the solution of diagonal decoupling problem by dynamic output feedback and constant precompensator the general case in this paper the solution to diagonal decoupling problem via dynamic output feedback and constant precompensator is presented in the general case the literature contains a number of particular solutions for this problem these solutions are valid for instance for square transfer matrices or they employ restricted dynamic output feedback schemes here we start with a nonsquare transfer matrix and consider a dynamic output feedback scheme where there is a proper compensator in the feedback loop accompanied by a constant gain precompensator in the feedforward path in this respect the problem being considered is the dynamic output feedback version of the well known morgan s problem the approach taken involves the characterization of open loop diagonalizers which admit a dynamic output feedback realization the key concepts involved in the solution are the generalized version of diagonal causality degree dominance and the set of attainable infinite zero orders an example is also included to illustrate the main result

on the singularity of adjacency matrices for random regular digraphs we prove that the non symmetric adjacency matrix of a uniform random d regular directed graph on n vertices is asymptotically almost surely invertible assuming min d n d ge c log 2n for a sufficiently large constant c 0 the proof makes use of a coupling of random regular digraphs formed by shuffling the neighborhood of a pair of vertices as well as concentration results for the distribution of edges proved in cook random struct algorithms arxiv 1410 5595 2014 we also apply our general approach to prove asymptotically almost surely invertibility of hadamard products varsigma mathrm circ varxi where varxi is a matrix of iid uniform pm 1 signs and varsigma is a 0 1 matrix whose associated digraph satisfies certain expansion properties

on the concentration of random multilinear forms and the universality of random block matrices the circular law asserts that if mathbf x n is a n times n matrix with iid complex entries of mean zero and unit variance then the empirical spectral distribution of frac 1 sqrt n mathbf x n converges almost surely to the uniform distribution on the unit disk as n tends to infinity answering a question of tao we prove the circular law for a general class of random block matrices with dependent entries the proof relies on an inverse type result for the concentration of linear operators and multilinear forms

optimal inverse littlewood offord theorems abstract let i i 1 n be iid bernoulli random variables taking values 1 with probability 1 2 given a multiset v of n integers v 1 v n we define the concentration probability as v sup x p v 1 1 v n n x a classical result of littlewood offord and erd s from the 1940s asserts that if the v i are non zero then v is o n 1 2 since then many researchers have obtained improved bounds by assuming various extra restrictions on v about 5 years ago motivated by problems concerning random matrices tao and vu introduced the inverse littlewood offord problem in the inverse problem one would like to characterize the set v given that v is relatively large in this paper we introduce a new method to attack the inverse problem as an application we strengthen the previous result of tao and vu obtaining an optimal characterization for v this immediately implies several classical theorems such as those of sarkozy and szemeredi and halasz the method also applies to the continuous setting and leads to a simple proof for the net theorem of tao and vu which plays a key role in their recent studies of random matrices all results extend to the general case when v is a subset of an abelian torsion free group and i are independent variables satisfying some weak conditions

smallest singular value of random matrices and geometry of random polytopes abstract we study the behaviour of the smallest singular value of a rectangular random matrix i e matrix whose entries are independent random variables satisfying some additional conditions we prove a deviation inequality and show that such a matrix is a good isomorphism on its image then we obtain asymptotically sharp estimates for volumes and other geometric parameters of random polytopes absolutely convex hulls of rows of random matrices all our results hold with high probability that is with probability exponentially in dimension close to 1

nearly optimal solutions for the chow parameters problem and low weight approximation of halfspaces the chow parameters of a boolean function f 1 1 n 1 1 are its np1 degree 0 and degree 1 fourier coefficients it has been known since 1961 lchow 1961 tannenbaum 1961r that the exact values of the chow parameters of any linear threshold function f uniquely specify f within the space of all boolean functions but until recently lo donnell and servedio 2011r nothing was known about efficient algorithms for reconstructing f exactly or approximately from exact or approximate values of its chow parameters we refer to this reconstruction problem as the chow parameters problem our main result is a new algorithm for the chow parameters problem which given sufficiently accurate approximations to the chow parameters of any linear threshold function f runs in time o n2 1 e o log2 1 e and with high probability outputs a representation of an ltf f that is e close to f in hamming distance the only previous algorithm lo donnell and servedio 2011r had running time poly n 22o 1 e2 as a byproduct of our approach we show that for any linear threshold function f over 1 1 n there is a linear threshold function f which is e close to f and has all weights that are integers of magnitude at most n 1 e o log2 1 e this significantly improves the previous best result of diakonikolas and servedio l2009r which gave a poly n 2o 1 e2 3 weight bound and is close to the known lower bound of max n 1 e log log 1 e lgoldberg 2006 servedio 2007r our techniques also yield improved algorithms for related problems in learning theory in addition to being significantly stronger than previous work our results are obtained using conceptually simpler proofs the two main ingredients underlying our results are 1 a new structural result showing that for f any linear threshold function and g any bounded function if the chow parameters of f are close to the chow parameters of g then f is close to g 2 a new boosting like algorithm that given approximations to the chow parameters of a linear threshold function outputs a bounded function whose chow parameters are close to those of f

a solution to the diagonalization problem by constant precompensator and dynamic output feedback a solution is presented for the problem of diagonalization row by row decoupling the problem is solved using a constant precompensator and a dynamic output feedback compensator of a p m linear time invariant system the solvability condition is compact and concerns the dimension of a single subspace defined via the concepts of essential rows and static kernels associated with the transfer matrix a characterization of the set of all solutions to the problem is also given in solving this dynamic feedback problem a complete solution to its state feedback counterpart namely the restricted state feedback problem of diagonalization is also presented

solution of the littlewood offord problem in high dimensions consider the 2 partial sums of arbitrary n vectors of length at least one in d dimensional euclidean space it is shown that as n goes to infinity no closed ball of diameter a contains more than a 1 o l ln2i out of these sums and this is best possible for a a small an exact formula is given

the circular law for random matrices we consider the joint distribution of real and imaginary parts of eigenvalues of random matrices with independent real entries with mean zero and unit variance we prove the convergence of this distribution to the uniform distribution on the unit disc without assumptions on the existence of a density for the distribution of entries we assume that the entries have a finite moment of order larger than two and consider the case of sparse matrices the results are based on previous work of bai rudelson and the authors extending results to a larger class of sparse matrices

labeled cuts in graphs in this paper we study labeled extensions of the classical s t mincut problem in which we are given a graph g v e two specific vertices s t v a set l of labels and a labeling e l of the edges the goal is to choose a subset l l of labels so that s and t become disconnected when deleting the edges with labels in l we give an algorithm with an o n 2 3 approximation factor guarantee which improves the o m approximation guarantee of zhang et al 2009 16 we also consider variants in which selected subsets of paths between s and t have to be removed instead of all paths these labeled cut problems are much harder than the classical mincut problem

decoupling with internal stability for unity output feedback systems abstract this paper presents for the first time a necessary and sufficient condition for solvability of the decoupling problem with internal stability for square plants by unit output feedback compensation it consists of some polynomial matrix coprimeness conditions and is concerned only with the greatest c divisor of the plant parameterizations of all decoupling compensators and achievable diagonal loop maps with internal stabilizability are given and the results are also generalized in a very natural way to the decoupling problem with arbitrary pole assignment examples are included for illustration

adjacency matrices of random digraphs singularity and anti concentration abstract let d n d be the set of all d regular directed graphs on n vertices let g be a graph chosen uniformly at random from d n d and m be its adjacency matrix we show that m is invertible with probability at least 1 c ln 3 d d for c d c n ln 2 n where c c are positive absolute constants to this end we establish a few properties of d regular directed graphs one of them a littlewood offord type anti concentration property is of independent interest let j be a subset of vertices of g with j n d let i be the indicator of the event that the vertex i is connected to j and define 1 2 n 0 1 n then for every v 0 1 n the probability that v is exponentially small this property holds even if a part of the graph is frozen

approximation algorithms for knapsack problems with cardinality constraints abstract we address a variant of the classical knapsack problem in which an upper bound is imposed on the number of items that can be selected this problem arises in the solution of real life cutting stock problems by column generation and may be used to separate cover inequalities with small support within cutting plane approaches to integer linear programs we focus our attention on approximation algorithms for the problem describing a linear storage polynomial time approximation scheme ptas and a dynamic programming based fully polynomial time approximation scheme fptas the main ideas contained in our ptas are used to derive ptas s for the knapsack problem and its multi dimensional generalization which improve on the previously proposed ptas s we finally illustrate better ptas s and fptas s for the subset sum case of the problem in which profits and weights coincide

the littlewood offord problem and invertibility of random matrices abstract we prove two basic conjectures on the distribution of the smallest singular value of random n n matrices with independent entries under minimal moment assumptions we show that the smallest singular value is of order n 1 2 which is optimal for gaussian matrices moreover we give a optimal estimate on the tail probability this comes as a consequence of a new and essentially sharp estimate in the littlewood offord problem for i i d random variables x k and real numbers a k determine the probability p that the sum k a k x k lies near some number v for arbitrary coefficients a k of the same order of magnitude we show that they essentially lie in an arithmetic progression of length 1 p

on the hardness of approximating minimization problems we prove results indicating that it is hard to compute efficiently good approximate solutions to the graph coloring set covering and other related minimization problems specifically there is an e 0 such that graph coloring cannot be approximated with ratio n e unless p np set covering cannot be approximated with ratio c log n for any c 1 4 unless np is contained in dtime n poly log n similar results follow for related problems such as clique cover fractional chromatic number dominating set and others

random matrices tail bounds for gaps between eigenvalues gaps or spacings between consecutive eigenvalues are a central topic in random matrix theory the goal of this paper is to study the tail distribution of these gaps in various random matrix models we give the first repulsion bound for random matrices with discrete entries and the first super polynomial bound on the probability that a random graph has simple spectrum along with several applications

inverse littlewood offord theorems and the condition number of random discrete matrices consider a random sum r v r nvn where 771 rin are independently and identically distributed i i d random signs and vi vn are integers the littlewood offord problem asks to maximize concentration probabilities such as p r i ih in this paper we develop an inverse littlewood offord theory somewhat in the spirit of freiman s inverse theory in additive combinatorics which starts with the hypothesis that a concentration probability is large and concludes that almost all of the v vn are efficiently contained in a generalized arithmetic progression as an application we give a new bound on the magnitude of the

john type theorems for generalized arithmetic progressions and iterated sumsets a classical theorem of fritz john allows one to describe a convex body up to constants as an ellipsoid in this article we establish similar descriptions for generalized i e multidimensional arithmetic progressions in terms of proper i e collision free generalized arithmetic progressions in both torsion free and torsion settings we also obtain a similar characterization of iterated sumsets in arbitrary abelian groups in terms of progressions thus strengthening and extending recent results of szemeredi and vu

approximate strong separation with application in fractional graph coloring and preemptive scheduling in this paper we show that approximation algorithms for the weighted independent set and s dimensional knapsack problem with ratio a can be turned into approximation algorithms with the same ratio for fractional weighted graph coloring and preemptive resource constrained scheduling in order to obtain these results we generalize known results by grotschel lovasz and schrijver on separation non emptiness test optimization and violation in the direction of approximability

long arithmetic progressions in sumsets thresholds and bounds l a a1 al ai ai ai aj among the most well known results in all of mathematics are vinogradov s theorem which says that 3p p is the set of primes contains all sufficiently large odd numbers and waring s conjecture proved by hilbert hardy and littlewood hua and many others which asserts that for any given r there is a number l such that l nr n denotes the set of rth powers contains all sufficiently large positive integers see 29 for an excellent exposition concerning these results in recent years a considerable amount of attention has been paid to the study of finite sumsets given a finite set a and a positive integer l the natural analogue of vinogadov waring results is to show that under proper conditions the sumset la l a contains a long arithmetic progression let us assume that a is a subset of the interval n 1 n where n is a large positive integer the concrete problem we would like to address is to estimate the minimum length of the longest arithmetic progression in la l a as a function of l n and a we denote this function by f a l n f a l n following the notation in 13 many estimates for f a l n have been discovered by bourgain freiman halberstam green ruzsa and sarkozy see section 3 but most of these results focus on sets with very high density namely a is close to n estimating f a l n seems much harder and not much was known prior to our study

inverse littlewood offord problems and the singularity of random symmetric matrices let mn denote a random symmetric n by n matrix whose upper diagonal entries are iid bernoulli random variables which take value 1 and 1 with probability 1 2 improving the earlier result by costello tao and vu 4 we show that mn is nonsingular with probability 1 o n c for any positive constant c the proof uses an inverse littlewood offord result for quadratic forms which is of interest of its own

resilience for the littlewood offord problem abstract consider the sum x i 1 n a i i where a a i i 1 n is a sequence of non zero reals and i i 1 n is a sequence of i i d rademacher random variables that is pr i 1 pr i 1 1 2 the classical littlewood offord problem asks for the best possible upper bound on the concentration probabilities pr x x in this paper we study a resilience version of the littlewood offord problem how many of the i is an adversary typically allowed to change without being able to force concentration on a particular value we solve this problem asymptotically and present a few interesting open problems

decoupling by restricted static state feedback the general case in this paper we tackle the general block decoupling problem for linear constant dynamical systems c a b with m inputs and p outputs with restricted static state feedback in other words with control laws of type u gfx gv we give a necessary and sufficient condition of existence for such laws which generalizes the one previously given in 2 for the simple case k p leq m where k denotes the number of blocks to be decoupled

invertibility of random matrices norm of the inverse let a be an n n matrix whose entries are independent copies of a centered random variable satisfying the subgaussian tail estimate we prove that the operator norm of a 1 does not exceed cn3 2 with probability close to 1

approximation algorithms for general packing problems and their application to the multicast congestion problem in this paper we present approximation algorithms based on a lagrangian decomposition via a logarithmic potential reduction to solve a general packing or min max resource sharing problem with m non negative convex constraints on a convex set b we generalize a method by grigoriadis et al to the case with weak approximate block solvers i e with only constant logarithmic or even worse approximation ratios given an accuracy varepsilon in 0 1 we show that our algorithm needs o m ln m varepsilon 2 ln varepsilon 1 calls to the block solver a bound independent of the data and the approximation ratio of the block solver for small approximation ratios the algorithm needs o m ln m varepsilon 2 calls to the block solver as an application we study the problem of minimizing the maximum edge congestion in a multicast communication network interestingly the block problem here is the classical steiner tree problem that can be solved only approximately we show how to use approximation algorithms for the steiner tree problem to solve the multicast congestion problem approximately

smooth analysis of the condition number and the least singular value let x be a complex random variable with mean zero and bounded variance let n n be the random matrix of size n whose entries are iid copies of x and let m be a fixed matrix of the same size the goal of this paper is to give a general estimate for the condition number and least singular value of the matrix m n n generalizing an earlier result of spielman and teng for the case when x is gaussian our investigation reveals an interesting fact that the core matrix m does play a role on tail bounds for the least singular value of m n n this does not occur in spielman teng studies when x is gaussian consequently our general estimate involves the norm m in the special case when m is relatively small this estimate is nearly optimal and extends or refines existing results

transfer function matrix approach to decoupling problem with stability a simple necessary and sufficient condition for solvability of decoupling with internal stability for unity output feedback for non singular plants is proposed

approximation algorithms for inventory problems with submodular or routing costs we consider the following two deterministic inventory optimization problems with non stationary demands submodular joint replenishment problem this involves multiple item types and a single retailer who faces demands over a finite planning horizon of t periods in each time period any subset of item types can be ordered incurring a joint ordering cost which is submodular moreover items can be held in inventory while incurring a holding cost the objective is to find a sequence of orders that satisfies all demands and minimizes the total ordering and holding costs inventory routing problem this involves a single depot that stocks items and multiple retailer locations facing demands over a finite planning horizon of t periods in each time period any subset of locations can be visited using a vehicle originating from the depot there is also cost incurred for holding items at any retailer the objective here is to satisfy all demands while minimizing the sum of routing and holding costs we present a unified approach that yields mathcal o left frac log t log log t right ologtloglogt factor approximation algorithms for both problems when the holding costs are polynomial functions a special case is the classic linear holding cost model wherein this is the first sub logarithmic approximation ratio for either problem

approximation algorithms for the m dimensional 0 1 knapsack problem worst case and probabilistic analyses abstract we describe a polynomial approximation scheme for an m constraint 0 1 integer programming problem m fixed based on the use of the dual simplex algorithm for linear programming we also analyse the asymptotic properties of a particular random model

a fully polynomial approximation algorithm for the 0 1 knapsack problem abstract a fully polynomial approximation algorithm is developed for the 0 1 knapsack problem the algorithm uses results of lawler and ibarra and kim a pseudo polynomial dynamic programming algorithm is first suggested which solves the problem in o nb log n time and o b space

random matrices universality of esds and the circular law given an n x n complex matrix a let mu a x y 1 n vertical bar 1 i n re lambda i x im lambda i y vertical bar be the empirical spectral distribution esd of its eigenvalues lambda i is an element of c i l n we consider the limiting distribution both in probability and in the almost sure convergence sense of the normalized esd mu 1 root n an of a random matrix a n a ij 1 i j n where the random variables a ij e a ij are i i d copies of a fixed random variable x with unit variance we prove a universality principle for such ensembles namely that the limit distribution in question is independent of the actual choice of x in particular in order to compute this distribution one can assume that x is real or complex gaussian as a related result we show how laws for this esd follow from laws for the singular value distribution of 1 root n a n zi for complex z as a corollary we establish the circular law conjecture both almost surely and in probability which asserts that mu 1 root n an converges to the uniform measure on the unit disc when the a ij have zero mean

approximating spectral sums of large scale matrices using stochastic chebyshev approximations computation of the trace of a matrix function plays an important role in many scientific computing applications including applications in machine learning computational physics e g lattice qua

polynomial time approximation schemes for geometric intersection graphs a disk graph is the intersection graph of a set of disks with arbitrary diameters in the plane for the case that the disk representation is given we present polynomial time approximation schemes ptass for the maximum weight independent set problem selecting disjoint disks of maximum total weight and for the minimum weight vertex cover problem in disk graphs these are the first known ptass for mathcal np hard optimization problems on disk graphs they are based on a novel recursive subdivision of the plane that allows applying a shifting strategy on different levels simultaneously so that a dynamic programming approach becomes feasible the ptass for disk graphs represent a common generalization of previous results for planar graphs and unit disk graphs they can be extended to intersection graphs of other disk like geometric objects such as squares or regular polygons also in higher dimensions

invertibility of random matrices unitary and orthogonal perturbations author s rudelson mark vershynin roman abstract we show that a perturbation of any fixed square matrix d by a random unitary matrix is well invertible with high probability a similar result holds for perturbations by random orthogonal matrices the only notable exception is when d is close to orthogonal as an application these results completely eliminate a hard to check condition from the single ring theorem by guionnet krishnapur and zeitouni

decoupling through specified input output channels with internal stability abstract a system with two sets of inputs u 1 u 2 and two sets of outputs y 1 y 2 is considered the objective is to construct a dynamic feedback compensator between u 1 and y 1 so that the closed loop transfer matrix between u 2 and y 2 is diagonal and nonsingular internal stability of the closed loop system is also required this problem can be regarded as the general version of the standard diagonal decoupling problem where the output to be controlled decoupled and the measured output are not distinguished along these lines it is shown that the diagonal causality degree dominance dcdd which is the key concept in standard diagonal decoupling problems also arises in the present context in a modified form the first modification is due to the constraint of internal stability which gives rise to a new notion diagonal stability and causality degree dominance dscdd these two notions are investigated in the first part of the paper the second modification comes from the crosschannel transfer matrices between u 1 and y 2 and u 2 and y 1 this yields another new notion joint essential stability functions the solvability conditions of the problem are expressed in terms of the properties of the subrings of the ring of stable and proper rational functions and their quotient rings these subrings are generated by joint essential stability functions of the cross channel transfer matrices the degrees of the joint essential stability functions are called the joint essential stability orders which are extensions of the well known essential orders and essential stability orders to the problem considered here

fractional and integral colourings letg v e be an undirected graph andc any vector in v g denote by gc resp gc the chromatic number resp fractional chromatic number ofg with respect toc we study graphs for which gc gc 1 we show that for the class of graphs satisfying gc gc a class generalizing perfect graphs an analogue of the duplication lemma does not hold we also describe a 2 vertex cut decomposition procedure related to the integer decomposition property we use this procedure to show that gc gc for series parallel graphs and gc gc 1 for graphs that do not have the 4 wheel as a minor

on the modification of a structure at infinity by a return of static state le bouclage d un systeme lineaire par un retour d etat statique non regulier c est a dire du type u fx gv avec g non inversible modifie en general sa structure et en particulier sa structure a l infini qui joue un role cle daps les problemes de commande l objet de cet article est d etudier ces modifications nous donnons une condition structurelle necessaire et sufsante pour qu une lists donnee soft la lists des ordres des zeros infinis du systeme boucle par un retour d etat statique nous calculons alors explicitement un retour d etat f g qui produit cette modification de structure a l infini

on the complexity of bandwidth allocation in radio networks we define and study an optimization problem that is motivated by bandwidth allocation in radio networks because radio transmissions are subject to interference constraints in radio networks physical space is a common resource that the nodes have to share in such a way that concurrent transmissions do not interfere the bandwidth allocation problem we study under these constraints is the following given bandwidth traffic demands between the nodes of the network the objective is to schedule the radio transmissions in such a way that the traffic demands are satisfied the problem is similar to a multicommodity flow problem where the capacity constraints are replaced by the more complex notion of non interfering transmissions we provide a formal specification of the problem that we call round weighting by modeling non interfering radio transmissions as independent sets we relate the complexity of round weighting to the complexity of various independent set problems e g maximum weight independent set vertex coloring fractional coloring from this relation we deduce that in general round weighting is hard to approximate within n 1 e n being the size of the radio network we also provide polynomial exact or approximation algorithms e g in the following two cases a when the interference constraints are specific for instance for a network whose vertices belong to the euclidean space or b when the traffic demands are directed towards a unique node in the network also called gathering analogous to single commodity flow

online codes for analog signals this paper revisits a classical scenario in communication theory a waveform sampled at regular intervals is to be encoded so as to minimize distortion in its reconstruction despite the noise this transformation must be online causal to enable real time signaling and should use no more power than the original signal the noise model we consider is an atomic norm convex relaxation of the standard discrete alphabet hamming weight bounded model namely adversarial ell 1 bounded in the block coding noncausal setting such encoding is possible due to the existence of large almost euclidean sections in ell 1 spaces a notion first studied in the work of dvoretzky in 1961 our main result is that an analogous result is achievable even casually equivalently our work may be seen as a lower triangular version of ell 1 dvoretzky theorems in terms of communication the guarantees are expressed in terms of certain time weighted norms the time weighted ell 2 norm imposed on the decoder forces increasingly accurate reconstruction of the distant past signal while the time weighted ell 1 norm on the noise ensures vanishing interference from distant past noise encoding is linear hence easy to implement in analog hardware decoding is performed by an lp analogous to those used in compressed sensing

set of open loop block diagonalizers of transfer matrices abstract the set of open loop right block diagonalizers of a given p m transfer matrix z with specified block sizes is studied a characterization of this set is obtained in terms of a maximal open loop block diagonalizer and a minimal proper basis of the kernel of z the relevance of the results to some non interacting control problems is pointed out in particular the results obtained yield a minimum delay solution to the block decoupling problem using dynamic state feedback

upper bound for intermediate singular values of random matrices abstract in this paper we prove that an n n matrix a with independent centered subgaussian entries satisfies s n 1 l a c 1 t l n with probability at least 1 exp c 2 t l this yields s n 1 l a c l n in combination with a known lower bound these results can be generalized to the rectangular matrix case

the smallest singular value of a shifted d regular random square matrix we derive a lower bound on the smallest singular value of a random d regular matrix that is the adjacency matrix of a random d regular directed graph specifically let c 1 d c n log 2 n and let mathcal m n d be the set of all n times n square matrices with 0 1 entries such that each row and each column of every matrix in mathcal m n d has exactly d ones let m be a random matrix uniformly distributed on mathcal m n d then the smallest singular value s n m of m is greater than n 6 with probability at least 1 c 2 log 2 d sqrt d where c c 1 and c 2 are absolute positive constants independent of any other parameters analogous estimates are obtained for matrices of the form m z mathrm id where mathrm id is the identity matrix and z is a fixed complex number

random doubly stochastic matrices the circular law hoi h nguyenabstract let x be a matrix sampled uniformly from the set of doubly stochastic matri ces of size n n we show that the empirical spectral distribution of the normalized matrixpn x ex converges almost surely to the circular law this con rms a conjecture ofchatterjee diaconis and sly