secret sharing based on a hard on average problem abstract the main goal of this work is to propose the design of secret sharing schemes based on hard on average problems it includes the description of a new multiparty protocol whose main application is key management in networks its unconditionally perfect security relies on a discrete mathematics problem classified as distnp complete under the average case analysis the so called distributional matrix representability problem thanks to the use of the search version of the mentioned decision problem the security of the proposed scheme is guaranteed although several secret sharing schemes connected with combinatorial structures may be found in the bibliography the main contribution of this work is the proposal of a new secret sharing scheme based on a hard on average problem which allows to enlarge the set of tools for designing more secure cryptographic applications

distributional word problem for groups this paper studies the word problem for finitely presented groups under the restriction that words can only be rewritten for a bounded number of times we obtain a similar result to the novikov boone theorem in the setting of average case np completeness the word problem we consider here is to decide when given a finite presentation of a group g words x y z and an integer k whether x 1yx z can be derived from z x 1yx z in the presentation of g in k steps we show that when each component of the instance is chosen uniformly at random the problem cannot be solved fast on average unless every np problem under every reasonable distribution on instances can be solved fast on average

an empirical analysis of scenario generation methods for stochastic optimization this work presents an empirical analysis of popular scenario generation methods for stochastic optimization including quasi monte carlo moment matching and methods based on probability metrics as well as a new method referred to as voronoi cell sampling solution quality is assessed by measuring the error that arises from using scenarios to solve a multi dimensional newsvendor problem for which analytical solutions are available in addition to the expected value the work also studies scenario quality when minimizing the expected shortfall using the conditional value at risk to quickly solve problems with millions of random parameters a reformulation of the risk averse newsvendor problem is proposed which can be solved via benders decomposition the empirical analysis identifies voronoi cell sampling as the method that provides the lowest errors with particularly good results for heavy tailed distributions a controversial finding concerns evidence for the ineffectiveness of widely used methods based on minimizing probability metrics under high dimensional randomness

controllability properties of perturbed distributed parameter systems abstract we analyse the action of perturbations on controllable distributed parameter systems we show that the most common controllability properties are not preserved under the action of perturbations whose norm is smaller than any prescribed tolerance v 0 in particular we show that small perturbations of the values of the coefficients of a system with delays can destroy the property of modal controllability finally a positive result which is important for application to the stabilization problem is presented

k adaptability in two stage robust binary programming over the last two decades robust optimization has emerged as a computationally attractive approach to formulate and solve single stage decision problems affected by uncertainty more recently robust optimization has been successfully applied to multistage problems with continuous recourse this paper takes a step toward extending the robust optimization methodology to problems with integer recourse which have largely resisted solution so far to this end we approximate two stage robust binary programs by their corresponding k adaptability problems in which the decision maker precommits to k second stage policies here and now and implements the best of these policies once the uncertain parameters are observed we study the approximation quality and the computational complexity of the k adaptability problem and we propose two mixed integer linear programming reformulations that can be solved with off the shelf software we demonstrate the effectiveness of our reformulations for stylized instances o

the complexity of manipulative attacks in nearly single peaked electorates many electoral control and manipulation problems which we will refer to in general as manipulative actions problems are np hard in the general case it has recently been noted that many of these problems fall into polynomial time if the electorate is single peaked i e is polarized along some axis issue however real world electorates are not truly single peaked there are usually some mavericks and so real world electorates tend merely to be nearly single peaked this paper studies the complexity of manipulative action algorithms for elections over nearly single peaked electorates we do this for many notions of nearness and for a broad range of election systems we provide instances where even one maverick jumps the manipulative action complexity up to np hardness but we also provide many instances where some number of mavericks can be tolerated without increasing the manipulative action complexity

membership problem for the modular group the modular group plays an important role in many branches of mathematics we show that the membership problem for the modular group is decidable in polynomial time to this end we develop a new syllable based version of the known subgroup graph approach the new approach can be used to prove additional results we demonstrate this by using it to prove that the membership problem for a free group remains decidable in polynomial time when elements are written in a normal form with exponents

rapid exponential feedback stabilization with unbounded control operators we study the wellposedness and the main features of a class of feedback control systems the involved control system is composed of the generator of a strongly continuous group for the free part and of an unbounded control operator so that the results can be applied to boundary or point control problems for partial differential equations of hyperbolic or petrowski type the feedback operator is explicit and one can achieve an arbitrary large decay rate for the closed loop system these results are proved under a controllability assumption and the proofs rely on general results about the algebraic riccati equation associated with the linear quadratic regulator problem

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

approximating k median via pseudo approximation we present a novel approximation algorithm for k median that achieves an approximation guarantee of 1 sqrt 3 epsilon improving upon the decade old ratio of 3 epsilon our improved approximation ratio is achieved by exploiting the power of pseudo approximation more specifically our approach is based on two components each of which we believe is of independent interest first we show that in order to give an alpha approximation algorithm for k median it is sufficient to give a pseudo approximation algorithm that finds an alpha approximate solution by opening k o 1 facilities this is a rather surprising result as there exist instances for which opening k 1 facilities may lead to a significantly smaller cost than that of opening only k facilities second we give such a pseudo approximation algorithm with alpha 1 sqrt 3 epsilon prior to our work it was not even known whether opening k o k facilities would help improve the approximation ratio

scenario tree generation for multiperiod financial optimization by optimal discretization multiperiod financial optimization is usually based on a stochastic model for the possible market situations there is a rich literature about modeling and estimation of continuous state financial processes but little attention has been paid how to approximate such a process by a discrete state scenario process and how to measure the pertaining approximation error in this paper we show how a scenario tree may be constructed in an optimal manner on the basis of a simulation model of the underlying financial process by using a stochastic approximation technique consistency relations for the tree may also be taken into account

on the sphere decoding algorithm i expected complexity the problem of finding the least squares solution to a system of linear equations where the unknown vector is comprised of integers but the matrix coefficient and given vector are comprised of real numbers arises in many applications communications cryptography gps to name a few the problem is equivalent to finding the closest lattice point to a given point and is known to be np hard in communications applications however the given vector is not arbitrary but rather is an unknown lattice point that has been perturbed by an additive noise vector whose statistical properties are known therefore in this paper rather than dwell on the worst case complexity of the integer least squares problem we study its expected complexity averaged over the noise and over the lattice for the sphere decoding algorithm of fincke and pohst we find a closed form expression for the expected complexity both for the infinite and finite lattice it is demonstrated in the second part of this paper that for a wide range of signal to noise ratios snrs and numbers of antennas the expected complexity is polynomial in fact often roughly cubic since many communications systems operate at noise levels for which the expected complexity turns out to be polynomial this suggests that maximum likelihood decoding which was hitherto thought to be computationally intractable can in fact be implemented in real time a result with many practical implications

finite dimensional controller design for infinite dimensional systems the circle criterion approach abstract the hilbert space generalization of the circle criterion is used for the finite dimensional compensator design for unstable infinite dimensional systems a design algorithm which generalizes the small gain based algorithm in 5 is presented

controllability and stability of a second order hyperbolic system with collocated sensor actuator a second order hyperbolic system with collocated sensor actuator is considered the semigroup generation is shown for the closed loop system under the feedback of a generic unbounded observation operator the equivalence between the exponential stability of the closed loop system and exact controllability of the open loop system is established in the general framework of well posed linear systems finally the conditions are weakened for the diagonal semigroups with finite dimensional inputs example of beam equation is presented to display the application

scenario reduction in stochastic programming an approach using probability metrics given a convex stochastic programming problem with a discrete initial probability distribution the problem of optimal scenario reduction is stated as follows determine a scenario subset of prescribed cardinality and a probability measure based on this set that is the closest to the initial distribution in terms of a natural or canonical probability metric arguments from stability analysis indicate that fortet mourier type probability metrics may serve as such canonical metrics efficient algorithms are developed that determine optimal reduced measures approximately numerical experience is reported for reductions of electrical load scenario trees for power management under uncertainty for instance it turns out that after 50 reduction of the scenario tree the optimal reduced tree still has about 90 relative accuracy

the computational complexity of generating random fractals we examine a number of models that generate random fractals the models are studied using the tools of computational complexity theory from the perspective of parallel computation diffusion limited aggregation and several widely used algorithms for equilibrating the ising model are shown to be highly sequential it is unlikely they can be simulated efficiently in parallel this is in contrast to mandelbrot percolation which can be simulated in constant parallel time our research helps shed light on the intrinsic complexity of these models relative to each other and to different growth processes that have been recently studied using complexity theory in addition the results may serve as a guide to simulation physics

a random np complete problem for inversion of 2d cellular automata in this paper we prove the co rnp completeness rnp random np of the following decision problem given a 2 dimensional cellular automaton a is a reversible when restricted to finite configurations extending a given row in order to prove this result we introduce a polynomial reduction from problems concerning finite tilings into problems concerning cellular automata then we add to tile sets and cellular automata two probability functions and we prove that these problems are not only co np complete but co rnp complete too

matrix transformation is complete for the average case in the theory of worst case complexity np completeness is used to establish that for all practical purposes the given np problem is not decidable in polynomial time in the theory of average case complexity average case completeness is supposed to play the role of np completeness however the average case reduction theory is still at an early stage and only a few average case complete problems are known we present the first algebraic problem complete for the average case under a natural probability distribution the problem is this given a unimodular matrix x of integers a set s of linear transformations of such unimodular matrices and a natural number n decide if there is a product of leq n not necessarily different members of s that takes x to the identity matrix

cosmological lower bound on the circuit complexity of a small problem in logic an exponential lower bound on the circuit complexity of deciding the weak monadic second order theory of one successor ws1s is proved circuits are built from binary operations or 2 input gates which compute arbitrary boolean functions in particular to decide the truth of logical formulas of length at most 610 in this second order language requires a circuit containing at least 10125 gates so even if each gate were the size of a proton the circuit would not fit in the known universe this result and its proof due to both authors originally appeared in 1974 in the ph d thesis of the first author in this article the proof is given the result is put in historical perspective and the result is extended to probabilistic circuits

stability of varepsilon approximate solutions to convex stochastic programs an analysis of convex stochastic programs is provided when the underlying probability distribution is subjected to small perturbations it is shown in particular that varepsilon approximate solution sets of convex stochastic programs behave lipschitz continuously with respect to certain distances of probability distributions that are generated by the relevant integrands it is shown that these results apply to linear two stage stochastic programs with random recourse we discuss the consequences on associating fortet mourier metrics to two stage models and on the asymptotic behavior of empirical estimates of such models respectively

a probability metrics approach for reducing the bias of optimality gap estimators in two stage stochastic linear programming monte carlo sampling based estimators of optimality gaps for stochastic programs are known to be biased when bias is a prominent factor estimates of optimality gaps tend to be large on average even for high quality solutions this diminishes our ability to recognize high quality solutions in this paper we present a method for reducing the bias of the optimality gap estimators for two stage stochastic linear programs with recourse via a probability metrics approach motivated by stability results in stochastic programming we apply this method to the averaged two replication procedure a2rp by partitioning the observations in an effort to reduce bias which can be done in polynomial time in sample size we call the resulting procedure the averaged two replication procedure with bias reduction a2rp b we provide conditions under which a2rp b produces strongly consistent point estimators and an asymptotically valid confidence interval we illustrate the effectiveness of our approach analytically on a newsvendor problem and test the small sample behavior of a2rp b on a number of two stage stochastic linear programs from the literature our computational results indicate that the procedure effectively reduces bias we also observe variance reduction in certain circumstances

complete distributional problems hard languages and resource bounded measure we say that a distribution is reasonable if there exists a constant s 0 such that x x n 1 ns we prove the following result which suggests that all distnp complete problems have reasonable distributions if np contains a dtime 2n bi immune set then every distnp complete set has a reasonable distribution it follows from work of mayordomo 19 that the consequent holds if the p measure of np is not zero cai and selman 6 defined a modification and extension of levin s notion of average polynomial time to arbitrary time bounds and proved that if l is p bi immune then l is distributionally hard meaning that for every polynomial time computable distribution the distributional problem l is not polynomial on the average we prove the following results which suggest that distributional hardness is closely related to more traditional notions of hardness 1 if np contains a distributionally hard set then np contains a p immune set 2 there exists a language l that is distributionally hard but not p bi immune if and only if p contains a set that is immune to all p printable sets the following corollaries follow readily 1 if the p measure of np is not zero then there exists a language l that is distributionally hard but not p bi immune 2 if the p2 measure of np is not zero then there exists a language l in np that is distributionally hard but not p bi immune

second order cone programming second order cone programming socp problems are convex optimization problems in which a linear function is minimized over the intersection of an affine linear manifold with the cartesian product of second order lorentz cones linear programs convex quadratic programs and quadratically constrained convex quadratic programs can all be formulated as socp problems as can many other problems that do not fall into these three categories these latter problems model applications from a broad range of fields from engineering control and finance to robust optimization and combinatorial optimization on the other hand semidefinite programming sdp that is the optimization problem over the intersection of an affine set and the cone of positive semidefinite matrices includes socp as a special case therefore socp falls between linear lp and quadratic qp programming and sdp like lp qp and sdp problems socp problems can be solved in polynomial time by interior point methods the computational effort per iteration required by these methods to solve socp problems is greater than that required to solve lp and qp problems but less than that required to solve sdp s of similar size and structure because the set of feasible solutions for an socp problem is not polyhedral as it is for lp and qp problems it is not readily apparent how to develop a simplex or simplex like method for socp while socp problems can be solved as sdp problems doing so is not advisable both on numerical grounds and computational complexity concerns for instance many of the problems presented in the survey paper of vandenberghe and boyd vb96 as examples of sdps can in fact be formulated as socps and should be solved as such in 2 3 below we give socp formulations for four of these examples the convex quadratically constrained quadratic programming qcqp problem problems involving fractional quadratic functions rutcor rutgers university e mail alizadeh rutcor rutgers edu research supported in part by the u s national science foundation grant ccr 9901991 ieor columbia university e mail gold ieor columbia edu research supported in part by the department of energy grant de fg02 92er25126 national science foundation grants dms 94 14438 cda 97 26385 and dms 01 04282

necessary conditions for the exact observability of systems on hilbert spaces abstract in this paper we introduce the zero class of admissible observation operators for which the norm of the output map tends to zero as the time tends to zero we show that for such an observation operator exact observability implies that the semigroup of the system is bounded from below in addition if the semigroup generator has no residual spectrum then the semigroup can be extended to a c 0 group

scenario reduction in stochastic programming given a convex stochastic programming problem with a discrete initial probability distribution the problem of optimal scenario reduction is stated as follows determine a scenario subset of prescribed cardinality and a probability measure based on this set that is the closest to the initial distribution in terms of a natural or canonical probability metric arguments from stability analysis indicate that fortet mourier type probability metrics may serve as such canonical metrics efficient algorithms are developed that determine optimal reduced measures approximately numerical experience is reported for reductions of electrical load scenario trees for power management under uncertainty for instance it turns out that after 50 reduction of the scenario tree the optimal reduced tree still has about 90 relative accuracy

on the implementation of huge random objects we initiate a general study of the feasibility of implementing huge random objects and demonstrate its applicability to a number of areas in which random objects occur naturally we highlight two types of measures of the quality of the implementation with respect to the desired specification the first type corresponds to various standard notions of indistinguishability applied to function ensembles whereas the second type is a novel notion that we call truthfulness intuitively a truthful implementation of a random object of type t must always be an object of type t and not merely be indistinguishable from a random object of type t our formalism allows for the consideration of random objects that satisfy some fixed property or have some fixed structure as well as the consideration of objects supporting complex queries for example we consider the truthful implementation of random hamiltonian graphs as well as supporting complex queries regarding such graphs e g providing the next vertex along a fixed hamiltonian path in such a graph

scenario reduction revisited fundamental limits and guarantees the goal of scenario reduction is to approximate a given discrete distribution with another discrete distribution that has fewer atoms we distinguish continuous scenario reduction where the new atoms may be chosen freely and discrete scenario reduction where the new atoms must be chosen from among the existing ones using the wasserstein distance as measure of proximity between distributions we identify those n point distributions on the unit ball that are least susceptible to scenario reduction i e that have maximum wasserstein distance to their closest m point distributions for some prescribed m n we also provide sharp bounds on the added benefit of continuous over discrete scenario reduction finally to our best knowledge we propose the first polynomial time constant factor approximations for both discrete and continuous scenario reduction as well as the first exact exponential time algorithms for continuous scenario reduction

average case complete problems many interesting combinatorial problems were found to be np complete since there is little hope to solve them fast in the worst case researchers look for algorithms which are fast just on average this matter is sensitive to the choice of a particular np complete problem and a probability distribution of its instances some of these tasks are easy and some not but one needs a way to distinguish the difficult on average problems such negative results could not only save positive efforts but may also be used in areas like cryptography where hardness of some problems is a frequent assumption it is shown in 1 that the tiling problem with uniform distribution of instances has no polynominal on average algorithm unless every np problem with every simple probability distribution has it

necessary condition of linear distributed parameter systems with exact controllability abstract in this paper we studied the necessary condition of distributed parameter system with exact controllability let 0 t be the control mapping we introduce new a class of control operators that is called the i class control which satisfy r 0 t r t t is closed set for t 0 if the system is exactly controllable in finite time then the semigroup t t must have closed range in particular if the generator of t t has compact resolvent its spectra distribute in a strip parallel to imaginary axis as an application we assert that the systems associated with the immediately norm continuous semigroups are never exact controllable for zero class or i class controls

exact recovery in the hypergraph stochastic block model a spectral algorithm abstract we consider the exact recovery problem in the hypergraph stochastic block model hsbm with k blocks of equal size more precisely we consider a random d uniform hypergraph h with n vertices partitioned into k clusters of size s n k hyperedges e are added independently with probability p if e is contained within a single cluster and q otherwise where 0 q p 1 we present a spectral algorithm which recovers the clusters exactly with high probability given mild conditions on n k p q and d our algorithm is based on the adjacency matrix of h which is a symmetric n n matrix whose u v th entry is the number of hyperedges containing both u and v to the best of our knowledge our algorithm is the first to guarantee exact recovery when the number of clusters k n

tilings recursivity and regularity we establish a first step towards a rice theorem for tilings for non trivial sets it is undecidable to know whether two different tile sets produce the same tilings of the plane then we study quasiperiodicity functions associated with tilings this function is a way to measure the regularity of tilings we prove that not only almost all recursive functions can be obtained as quasiperiodicity functions but also a function which overgrows any recursive function

generic case complexity decision problems in group theory and random walks abstract we give a precise definition of generic case complexity and show that for a very large class of finitely generated groups the classical decision problems of group theory the word conjugacy and membership problems all have linear time generic case complexity we prove such theorems by using the theory of random walks on regular graphs

data driven distributionally robust optimization using the wasserstein metric performance guarantees and tractable reformulations we consider stochastic programs where the distribution of the uncertain parameters is only observable through a finite training dataset using the wasserstein metric we construct a ball in the space of multivariate and non discrete probability distributions centered at the uniform distribution on the training samples and we seek decisions that perform best in view of the worst case distribution within this wasserstein ball the state of the art methods for solving the resulting distributionally robust optimization problems rely on global optimization techniques which quickly become computationally excruciating in this paper we demonstrate that under mild assumptions the distributionally robust optimization problems over wasserstein balls can in fact be reformulated as finite convex programs in many interesting cases even as tractable linear programs leveraging recent measure concentration results we also show that their solutions enjoy powerful finite sample performance guarantees our theoretical results are exemplified in mean risk portfolio optimization as well as uncertainty quantification

local search heuristics for k median and facility location problems we analyze local search heuristics for the metric k median and facility location problems we define the locality gap of a local search procedure for a minimization problem as the maximum ratio of a locally optimum solution obtained using this procedure to the global optimum for k median we show that local search with swaps has a locality gap of 5 furthermore if we permit up to p facilities to be swapped simultaneously then the locality gap is 3 2 p this is the first analysis of a local search for k median that provides a bounded performance guarantee with only k medians this also improves the previous known 4 approximation for this problem for uncapacitated facility location we show that local search which permits adding dropping and swapping a facility has a locality gap of 3 this improves the bound of 5 given by m korupolu c plaxton and r rajaraman analysis of a local search heuristic for facility location problems technical report 98 30 dimacs 1998 we also consider a capacitated facility location problem where each facility has a capacity and we are allowed to open multiple copies of a facility for this problem we introduce a new local search operation which opens one or more copies of a facility and drops zero or more facilities we prove that this local search has a locality gap between 3 and 4

average case quantum query complexity we compare classical and quantum query complexities of total boolean functions it is known that for worst case complexity the gap between quantum and classical can be at most polynomial we show that for average case complexity under the uniform distribution quantum algorithms can be exponentially faster than classical algorithms under non uniform distributions the gap can even be super exponential we also prove some general bounds for average case complexity and show that the average case quantum complexity of majority under the uniform distribution is nearly quadratically better than the classical complexity a preliminary version of this paper appeared in the proceedings of the 17th annual symposium on theoretical aspects of computer science stacs 2000 lecture notes in computer science vol 1770 berlin springer