to be the minimum positive value of a1 ak b1 bk where a1 a2 ak b1 b2 bk are positive integers no larger than n define r n k to be logr n k it is important to find tight bounds for r n k and r n k in connection to the sum of square roots problem a famous open problem in computational geometry the current best lower bound and upper bound are far apart in this paper we prove an upper bound of 2o n logn for r n k which is better than the best known result o 22klogn whenever n cklogk for some constant c in particular our result implies an algorithm subexponential in k i e with time complexity 2o k logn o 1 to compare two sums of k square roots of integers of value o klogk we then present an algorithm to find r n k exactly in nk o k time and in n k 2 o k space as an example we are able to compute r 100 7 exactly in a few hours on a single pc the numerical data indicate that the root separation bound is very far away from the true value of r n k

a parallel time hardware tradeoff t h o 2 sup n 2 for the knapsack problem a parallel algorithm for solving the knapsack problem on a single instruction multiple data machine with shared memory is presented the shared memory allows concurrent reading while concurrent writing is forbidden the knapsack problem is of size n which the algorithm solves in time t o n 2 sup n 2 sup epsilon when p o 2 sup n 2 sup 1 epsilon 0

some convergence results in perimeter controlled topology optimization abstract the computation of optimal topologies of elastic continuum structures using a constraint on the perimeter is investigated predicting macroscopic black white topologies without the use of homogenization techniques this approach is presently one of the most attractive approaches in topology optimization mathematical justifications are given for both the relaxation of the discrete value constraint on the design variable and for the finite element discretizations it turns out that the way in which the perimeter has been calculated to date the numerical results will not approximate the intended original problem but one with a taxi cab perimeter which measures lengths of structural edges after projection onto the coordinate axes

on the average number of steps of the simplex method of linear programming the goal is to give some theoretical explanation for the efficiency of the simplex method of george dantzig fixing the number of constraints and using dantzig s self dual parametric algorithm we show that the number of pivots required to solve a linear programming problem grows in proportion to the number of variables on the average

classical randomness in quantum measurements similarly to quantum states also quantum measurements can be mixed corresponding to a random choice within an ensemble of measuring apparatuses such mixing is equivalent to a sort of hidden variable which produces a noise of purely classical nature it is then natural to ask which apparatuses are indecomposable i e do not correspond to any random choice of apparatuses this problem is interesting not only for foundations but also for applications since most optimization strategies give optimal apparatuses that are indecomposable mathematically the problem is posed describing each measuring apparatus by a positive operator valued measure povm which gives the statistics of the outcomes for any input state the povms form a convex set and in this language the indecomposable apparatuses are represented by extremal points the analogous of pure states in the convex set of states differently from the case of states however indecomposable povms are not necessarily rank one e g von neumann measurements in this paper we give a complete classification of indecomposable apparatuses for discrete spectrum by providing different necessary and sufficient conditions for extremality of povms along with a simple general algorithm for the decomposition of a povm into extremals as an interesting application informationally complete measurements are analysed in this respect the convex set of povms is fully characterized by determining its border in terms of simple algebraic properties of the corresponding povms

fast finite methods for a system of linear inequalities abstract 1 several algorithms are known that solve a system of m linear inequalities in n variables using a polynomial number of steps in m and n for all inputs the simplex method runs in polynomial time in the average case proven and requires only linear in m number of steps in practice we consider a family of finite algorithms some of them involve o m n log3 n iteration steps in the average case under a certain additional assumption about the behavior of the computed approximations to a solution and only o m m n arithmetical operations are required per step the theoretical study is confirmed by some numerical experiments also two methods are presented that enable us to turn any converging algorithm for a system of linear inequalities with real inputs into an algorithm that converges in a finite number of steps

on space efficient algorithms for certain np complete problems some recent results claimed the existence of a class of algorithms for certain np complete problems with running time o n1g k 2n 2 and storage requirements o k 2n k for 2 k n in this note we show that those results do not hold implying that an algorithm with time o n 2n 2 and space o 2n 4 is still the best known solution for such class of np complete problems

computational simulation of trabecular adaptation progress in human proximal femur during growth abstract there are a large number of clinical and experimental studies that analyzed trabecular architecture as a result of bone adaptation however only a limited amount of quantitative data is currently available on the progress of trabecular adaptation during growth in this paper we proposed a two step numerical simulation method that predicts trabecular adaptation progress during growth using a recently developed topology optimization algorithm design space optimization dso under the hypothesis that the mechanisms of dso are functionally equivalent to those of bone adaptation we applied the proposed scheme to trabecular adaptation simulation in human proximal femur for the simulation the full trabecular architecture in human proximal femur was represented by a two dimensional fe model with 50 m resolution in step 1 we determined a reference value that regulates trabecular adaptation in human proximal femur in step 2 we simulated trabecular adaptation in human proximal femur during growth with the reference value derived in step 1 we analyzed the architectural and mechanical properties of trabecular patterns through iterations from the comparison with experimental data in the literature we showed that in the early growth stage trabecular adaptation was achieved mainly by increasing bone volume fraction or trabecular thickness while in the later stage of the development the trabecular architecture gained higher structural efficiency by increasing structural anisotropy with a relatively low level of bone volume fraction or trabecular thickness we demonstrated that the proposed numerical framework predicted the growing progress of trabecular bone that has a close correlation with experimental data

generalized relative entropies and the capacity of classical quantum channels we provide lower and upper bounds on the information transmission capacity of one single use of a classical quantum channel the lower bound is expressed in terms of the hoeffding capacity which we define similarly to the holevo capacity but replacing the relative entropy with the hoeffding distance similarly our upper bound is in terms of a quantity obtained by replacing the relative entropy with the recently introduced max relative entropy in the definition of the divergence radius of a channel

some remarks on c convexity abstract recently the study of completely positive maps has become important to the results of brown douglas and fillmore on ext a c algebra attempts to solve questions related to ext have often turned into questions about the matrix algebras mn in this paper we wish to discuss a notion of c convexity related to completely positive linear maps to state some facts about c convexity and to ask some questions about c convexity to a large degree the tone of this paper is expository

on the convex structure of process positive operator valued measures measurements on quantum channels are described by so called process positive operator valued measures or process povms we study implementing schemes of extremal process povms as it turns out the corresponding measurement must satisfy certain extremality property which is stronger than the usual extremality given by the convex structure this property motivates the introduction and investigation of the a convex structure of povms which generalizes both the usual convex and c convex structures we show that extremal points and faces of the set of process povms are closely related to a extremal points and a faces of povms for a certain subalgebra a we also give a characterization of a extremal and a pure povms

completely positive divisibility does not mean markovianity in the classical domain it is well known that divisibility does not imply that a stochastic process is markovian however for quantum processes divisibility is often considered to be synonymous with markovianity we show that completely positive divisible quantum processes can still involve non markovian temporal correlations that we then fully classify using the recently developed process tensor formalism which generalizes the theory of stochastic processes to the quantum domain

a t o 2 n 2 s o 2 n 4 algorithm for certain np complete problems in this paper we develop a general purpose algorithm that can solve a number of np complete problems in time t o 2 n 2 and space s o 2 n 4 the algorithm can be generalized to a fam

random polytopes their definition generation and aggregate properties the definition of random polytope adopted in this paper restricts consideration to those probability measures satisfying two properties first the measure must induce an absolutely continuous distribution over the positions of the bounding hyperplanes of the random polytope and second it must result in every point in the space being equally as likely as any other point of lying within the random polytope an efficient monte carlo method for their computer generation is presented together with analytical formulas characterizing their aggregate properties in particular it is shown that the expected number of extreme points for such random polytopes increases monotonically in the number of constraints to the limiting case of a polytope topologically equivalent to a hypercube the implied upper bound of 2n wheren is the dimensionality of the space is significantly less than mcmullen s attainable bound on the maximal number of vertices even for a moderate number of constraints

extending quantum operations for a given set of input output pairs of quantum states or observables we ask the question whether there exists a physically implementable transformation that maps each of the inputs to the corresponding output the physical maps on quantum states are trace preserving completely positive maps but we also consider variants of these requirements we generalize the definition of complete positivity to linear maps defined on arbitrary subspaces then formulate this notion as a semidefinite program and relate it by duality to approximative extensions of this map this gives a characterization of the maps which can be approximated arbitrarily well as the restriction of a map that is completely positive on the whole algebra also yielding the familiar extension theorems on operator spaces for quantum channel extensions and extensions by probabilistic operations we obtain semidefinite characterizations and we also elucidate the special case of abelian inputs or outputs finally revisiting a theorem by alberti and uhlmann we provide simpler and more widely applicable conditions for certain extension problems on qubits and by using a semidefinite programming formulation we exhibit counterexamples to seemingly reasonable but false generalizations of the alberti uhlmann theorem

comparison of quantum statistical models equivalent conditions for sufficiency a family of probability distributions i e a statistical model is said to be sufficient for another if there exists a transition matrix transforming the probability distributions in the former to the probability distributions in the latter the blackwell sherman stein bss theorem provides necessary and sufficient conditions for one statistical model to be sufficient for another by comparing their information values in statistical decision problems in this paper we extend the bss theorem to quantum statistical decision theory where statistical models are replaced by families of density matrices defined on finite dimensional hilbert spaces and transition matrices are replaced by completely positive trace preserving maps i e coarse grainings the framework we propose is suitable for unifying results that previously were independent like the bss theorem for classical statistical models and its analogue for pairs of bipartite quantum states recently proved by shmaya an important role in this paper is played by statistical morphisms namely affine maps whose definition generalizes that of coarse grainings given by petz and induces a corresponding criterion for statistical sufficiency that is weaker and hence easier to be characterized than petz s

design space optimization using a numerical design continuation method summary a generalized optimization problem in which design space is also a design to be found is de9ned and a numerical implementation method is proposed in conventional optimization only a portion of structural parameters is designated as design variables while the remaining set of other parameters related to the design space are often taken for granted a design space is speci9ed by the number of design variables and the layout or con9guration to solve this type of design space problems a simple initial design space is selected and gradually improved while the usual design variables are being optimized to make the design space evolve into a better one one may increase the number of design variables but in this transition there are discontinuities in the objective and constraint functions accordingly the sensitivity analysis methods based on continuity will not apply to this discontinuous stage to overcome the di culties a numerical continuation scheme is proposed based on a new concept of a pivot phase design space two new categories of structural optimization problems are formulated and concrete examples shown copyright 2001 john wiley sons ltd

the operational meaning of min and max entropy in this paper we show that the conditional min entropy h min a b of a bipartite state rhoab is directly related to the maximum achievable overlap with a maximally entangled state if only local actions on the b part of rhoab are allowed in the special case where a is classical this overlap corresponds to the probability of guessing a given b in a similar vein we connect the conditional max entropy h max a b to the maximum fidelity of rhoab with a product state that is completely mixed on a in the case where a is classical this corresponds to the security of a when used as a secret key in the presence of an adversary holding b because min and max entropies are known to characterize information processing tasks such as randomness extraction and state merging our results establish a direct connection between these tasks and basic operational problems for example they imply that the logarithm of the probability of guessing a given b is a lower bound on the number of uniform secret bits that can be extracted from a relative to an adversary holding b

extremal generalized quantum measurements a measurement on a section k of the set of states of a finite dimensional c algebra is defined as an affine map from k to a probability simplex special cases of such sections are used in description of quantum networks in particular quantum channels measurements on a section correspond to equivalence classes of the so called generalized povms which are called quantum testers in the case of networks we find extremality conditions for measurements on k and characterize generalized povms such that the corresponding measurement is extremal these results are applied to the set of channels we find explicit extremality conditions for two outcome measurements on qubit channels and give an example of an extremal qubit 1 tester such that the corresponding measurement is not extremal

an o 2 0 304n algorithm for solving maximum independent set problem a faster algorithm for finding a maximum independent set in a graph is presented the algorithm is an improved version of the one by tarjan and trojanowski 7 a technique to further accelerate this algorithm is also described

conditions for optimal input states for discrimination of quantum channels we find optimality conditions for testers in discrimination of quantum channels these conditions are obtained using semidefinite programming and are similar to optimality conditions for discrimination of quantum states we get a simple condition for existence of an optimal tester with any given input state with maximal schmidt rank in particular with a maximally entangled input state in the case when maximally entangled state is not optimal an upper bound on the optimal success probability is obtained the results for discrimination of two channels are applied to covariant channels qubit channels unitary channels and simple projective measurements

divisibility and information flow notions of quantum markovianity for noninvertible dynamical maps we analyze the relation between completely positive cp divisibility and the lack of information backflow for an arbitrary not necessarily invertible dynamical map it is well known that cp divisibility always implies a lack of information backflow moreover these two notions are equivalent for invertible maps in this letter it is shown that for a map which is not invertible the lack of information backflow always implies the existence of a completely positive propagator which however needs not be trace preserving interestingly for a wide class of image nonincreasing dynamical maps this propagator becomes trace preserving as well and hence the lack of information backflow implies cp divisibility this result sheds new light into the structure of the time local generators giving rise to cp divisible evolutions we show that if the map is not invertible then positivity of dissipation decoherence rates is no longer necessary for cp divisibility

polynomial average time satisfiability problems a class of random conjunctive normal form cnf predicates is characterized by v the number of variables from which literals may be formed giving 2v literals p v the probability that a given literal is in a random clause and t v the number of random independently selected clauses in a random predicate determining satisfiability on such a class using backtracking takes polynomial average time under the following conditions 1 limv vp v 0 and t v ln2 v ln v l p v and 2 lim vp v lim p v 0 and t v ln2 ep v exp 2vp v for any fixed 0 additional cases are also covered an analysis by goldberg showed that problems for which p v is constant can be solved in polynomial average time by a simplified davis putnam procedure we give an algorithm for solving problems with t v ln lnv ln3 in polynomial worst case time thus all problem sets for which p v or t v grows extremely rapidly or extremely slowly can be solved in polynomial average time we also show that backtracking requires exponential average time when limvp v 0 and t v ln2 v ln v 1 p v and when limvp v limv vp v 0 and t v ln2 ge ep v exp 2vp v

clean positive operator valued measures in quantum mechanics the statistics of the outcomes of a measuring apparatus is described by a positive operator valued measure povm a quantum channel transforms povms into povms generally irreversibly thus losing some of the information retrieved from the measurement this poses the problem of which povms are undisturbed i e they are not irreversibly connected to another povm we will call such povms clean in a sense the clean povms would be perfect since they would not have any additional extrinsical noise quite unexpectedly it turns out that such a cleanness property is largely unrelated to the convex structure of povms and there are clean povms that are not extremal and vice versa in this article we solve the cleannes classification problem for number n of outcomes n d d dimension of the hilbert space and we provide a set of either necessary or sufficient conditions for n d along with an iff condition for the case of informationally complete povms for n d2

average polynomial time complexity of some np complete problems abstract we define clique n n to be the problem clique restricted to the set of graphs g v e such that e n v in this paper it is proved that when n n n e e a rational number 0 clique n n is an np complete problem and under the assumption of a uniform distribution on the set of instances clique n n can be solved by a deterministic algorithm in average and almost everywhere polynomial time some cases of nonuniform distributions on the set of instances for which the usual problem clique is solved in average polynomial time are also considered

a complete anytime algorithm for number partitioning abstract given a set of numbers the two way number partitioning problem is to divide them into two subsets so that the sum of the numbers in each subset are as nearly equal as possible the problem is np complete based on a polynomial time heuristic due to karmarkar and karp we present a new algorithm called complete karmarkar karp ckk that optimally solves the general number partitioning problem and significantly outperforms the best previously known algorithms for large problem instances for numbers with twelve significant digits or less ckk can optimally solve two way partitioning problems of arbitrary size in practice for numbers with greater precision ckk first returns the karmarkar karp solution then continues to find better solutions as time allows over seven orders of magnitude improvement in solution quality is obtained in less than an hour of running time rather than building a single solution one element at a time or modifying a complete solution ckk constructs subsolutions and combines them together in all possible ways this approach may be effective for other np hard problems as well

base norms and discrimination of generalized quantum channels we introduce and study norms in the space of hermitian matrices obtained from base norms in positively generated subspaces these norms are closely related to discrimination of so called generalized quantum channels including quantum states channels and networks we further introduce generalized quantum decision problems and show that the maximal average payoffs of decision procedures are again given by these norms we also study optimality of decision procedures in particular we obtain a necessary and sufficient condition under which an optimal 1 tester for discrimination of quantum channels exists such that the input state is maximally entangled

hilbert s projective metric in quantum information theory we introduce and apply hilbert s projective metric in the context of quantum information theory the metric is induced by convex cones such as the sets of positive separable or positive partial transpose operators it provides bounds on measures for statistical distinguishability of quantum states and on the decrease of entanglement under protocols involving local quantum operations and classical communication or under other cone preserving operations the results are formulated in terms of general cones and base norms and lead to contractivity bounds for quantum channels for instance improving ruskai s trace norm contraction inequality a new duality between distinguishability measures and base norms is provided for two given pairs of quantum states we show that the contraction of hilbert s projective metric is necessary and sufficient for the existence of a probabilistic quantum operation that maps one pair onto the other inequalities between hilbert s projective metric and the chernoff bound the

exact semidefinite formulations for a class of random and non random nonconvex quadratic programs we study a class of quadratically constrained quadratic programs qcqps called diagonal qcqps which contain no off diagonal terms x j x k for j ne k and we provide a sufficient condition on the problem data guaranteeing that the basic shor semidefinite relaxation is exact our condition complements and refines those already present in the literature and can be checked in polynomial time we then extend our analysis from diagonal qcqps to general qcqps i e ones with no particular structure by reformulating a general qcqp into diagonal form we establish new polynomial time checkable sufficient conditions for the semidefinite relaxations of general qcqps to be exact finally these ideas are extended to show that a class of random general qcqps has exact semidefinite relaxations with high probability as long as the number of constraints grows no faster than a fixed polynomial in the number of variables to the best of our knowledge this is the first result establishing the exactness of the semidefinite relaxation for random general qcqps

bone remodeling and structural optimization bone remodeling has been viewed both as a process which adapts bone tissue to the mechanical environment at each point in the structure and as a process which optimally adjusts the tissue distribution within bones to bear the loads placed on them we have developed a connection between these two views of bone remodeling in a restricted sense we start with a remodeling rate equation based on strain energy density we then define an indicator function which is a weighted sum of total strain energy and a measure of bone mass and we show that finding bone density distributions in which the remodeling rate equation predicts no changes with time is the same as finding density distributions in which the indicator function is insensitive to small changes in density the set point in the remodeling rate equation corresponds to a parameter in the indicator function which determines the relative importance of bone mass and strain energy in the optimization indicator function we have not assessed whether the density distributions which make the density rate of change zero are actually local or global minima for the indicator function in this study but a related study shows that there is a single unique minimum for the indicator function developed here implying that a unique solution exists for the bone remodeling rate equations considered in this study

improved asymptotic analysis of the average number of steps performed by the self dual simplex algorithm in this paper we analyze the average number of steps performed by the self dual simplex algorithm for linear programming under the probabilistic model of spherical symmetry the model was proposed by smale consider a problem ofn variables withm constraints smale established that for every number of constraintsm there is a constantc m such that the number of pivot steps of the self dual algorithm m n is less thanc m lnn m m 1 we improve upon this estimate by showing that m n is bounded by a function ofm only the symmetry of the function inm andn implies that m n is in fact bounded by a function of the smaller ofm andn

constructive method for detecting the information backflow of non markovian dynamics we investigate the relation between two approaches to the characterization of quantum markovianity divisibility and lack of information backflow we show that a bijective dynamical map is completely positive divisible if and only if a monotonic nonincrease of distinguishability is observed for two equiprobable states of the evolving system and an ancilla moreover our proof is constructive given any such map that is not completely positive divisible we give an explicit construction of two states that when taken with the same a priori probability exhibit information backflow finally while an ancilla is necessary for the equivalence to hold in general we show that it is always possible to witness the non markovianity of bijective maps without using any entanglement between the system and ancilla

a counterexample to the hirsch conjecture the hirsch conjecture 1957 stated that the graph of a d dimensional polytope with n facets cannot have combinatorial diameter greater than n d that is any two vertices of the polytope can be connected by a path of at most n d edges this paper presents the rst counterexample to the conjecture our polytope has dimension 43 and 86 facets it is obtained from a 5 dimensional polytope with 48 facets that violates a certain generalization of the d step conjecture of klee and walkup

on a measure of distance for quantum strategies the present paper studies an operator norm that captures the distinguishability of quantum strategies in the same sense that the trace norm captures the distinguishability of quantum states or the diamond norm captures the distinguishability of quantum channels characterizations of its unit ball and dual norm are established via strong duality of a semidefinite optimization problem a full formal proof of strong duality is presented for the semidefinite optimization problem in question this norm and its properties are employed to generalize a state discrimination result of gutoski and watrous in proceedings of the 22nd symposium on theoretical aspects of computer science stacs 05 lecture notes in computer science vol 3404 springer 2005 pp 605 616 the generalized result states that for any two convex sets s0 s1 of strategies there exists a fixed interactive measurement scheme that successfully distinguishes any choice of s0 s0 from any choice of s1 s1 with bias proportional to the minim

a measure conquer approach for the analysis of exact algorithms for more than 40 years branch reduce exponential time backtracking algorithms have been among the most common tools used for finding exact solutions of np hard problems despite that the way to analyze such recursive algorithms is still far from producing tight worst case running time bounds motivated by this we use an approach that we call measure conquer as an attempt to step beyond such limitations the approach is based on the careful design of a nonstandard measure of the subproblem size this measure is then used to lower bound the progress made by the algorithm at each branching step the idea is that a smarter measure may capture behaviors of the algorithm that a standard measure might not be able to exploit and hence lead to a significantly better worst case time analysis in order to show the potentialities of measure conquer we consider two well studied np hard problems minimum dominating set and maximum independent set for the first problem we consider the current best algorithm and prove thanks to a better measure a much tighter running time bound for it for the second problem we describe a new simple algorithm and show that its running time is competitive with the current best time bounds achieved with far more complicated algorithms and standard analysis our examples show that a good choice of the measure made in the very first stages of exact algorithms design can have a tremendous impact on the running time bounds achievable

min and max relative entropies and a new entanglement monotone two new relative entropy quantities called the min and max relative entropies are introduced and their properties are investigated the well known min and max entropies introduced by renner are obtained from these we define a new entanglement monotone which we refer to as the max relative entropy of entanglement and which is an upper bound to the relative entropy of entanglement we also generalize the min and max relative entropies to obtain smooth min and max relative entropies these act as parent quantities for the smooth renyi entropies eth zurich ph d dissertation 2005 and allow us to define the analogues of the mutual information in the smooth renyi entropy framework further the spectral divergence rates of the information spectrum approach are shown to be obtained from the smooth min and max relative entropies in the asymptotic limit

new algorithms for exact satisfiability the exact satisfiability problem is to determine if a cnf formula has a truth assignment satisfying exactly one literal in each clause exact 3 satisfiability is the version in which each clause contains at most three literals in this paper we present algorithms for exact satisfiability and exact 3 satisfiability running in time o 20 2325n and o 20 1379n respectively the previously best algorithms have running times o 20 2441n for exact satisfiability methods oper res 43 1981 419 431 and o 20 1626n for exact 3 satisfiability annals of mathematics and artificial intelligence 43 1 2005 173 193 and zapiski nauchnyh seminarov pomi 293 2002 118 128 we extend the case analyses of these papers and observe that a formula not satisfying any of our cases has a small number of variables for which we can try all possible truth assignments and for each such assignment solve the remaining part of the formula in polynomial time

a sharp upper bound for the expected number of shadow vertices in lp polyhedra under orthogonal projection on two dimensional planes consider a set a1 am u v n satisfying a mild nondegeneracy condition when the polyhedron x x a1tx 1 amtx 1 n is orthogonally projected on the two dimensional plane linu v then some of its vertices may be mapped on vertices of the two dimensional image of x these vertices of x will be called shadow vertices under that projection their number will be denoted by s this figure s has an outstanding relevance for the complexity of a special simplex method the dimension by dimension method for solving lps of the type maximize vtx s t a1tx 1 amtx 1 where x v a1 am n and m n this method applies the shadow vertex algorithm in n 1 stages of increasing dimension by constructing a simplex path that visits only shadow vertices in each stage specific versions of the vectors u and v are used to define the projection plane and to control the application of the shadow vertex algorithm so s delivers an upper bound for the length of such a path in one stage we want to study the average case complexity of these algorithms and are therefore interested in em ns the expected value of s for that purpose we assume a distribution of the linear programming problems corresponding to the rotation symmetry model rsm the vectors a1 am u v are distributed on n 0 independently identically and symmetrically under rotations in this paper we improve the upper bounds for em ns from our polynomiality proofs of 1982 and 1987 there we had found an upper bound of m1 n 1 i n3 i const valid for all rsm distributions and all pairs m n based on a refined evaluation technique for space angles on a reorganization and on substantial modifications of the old proof we are now able to confirm a bound em ns m1 n 1 i n2 i const for all pairs m n with m n this result had been desired since 1982 because then a bound m1 n 1 i n2 i const had been derived exclusively for the situation where n is fixed and m tends to infinity all the time the question had been open whether the asymptotical behaviour would be better than the behaviour in moderate dimensions or whether the bound for the moderate case was crude in addition we know from an asymptotic analysis in 1987 that for a special rsm distribution namely the uniform distribution on the unit sphere there is a lower bound for em ns of type m1 n 1 i n2 i const the combination of this result and the new upper bound shows that the new bound is sharp

polynomial expected behavior of a pivoting algorithm for linear complementarity and linear programming problems we show that a particular pivoting algorithm which we call the lexicographic lemke algorithm takes an expected number of steps that is bounded by a quadratic inn when applied to a random linear complementarity problem of dimensionn we present two probabilistic models both requiring some nondegeneracy and sign invariance properties the second distribution is concerned with linear complementarity problems that arise from linear programming in this case we give bounds that are quadratic in the smaller of the two dimensions of the linear programming problem and independent of the larger similar results have been obtained by adler and megiddo

barycentric decomposition of quantum measurements in finite dimensions we analyze the convex structure of the set of positive operator valued measures povms representing quantum measurements on a given finite dimensional quantum system with outcomes in a given locally compact hausdorff space the extreme points of the convex set are operator valued measures concentrated on a finite set of k d2 points of the outcome space d being the dimension of the hilbert space we prove that for second countable outcome spaces any povm admits a choquet representation as the barycenter of the set of extreme points with respect to a suitable probability measure in the general case krein milman theorem is invoked to represent povms as barycenters of a certain set of povms concentrated on k d2 points of the outcome space

a homogenization sampling procedure for calculating trabecular bone effective stiffness and tissue level stress a homogenization sampling procedure is introduced which allows computation of effective trabecular bone stiffness and individual trabecula level stress based on precise models of trabecular bone architecture three dimensional digitized images of 53 trabecular bone specimens with a resolution of 50 f per voxel were directly converted into three dimensional finite element meshes by making each voxel an 8 node isoparametric brick element owing to the large mesh of 8ooo elements an element by element pre conditioned conjugate gradient ebepcg program was written to solve the local homogenization finite element equations predicted effective stiffness measures correlated well with experimental results r2 z 0 73 the predicted effective stiffnesses tended to under estimate the experimental values average absolute errors in effective stiffness estimates ranged between 31 and 38 for the sampling procedure compared to a range 49 150 for a regression fit to volume fraction squared trabecula level stress ranged between 200 and 300 times that predicted by analyzing trabecular bone as a continuum both tensile and compressive tissue stresses were engendered by a continuum compressive stress trabecula level strain energy density sed ranged between 0 and 100 times the continuum sed value for two trabecular specimens in conclusion the homogenization sampling procedure consistently predicted the influence of trabecular bone architecture on effective stiffness it can also provide trabecular tissue stress and strain estimates for arbitrary global loading of whole bones tissue stresses and strains showed large variations compared to corresponding continuum level quantities

frequency response algorithms for h sub infinity optimization with time domain constraints a very broad framework for control system design is considered that encompasses frequency response methodologies for h sub infinity optimization that solve various aspects of the control design problem and that are less well known that state space methods the focus is on linear programming lawson s algorithm and trefethen s algorithm a modified lawson s algorithm is proposed and related to trefethan s method the modified algorithm is shown to be significantly faster than linear programming and lawson s algorithm it is also shown how to extend the modified lawson s algorithm so as to handle time domain constraints in addition to frequency domain specifications which distinguishes it from other h sub infinity optimization methods some steps are taken toward dealing with time domain constraints within an h sub infinity optimization framework

joint measurability of quantum effects and the matrix diamond in this work we investigate the joint measurability of quantum effects and connect it to the study of free spectrahedra free spectrahedra typically arise as matricial relaxations of linear matrix inequalities an example of a free spectrahedron is the matrix diamond which is a matricial relaxation of the l1 ball we find that joint measurability of binary positive operator valued measures is equivalent to the inclusion of the matrix diamond into the free spectrahedron defined by the effects under study this connection allows us to use results about inclusion constants from free spectrahedra to quantify the degree of incompatibility of quantum measurements in particular we completely characterize the case in which the dimension is exponential in the number of measurements conversely we use techniques from quantum information theory to obtain new results on spectrahedral inclusion for the matrix diamond