low weight superimposed codes and related combinatorial structures bounds and applications abstract a k n superimposed code is a well known and widely used combinatorial structure that can be represented by a t n binary matrix such that for any k columns of the matrix and for any column c chosen among these k columns there exists a row in correspondence of which column c has an entry equal to 1 and the remaining k 1 columns have entries equal to 0 due to the many situations in which superimposed codes find applications there is an abundant literature that studies the problem of constructing k n superimposed codes with a small number t of rows motivated by applications to conflict free communication in multiple access networks group testing and data security we study the problem of constructing superimposed codes that have the additional constraints that the number of 1 s in each column of the matrix is constant and equal to an input parameter w our results improve on the known literature in the area we also extend our findings to other important combinatorial structures like selectors generalized superimposed codes and z error correcting superimposed codes

second order convex splitting schemes for periodic nonlocal cahn hilliard and allen cahn equations we devise second order accurate unconditionally uniquely solvable and unconditionally energy stable schemes for the nonlocal cahn hilliard nch and nonlocal allen cahn nac equations for a large class of interaction kernels we present numerical evidence that both schemes are convergent we solve the nonlinear equations resulting from discretization using an efficient nonlinear multigrid method and demonstrate the performance of our algorithms by simulating nucleation and crystal growth for several different choices of interaction kernels

multicommodity max flow min cut theorems and their use in designing approximation algorithms in this paper we establish max flow min cut theorems for several important classes of multicommodity flow problems in particular we show that for any n node multicommodity flow problem with uniform demands the max flow for the problem is within an o log n factor of the upper bound implied by the min cut the result which is existentially optimal establishes an important analogue of the famous 1 commodity max flow min cut theorem for problems with multiple commodities the result also has substantial applications to the field of approximation algorithms for example we use the flow result to design the first polynomial time polylog n times optimal approximation algorithms for well known np hard optimization problems such as graph partitioning min cut linear arrangement crossing number vlsi layout and minimum feedback arc set applications of the flow results to path routing problems network reconfiguration communication in distributed networks scientific computing and rapidly mixing markov chains are also described in the paper categories and subject descriptors f 2 2 analysis of algorithms and problem complexity

expander flows geometric embeddings and graph partitioning we give a o slog n approximation algorithm for the sparsest cut edge expansion balanced separator and graph conductance problems this improves the o log n approximation of leighton and rao 1988 we use a well known semidefinite relaxation with triangle inequality constraints central to our analysis is a geometric theorem about projections of point sets in rd whose proof makes essential use of a phenomenon called measure concentration we also describe an interesting and natural approximate certificate for a graph s expansion which involves embedding an n node expander in it with appropriate dilation and congestion we call this an expander flow

approximate max flow min multi cut theorems and their applications consider the multicommodity flow problem in which the object is to maximize the sum of commodities routed we prove the following approximate max flow min multicut theorem dst frac mbox rm min multicut o log k leq mbox rm max flow leq mbox rm min multicut noindent where k is the number of commodities our proof is constructive it enables us to find a multicut within o log k of the max flow and hence also the optimal multicut in addition the proof technique provides a unified framework in which one can also analyse the case of flows with specified demands of leighton and rao and klein et al and thereby obtain an improved bound for the latter problem

a second order accurate numerical method for a fractional wave equation we study a generalized crank nicolson scheme for the time discretization of a fractional wave equation in combination with a space discretization by linear finite elements the scheme uses a non uniform grid in time to compensate for the singular behaviour of the exact solution at t 0 with appropriate assumptions on the data and assuming that the spatial domain is convex or smooth we show that the error is of order k2 h2 where k and h are the parameters for the time and space meshes respectively

new bounds on the number of tests for disjunct matrices given n items with at most d of which being positive instead of testing these items individually the theory of combinatorial group testing aims to identify all positive items using as few tests as possible this paper is devoted to a fundamental and thirty year old problem in the nonadaptive group testing theory a binary matrix is called d disjunct if the boolean sum of arbitrary d columns does not contain another column not in this collection let t d denote the minimal t such that there exists a t times n d disjunct matrix with n t t d can also be viewed as the minimal t such that there exists a nonadaptive group testing scheme which is better than the trivial one that tests each item individually it was known that t d ge binom d 2 2 and was conjectured that t d ge d 1 2 in this paper we narrow the gap by proving t d d 2 ge 15 sqrt 33 24 a quantity in 6 7 7 8

an o log k approximate min cut max flow theorem and approximation algorithm it is shown that the minimum cut ratio is within a factor of o log k of the maximum concurrent flow for k commodity flow instances with arbitrary capacities and demands this improves upon the previously best known bound of o log2 k and is existentially tight up to a constant factor an algorithm for finding a cut with ratio within a factor of o log k of the maximum concurrent flow and thus of the optimal min cut ratio is presented

logarithmic hardness of the undirected edge disjoint paths problem we show that there is no logf e m approximation for the undirected edge disjoint paths problem unless np zptime npolylog n where m is the size of the graph and e is any positive constant this hardness result also applies to the undirected all or nothing multicommodity flow problem and the undirected node disjoint paths problem

edge disjoint paths in planar graphs with constant congestion we study the maximum edge disjoint paths problem in undirected planar graphs given a graph g and node pairs demands s 1t 1 s 2t 2 dots s kt k the goal is to maximize the number of demands that can be connected routed by edge disjoint paths the natural multicommodity flow relaxation has an omega sqrt n integrality gap where n is the number of nodes in g motivated by this we consider solutions with small constant congestion c 1 that is solutions in which up to c paths are allowed to use an edge alternatively each edge has a capacity of c in previous work we obtained an o log n approximation with congestion 2 via the flow relaxation this was based on a method of decomposing into well linked subproblems in this paper we obtain an o 1 approximation with congestion 4 to obtain this improvement we develop an alternative decomposition that is specific to planar graphs the decomposition produces instances that we call okamura seymour os instances these have the property that all terminals lie on a single face another ingredient we develop is a constant factor approximation for the all or nothing flow problem on os instances via the flow relaxation

edge disjoint paths in moderately connected graphs we study the edge disjoint paths edp problem in undirected graphs given a graph g with n nodes and a set mathcal t of pairs of terminals connect as many terminal pairs as possible using paths that are mutually edge disjoint this leads to a variety of classic np complete problems for which approximability is not well understood we show a polylogarithmic approximation algorithm for the undirected edp problem in general graphs with a moderate restriction on graph connectivity we require the global minimum cut of g to be omega log 5n previously constant or polylogarithmic approximation algorithms were known for trees with parallel edges expanders grids grid like graphs and most recently even degree planar graphs these graphs either have special structure e g they exclude minors or have large numbers of short disjoint paths our algorithm extends previous techniques in that it applies to graphs with high diameters and asymptotically large minors

edge disjoint paths in expander graphs given a graph g v e and a set of kappa pairs of vertices in v we are interested in finding for each pair ai bi a path connecting ai to bi such that the set of kappa paths so found is edge disjoint for arbitrary graphs the problem is cal np complete although it is in cal p if kappa is fixed we present a polynomial time randomized algorithm for finding edge disjoint paths in an r regular expander graph g we show that if g has sufficiently strong expansion properties and r is sufficiently large then all sets of kappa omega n log n pairs of vertices can be joined this is within a constant factor of best possible

a polylogarithmic approximation algorithm for edge disjoint paths with congestion 2 in the edge disjoint paths with congestion problem edpwc we are given an undirected n vertex graph g a collection m s1 t1 sk tk of pairs of vertices called demand pairs and an integer c the goal is to connect the maximum possible number of the demand pairs by paths so that the maximum edge congestion the number of paths sharing any edge is bounded by c when the maximum allowed congestion is c 1 this is the classical edge disjoint paths problem edp the best current approximation algorithm for edp achieves an o n approximation by rounding the standard multi commodity flow relaxation of the problem this matches the n lower bound on the integrality gap of this relaxation we show an o poly log k approximation algorithm for edpwc with congestion c 2 by rounding the same multi commodity flow relaxation this gives the best possible congestion for a sub polynomial approximation of edpwc via this relaxation our results are also close to optimal in terms of the number of pairs routed since edpwc is known to be hard to approximate to within a factor of log n 1 c 1 for any constant congestion c prior to our work the best approximation factor for edpwc with congestion 2 was o n3 7 and the best algorithm achieving a polylogarithmic approximation required congestion 14

a second order and nonuniform time stepping maximum principle preserving scheme for time fractional allen cahn equations abstract in this work we present a second order nonuniform time stepping scheme for the time fractional allen cahn equation we show that the proposed scheme preserves the discrete maximum principle and by using the convolution structure of consistency error we present sharp maximum norm error estimates which reflect the temporal regularity as our analysis is built on nonuniform time steps we may resolve the intrinsic initial singularity by using the graded meshes moreover we propose an adaptive time stepping strategy for large time simulations numerical experiments are presented to show the effectiveness of the proposed scheme this seems to be the first second order maximum principle preserving scheme for the time fractional allen cahn equation

graph partitioning using single commodity flows we show that the sparsest cut in graphs with n vertices and m edges can be approximated within o log2 n factor in o m n3 2 time using polylogarithmic single commodity max flow computations previous algorithms are based on multicommodity flows that take time o m n2 our algorithm iteratively employs max flow computations to embed an expander flow thus providing a certificate of expansion our technique can also be extended to yield an o log2 n pseudo approximation algorithm for the edge separator problem with a similar running time

polynomial bounds for the grid minor theorem one of the key results in robertson and seymour s seminal work on graph minors is the grid minor theorem also called the excluded grid theorem the theorem states that for every grid h every graph whose treewidth is large enough relative to vv h v contains h as a minor this theorem has found many applications in graph theory and algorithms let f k denote the largest value such that every graph of treewidth k contains a grid minor of size f k f k the best previous quantitative bound due to recent work of kawarabayashi and kobayashi and leaf and seymour shows that f k e slog k log log k in contrast the best known upper bound implies that f k e o sk log k in this article we obtain the first polynomial relationship between treewidth and grid minor size by showing that f k e k for some fixed constant 0 and describe a randomized algorithm whose running time is polynomial in vv g v and k that with high probability finds a model of such a grid minor in g

new hardness results for routing on disjoint paths in the classical node disjoint paths sf ndp problem the input consists of an undirected n vertex graph g and a collection mathcal m s 1 t 1 ldots s k t k of pairs of its vert

existence and construction of edge disjoint pathson expander graphs given an expander graph g v e and a set of q disjoint pairs of vertices in v the authors are interested in finding for each pair a i b i a path connecting a i to b i such that the set of q paths so found is edge disjoint for general graphs the related decision problem is np complete the authors prove sufficient conditions for the existence of edge disjoint paths connecting any set of q leq n log n kappa disjoint pairs of vertices on any n vertex bounded degree expander where kappa depends only on the expansion properties of the input graph and not on n furthermore a randomized o n 3 time algorithm and a random cal nc algorithm for constructing these paths is presented previous existence proofs and construction algorithms allowed only up to n epsilon pairs for some epsilon ll frac 1 3 and strong expanders d peleg and e upfal combinatorica 9 1989 pp 289 313 in passing an algorithm is developed for splitting a sufficiently strong expander into two edge disjoint spanning expanders

born again group testing multiaccess communications a brief summary of the basic notions of group testing is presented together with a brief historical account one of the early papers on group testing is shown to include a description of the tree search polling algorithm of hayes the classical group testing problem is formulated including a criterion for optimality of test plans a restricted class of tests called nested testing is described and a complete description for an optimal nested strategy is given for both a finite number and an infinite number of bernoulli distributed random variables a generalization of group testing applicable to the random access communications problem is presented

approximation algorithms for disjoint paths and related routing and packing problems given a network and a set of connection requests on it we consider the maximum edge disjoint paths and related generalizations and routing problems that arise in assigning paths for these requests we present improved approximation algorithms and or integrality gaps for all problems considered the central theme of this work is the underlying multicommodity flow relaxation applications of these techniques to approximating families of packing integer programs are also presented

approximating disjoint path problems using packing integer programs in a packing integer program we are given a matrix a and column vectors b c with nonnegative entries we seek a vector x of nonnegative integers which maximizes c t x subject to ax leq b the edge and vertex disjoint path problems together with their unsplittable flow generalization are np hard problems with a multitude of applications in areas such as routing scheduling and bin packing these two categories of problems are known to be conceptually related but this connection has largely been ignored in terms of approximation algorithms we explore the topic of approximating disjoint path problems using polynomial size packing integer programs motivated by the disjoint paths applications we introduce the study of a class of packing integer programs called column restricted we develop improved approximation algorithms for column restricted programs a result that we believe is of independent interest additional approximation algorithms for disjoint paths are presented that are simple to implement and achieve good performance when the input has a special structure