a near tight lower bound on the time complexity of distributed minimum weight spanning tree construction this paper presents a lower bound of omega d sqrt n log n on the time required for the distributed construction of a minimum weight spanning tree mst in weighted n vertex networks of diameter d omega log n in the bounded message model this establishes the asymptotic near optimality of existing time efficient distributed algorithms for the problem whose complexity is o d sqrt n log n

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

local edge colouring of yao like subgraphs of unit disk graphs the focus of the present paper is on providing a local deterministic algorithm for colouring the edges of yao like subgraphs of unit disk graphs these are geometric graphs such that for some positive integers l k the following property holds at each node v if we partition the unit circle centered at v into 2k equally sized wedges then each wedge can contain at most l points different from v we assume that the nodes are location aware i e they know their cartesian coordinates in the plane the algorithm presented is local in the sense that each node can receive information emanating only from nodes which are at most a constant depending on k and l but not on the size of the graph number of hops measured in the original underlying unit disk graph away from it and hence the algorithm terminates in a constant number of steps the number of colours used is 2kl 1 and this is optimal for local algorithms since the maximal degree is 2kl and a colouring with 2kl colours can only be constructed by a global algorithm thus showing that in this class of graphs the price for locality is only one additional colour

scheduling arrivals to a stochastic service delivery system using copositive cones in this paper we investigate a stochastic appointment scheduling problem in an outpatient clinic with a single doctor the number of patients and their sequence of arrivals are fixed and the scheduling problem is to determine an appointment time for each patient the service durations of the patients are stochastic and only the mean and covariance estimates are known we do not assume any exact distributional form of the service durations and we solve for distributionally robust schedules that minimize the expectation of the weighted sum of patients waiting time and the doctor s overtime we formulate this scheduling problem as a convex conic optimization problem with a tractable semidefinite relaxation our model can be extended to handle additional support constraints of the service durations using the primal dual optimality conditions we prove several interesting structural properties of the optimal schedules we develop an efficient semidefinite relaxation of the conic program and show that we c

on the relationship between continuous and discrete time quantum walk quantum walk is one of the main tools for quantum algorithms defined by analogy to classical random walk a quantum walk is a time homogeneous quantum process on a graph both random and quantum walks can be defined either in continuous or discrete time but whereas a continuous time random walk can be obtained as the limit of a sequence of discrete time random walks the two types of quantum walk appear fundamentally different owing to the need for extra degrees of freedom in the discrete time case

about randomised distributed graph colouring and graph partition algorithms we present and analyse a very simple randomised distributed vertex colouring algorithm for arbitrary graphs of size n that halts in time o logn with probability 1 o n 1 each message containing 1 bit its bit complexity per channel is o logn from this algorithm we deduce and analyse a randomised distributed vertex colouring algorithm for arbitrary graphs of maximum degree d and size n that uses at most d 1 colours and halts in time o logn with probability 1 o n 1 we also obtain a partition algorithm for arbitrary graphs of size n that builds a spanning forest in time o logn with probability 1 o n 1 we study some parameters such as the number the size and the radius of trees of the spanning forest

bit complexity of breaking and achieving symmetry in chains and rings we consider a failure free asynchronous message passing network with n links where the processors are arranged on a ring or a chain the processors are identically programmed but have distinct identities taken from 0 1 m 1 we investigate the communication costs of three well studied tasks consensus leader and maxf finding the maximum identity we show that in chain and ring topologies the message complexities of all three tasks are the same hence we study a finer measure of complexity the number of transmitted bits required to solve a task t denoted bitc t we prove several new lower bounds and some simple upper bounds that imply the following results for the two processors case bitc consensus e 2 and bitc leader e bitc maxf e 2log2 m o 1 where the gap between the lower and upper bounds is almost always 1 for a chain bitc consensus e n bitc leader e n p log m and bitc maxf e n log m for the ring topology we prove the lower bound of n log m for leader and hence maxf we consider also a chain where the intermediate processors have no identities we prove that bitc leader e n log m which is equal to n times the bit complexity of the problem for two processors for the specific case when the chain length is even we prove that bitc leader e n for both above settings in addition we show that for any algorithm solving maxf there exists an input for which every execution has the bit complexity n log m this is not the case for leader in our proofs we use both methods of distributed computing and of communication complexity theory establishing new links between the two areas

simple neural like p systems for maximal independent set selection membrane systems p systems are distributed computing models inspired by living cells where a collection of processors jointly achieves a computing task the problem of maximal independent set mis selection in a graph is to choose a set of nonadjacent nodes to which no further nodes can be added in this letter we design a class of simple neural like p systems to solve the mis selection problem efficiently in a distributed way this new class of systems possesses two features that are attractive for both distributed computing and membrane computing first the individual processors do not need any information about the overall size of the graph second they communicate using only one bit messages

deterministic distributed vertex coloring in polylogarithmic time consider an n vertex graph g v e of maximum degree and suppose that each vertex v v hosts a processor the processors are allowed to communicate only with their neighbors in g the communication is synchronous that is it proceeds in discrete rounds in the distributed vertex coloring problem the objective is to color g with 1 or slightly more than 1 colors using as few rounds of communication as possible the number of rounds of communication will be henceforth referred to as running time efficient randomized algorithms for this problem are known for more than twenty years alon et al 1986 luby 1986 specifically these algorithms produce a 1 coloring within o log n time with high probability on the other hand the best known deterministic algorithm that requires polylogarithmic time employs o 2 colors this algorithm was devised in a seminal focs 87 paper by linial 1987 its running time is o log n in the same article linial asked whether one can color with significantly less than 2 colors in deterministic polylogarithmic time by now this question of linial became one of the most central long standing open questions in this area in this article we answer this question in the affirmative and devise a deterministic algorithm that employs 1 o 1 colors and runs in polylogarithmic time specifically the running time of our algorithm is o f log log n for an arbitrarily slow growing function f 1 we can also produce an o 1 coloring in o log log n time for an arbitrarily small constant 0 and an o coloring in o e log n time for an arbitrarily small constant e 0 our results are in fact far more general than this in particular for a graph of arboricity a our algorithm produces an o a1 coloring for an arbitrarily small constant 0 in time o log a log n

distributed δ 1 coloring in sublogarithmic rounds we give a new randomized distributed algorithm for 1 coloring in the local model running in o log 2o log log n rounds in a graph of maximum degree this implies that the 1 coloring problem is easier than the maximal independent set problem and the maximal matching problem due to their lower bounds of min log n log log n log log log by kuhn moscibroda and wattenhofer podc 04 our algorithm also extends to list coloring where the palette of each node contains 1 colors we extend the set of distributed symmetry breaking techniques by performing a decomposition of graphs into dense and sparse parts

optimal two stage algorithms for group testing problems group testing refers to the situation in which one is given a set of objects cal o an unknown subset cal p subseteq cal o and the task of determining cal p by asking queries of the type does cal p intersect cal q where cal q is a subset of cal o group testing is a basic search paradigm that occurs in a variety of situations such as quality control testing searching in storage systems multiple access communications and data compression among others group testing procedures have been recently applied in computational molecular biology where they are used for screening libraries of clones with hybridization probes and sequencing by hybridization motivated by particular features of group testing algorithms used in biological screening we study the efficiency of two stage group testing procedures our main result is the first optimal two stage algorithm that uses a number of tests of the same order as the information theoretic lower bound on the problem we also provide efficient algorithms for the case in which there is a bernoulli probability distribution on the possible sets cal p and an optimal algorithm for the case in which the outcome of tests may be unreliable because of the presence of inhibitory items in cal o our results depend on a combinatorial structure introduced in this paper we believe that it will prove useful in other contexts too

segal sugawara vectors for the lie algebra of type g2 explicit formulas for segal sugawara vectors associated with the simple lie algebra mathfrak g of type g 2 are found by using computer assisted calculations this leads to a direct proof of the feigin frenkel theorem describing the center of the corresponding affine vertex algebra at the critical level as an application we give an explicit solution of vinberg s quantization problem by providing formulas for generators of maximal commutative subalgebras of u mathfrak g we also calculate the eigenvalues of the hamiltonians on the bethe vectors in the gaudin model associated with mathfrak g

the local nature of list colorings for graphs of high girth we consider list coloring problems for graphs mathcal g of girth larger than c log delta 1 n where n and delta geq3 are respectively the order and the maximum degree of mathcal g and c is a suitable constant first we determine that the edge and total list chromatic numbers of these graphs are chi l mathcal g delta and chi l mathcal g delta 1 this proves that the general conjectures of bollobas and harris graphs combin 1 1985 pp 115 127 behzad the total chromatic number in combinatorial mathematics and its applications proc conf oxford 1969 academic press london 1971 pp 1 8 vizing diskret analiz 3 1964 pp 25 30 and juvan mohar and skrekovski combin probab comput 7 1998 pp 181 188 hold for this particular class of graphs moreover our proofs exhibit a certain degree of locality which we exploit to obtain an efficient distributed algorithm able to compute both kinds of optimal list colorings also using an argument similar to one of erdos we show that our algorithm can compute k list vertex colorings of graphs having girth larger than c log k 1 n

on spanners and lightweight spanners of geometric graphs we consider the problem of computing spanners of euclidean and unit disk graphs embedded in the two dimensional euclidean plane we are particularly interested in spanners that possess useful properties such as planarity bounded degree and or light weight such spanners have been extensively studied in the area of computational geometry and have been used as the building block for constructing efficient and reliable wireless network communication topologies we study the above problem under two computational models the centralized and the distributed model in the distributed model we focus on algorithms that are local such algorithms are suitable for the relevant applications e g wireless computing under the centralized model we present an o n lg n time algorithm that computes a bounded degree plane spanner of a complete euclidean graph where n is the number of points in the graph both upper bounds on the degree and the stretch factor significantly improve the previous bounds we extend this algorithm to compute a bounded degree plane lightweight spanner of a complete euclidean graph under the distributed model we give the first local algorithm for computing a spanner of a unit disk graph that is of bounded degree and plane the upper bounds on the degree stretch factor and the locality of the algorithm dramatically improve the previous results as shown in the paper this algorithm can also be extended to compute a bounded degree plane lightweight spanner of a unit disk graph our algorithms rely on structural and geometric results that we develop in this paper

locality in distributed graph algorithms this paper concerns a number of algorithmic problems on graphs and how they may be solved in a distributed fashion the computational model is such that each node of the graph is occupied by a processor which has its own id processors are restricted to collecting data from others which are at a distance at most t away from them in t time units but are otherwise computationally unbounded this model focuses on the issue of locality in distributed processing namely to what extent a global solution to a computational problem can be obtained from locally available data three results are proved within this model a 3 coloring of an n cycle requires time omega log n this bound is tight by previous work of cole and vishkin any algorithm for coloring the d regular tree of radius r which runs for time at most 2r 3 requires at least omega sqrt d colors in an n vertex graph of largest degree delta an o delta 2 coloring may be found in time o log n

optimal bit complexity randomised distributed mis and maximal matching algorithms for anonymous rings we present and analyse las vegas distributed algorithms which compute a mis or a maximal matching for anonymous rings their bit complexity and time complexity are o logn with high probability these algorithms are optimal modulo a multiplicative constant beyond the complexity results the interest of this work stands in the description and the analysis of these algorithms which may be easily generalised furthermore these results show a separation between the complexity of the mis problem and of the maximal matching problem on the one hand and the colouring problem on the other colouring can be computed only in w logn rounds on rings with high probability while mis is shown to have a faster algorithm this is in contrast to other models in which mis is at least as hard as colouring

appointment sequencing why the smallest variance first rule may not be optimal abstract we study the design of a healthcare appointment system with a single physician and a group of patients whose service durations are stochastic the challenge is to find the optimal arrival sequence for a group of mixed patients such that the expected total cost of patient waiting time and physician overtime is minimized while numerous simulation studies report that sequencing patients by increasing order of variance of service duration smallest variance first or svf rule performs extremely well in many environments analytical results on optimal sequencing are known only for two patients in this paper we shed light on why it is so difficult to prove the optimality of the svf rule in general we first assume that the appointment intervals are fixed according to a given template and analytically investigate the optimality of the svf rule in particular we show that the optimality of the svf rule depends on two important factors the number of patients in the system and the shape of service time distributions the svf rule is more likely to be optimal if the service time distributions are more positively skewed but this advantage gradually disappears as the number of patients increases these results partly explain why the optimality of the svf rule can only be proved for a small number of patients and why in practice the svf rule is usually observed to be superior since most empirical distributions of the service durations are positively skewed like log normal distributions the insights obtained from our analytical model apply to more general settings including the cases where the service durations follow log normal distributions and the appointment intervals are optimized

the distributed bit complexity of the ring in moran and warmuth 1993 siam j comput 22 no 2 379 399 it was shown that computing any non constant function on a ring of n of processors requires n log bits and this bound is tight this model assumed that all the processors in the ring are identical anonymous i e all processors run the same program and the only parameter of the program is the input to the processor in a relaxed model the anonymity is broken by providing each processor with a distinct identity which becomes a second parameter of the program that is executed by all processors if the set of possible identities grows doubly exponentially in n then by a reduction to the anonymous case one can show that the lower bound holds as well in this paper we show that the lower bound of n log n bits for computing any non constant function holds even if the set of possible identities is very small that is of sizer n1 for any positive

deterministic broadcasting time with partial knowledge of the network we consider the time of deterministic broadcasting in networks whose nodes have limited knowledge of network topology each node v knows only the part of the network within knowledge radiusr from it i e it knows the graph induced by all nodes at distance at most r from v apart from that each node knows the maximum degree of the network one node of the network called the source has a message which has to reach all other nodes we adopt the widely studied communication model called the one way model in which in every round each node can communicate with at most one neighbor and in each pair of nodes communicating in a given round one can only send a message while the other can only receive it this is the weakest of all store and forward models for point to point networks and hence our algorithms work for other models as well in at most the same time we show trade offs between knowledge radius and time of deterministic broadcasting when the knowledge radius is small i e when nodes are only aware of their close vicinity while for knowledge radius 0 minimum broadcasting time is e where e is the number of edges in the network broadcasting can be usually completed faster for positive knowledge radius our main results concern knowledge radius 1 we develop fast broadcasting algorithms and analyze their execution time we also prove lower bounds on broadcasting time showing that our algorithms are close to optimal

randomised distributed mis and colouring algorithms for rings with oriented edges in o log ź n bit rounds we present and analyse las vegas distributed algorithms which compute a mis or a colouring for anonymous rings with an arbitrary orientation of the edges their bit complexity and time complexity are o log n with high probability these algorithms are optimal modulo a multiplicative constant

improved distributed algorithms for coloring interval graphs with application to multicoloring trees we give a distributed 1 epsilon approximation algorithm for the minimum vertex coloring problem on interval graphs which runs in the mathcal local model and operates in mathrm o frac 1 epsilon log n rounds if nodes are aware of their interval representations then the algorithm can be adapted to the mathcal congest model using the same number of rounds prior to this work only constant factor approximations using mathrm o log n rounds were known 12 linial s ring coloring lower bound implies that the dependency on log n cannot be improved we further prove that the dependency on frac 1 epsilon is also optimal

arbitrary throughput versus complexity tradeoffs in wireless networks using graph partitioning several policies have recently been proposed for attaining the maximum throughput region or a guaranteed fraction thereof through dynamic link scheduling among these policies the ones that attain the maximum throughput region require a computation time which is linear in the network size and the ones that require constant or logarithmic computation time attain only certain fractions of the maximum throughput region in contrast in this paper we propose policies that can attain any desirable fraction of the maximum throughput region using a computation time that is largely independent of the network size first using a combination of graph partitioning techniques and lyapunov arguments we propose a simple policy for tree topologies under the primary interference model that requires each link to exchange only 1 bit information with its adjacent links and approximates the maximum throughput region using a computation time that depends only on the maximum degree of nodes and the approximation factor then we develop a framework for attaining arbitrary close approximations for the maximum throughput region in arbitrary networks and use this framework to obtain any desired tradeoff between throughput guarantees and computation times for a large class of networks and interference models specifically given any epsiv 0 the maximum throughput region can be approximated in these networks within a factor of 1 epsiv using a computation time that depends only on the maximum node degree and epsiv

exact leaf powers we define and study the new notion of exact k leaf powers where a graph g v g e g is an exact k leaf power if and only if there exists a tree t v t e t an exact k leaf root of g whose set of leaves equals v g such that email protected e g holds for u email protected v g if and only if the distance of u and v in t is exactly k this new notion is closely related to but different from leaf powers and neighbourhood subtree tolerance graphs we prove characterizations of exact 3 and 4 leaf powers which imply that such graphs can be recognized in linear time and that also the corresponding exact leaf roots can be found in linear time furthermore we characterize all exact 5 leaf roots of chordless cycles and derive several properties of exact 5 leaf powers

distributed coloring algorithms for triangle free graphs vertex coloring is a central concept in graph theory and an important symmetry breaking primitive in distributed computing whereas degree graphs may require palettes of 1 colors in the worst case it is well known that the chromatic number of many natural graph classes can be much smaller in this paper we give new distributed algorithms to find k coloring in graphs of girth 4 triangle free graphs girth 5 and trees the parameter k can be at most 1 4 o 1 ln in triangle free graphs and at most 1 o 1 ln in girth 5 graphs and trees where o 1 is a function of specifically for sufficiently large we can find such a coloring in o k log n time moreover for any we can compute such colorings in roughly logarithmic time for triangle free and girth 5 graphs and in o log log log n time on trees as a byproduct our algorithm shows that the chromatic number of triangle free graphs is at most 4 o 1 ln which improves on jamall s recent bound of 67 o 1 ln finally we show that 1 coloring for triangle free graphs can be obtained in sublogarithmic time for any

constructions of generalized superimposed codes with applications to group testing and conflict resolution in multiple access channels in this paper we introduce a parameterized generalization of the well known superimposed codes we give algorithms for their construction and provide non existential results we apply our new combinatorial structures to the efficient solution of new group testing problems and access coordination issues in multiple access channels

appointment scheduling with discrete random durations we consider the problem of determining an optimal appointment schedule for a given sequence of jobs e g medical procedures on a single processor e g operating room examination facility physician to minimize the expected total underage and overage costs when each job has a random processing duration given by a joint discrete probability distribution simple conditions on the cost rates imply that the objective function is submodular and l convex then there exists an optimal appointment schedule that is integer and can be found in polynomial time our model can handle a given due date for the total processing e g end of day for an operating room after which overtime is incurred as well as no shows and some emergencies

patterns from nature a well established problem in global optimization is the problem of colouring the vertices of an arbitrary graph using the minimal number of colours such that adjacent vertices are assigned different colours one way to restrict the number of colours used is to allow only greedy colourings a greedy colouring is an assignment of colours to the vertices of a graph that can be obtained by an algorithm that considers each vertex in turn and assigns the first colour that is not already assigned to some neighbour an optimal colouring can always be obtained in this way by choosing an appropriate order on the vertices recently a new bio inspired approach to distributed pattern formation has been proposed based on modelling the neurological development of the fruit fly building on that approach we propose a new simple randomised algorithm for distributed greedy colouring using only local processing at the vertices and messages along the edges in our approach the processors exchange only simple messages representing potential colour values and each processor has minimal graph knowledge we discuss two variations of this algorithm and investigate their time complexity and message complexity both theoretically and experimentally in addition we show experimentally that the number of colours used turns out to be optimal or near optimal for many standard graph colouring benchmarks thus for distributed networks our algorithm serves as an effective heuristic approach to computing a colouring with a small number of colours

a fast network decomposition algorithm and its applications to constant time distributed computation abstract a partition c 1 c 2 c q of g v e into clusters of strong respectively weak diameter d such that the supergraph obtained by contracting each c i is l colorable is called a strong resp weak d l network decomposition network decompositions were introduced in a seminal paper by awerbuch goldberg luby and plotkin in 1989 awerbuch et al showed that strong d l network decompositions with d l exp o log n log log n can be computed in distributed deterministic time o d even more importantly they demonstrated that network decompositions can be used for a great variety of applications in the message passing model of distributed computing the result of awerbuch et al was improved by panconesi and srinivasan in 1992 in the latter result d l exp o log n and the running time is o d as well in another remarkable breakthrough linial and saks in 1992 showed that weak o log n o log n network decompositions can be computed in distributed randomized time o log 2 n much more recently barenboim 2012 devised a distributed randomized constant time algorithm for computing strong network decompositions with d o 1 however the parameter l in his result is o n 1 2 in this paper we drastically improve the result of barenboim and devise a distributed randomized constant time algorithm for computing strong o 1 o n network decompositions as a corollary we derive a constant time randomized o n approximation algorithm for the distributed minimum coloring problem improving the previously best known o n 1 2 approximation guarantee we also derive other improved distributed algorithms for a variety of problems most notably for the extremely well studied distributed minimum dominating set problem currently there is no known deterministic polylogarithmic time algorithm we devise a deterministic polylogarithmic time approximation algorithm for this problem addressing an open problem of lenzen and wattenhofer 2010

local algorithms for edge colorings in udgs in this paper we consider two problems the edge coloring and the strong edge coloring problems on unit disk graphs udgs both problems have important applications in wireless sensor networks as they can be used to model link scheduling problems in such networks it is well known that both problems are np complete and approximation algorithms for them have been extensively studied under the centralized model of computation centralized algorithms however are not suitable for ad hoc wireless sensor networks whose devices typically have limited resources and lack the centralized coordination we develop local distributed approximation algorithms for the edge coloring and the strong edge coloring problems on unit disk graphs for the edge coloring problem our local distributed algorithm has approximation ratio 2 and locality 50 for the strong edge coloring problem on udgs we present two local distributed algorithms with different tradeoffs between their approximation ratio and locality the first algorithm has ratio 128 and locality 22 whereas the second algorithm has ratio 10 and locality 180

on the time and the bit complexity of distributed randomised anonymous ring colouring we present and analyse a very simple randomised distributed vertex colouring algorithm for ring graphs its time complexity is log 2n o logn on average and 2log 2n o logn with probability 1 o n 1 since each message contains one bit we deduce the same values for its bit complexity then we combine this algorithm with another and we obtain a 3 colouring algorithm for ring graphs thanks to an overlapping we obtain once more the same values for the time complexities on average and with probability 1 o n 1 the same results hold for the bit complexity these results are obtained using the mellin transform we establish lower bounds on average and with probability 1 o n 1 for the distributed randomised anonymous ring colouring problem we prove that our algorithms match these lower bounds modulo a negligible additive function negligible with respect to log 2n we assume that the ring is anonymous unique identities are not available to distinguish the processes we only assume that each vertex distinguishes between its neighbours furthermore we do not assume that the size or an upper bound on the size of the ring is known

the locality of distributed symmetry breaking symmetry breaking problems are among the most well studied in the field of distributed computing and yet the most fundamental questions about their complexity remain open in this article we work in the local model where the input graph and underlying distributed network are identical and study the randomized complexity of four fundamental symmetry breaking problems on graphs computing miss maximal independent sets maximal matchings vertex colorings and ruling sets a small sample of our results includes the following an mis algorithm running in o log2 2o log log n time where is the maximum degree this is the first mis algorithm to improve on the 1986 algorithms of luby and alon babai and itai when log n l l 2 log n and comes close to the log log log lower bound of kuhn moscibroda and wattenhofer a maximal matching algorithm running in o log log 4log n time this is the first significant improvement to the 1986 algorithm of israeli and itai moreover its dependence on is nearly optimal a 1 coloring algorithm requiring o log 2o log log n time improving on an o log log n time algorithm of schneider and wattenhofer a method for reducing symmetry breaking problems in low arboricity degeneracy graphs to low degree graphs roughly speaking the arboricity or degeneracy of a graph bounds the density of any subgraph corollaries of this reduction include an o log n time maximal matching algorithm for graphs with arboricity up to 2 log n and an o log 2 3n time mis algorithm for graphs with arboricity up to 2 log n 1 3 each of our algorithms is based on a simple but powerful technique for reducing a randomized symmetry breaking task to a corresponding deterministic one on a poly log n size graph

a primal dual bicriteria distributed algorithm for capacitated vertex cover in this paper we consider the capacitated vertex cover problem which is the variant of vertex cover where each node is allowed to cover a limited number of edges we present an efficient deterministic distributed approximation algorithm for the problem our algorithm computes a 2 epsilon approximate solution which violates the capacity constraints by a factor of 4 epsilon in a polylogarithmic number of communication rounds on the other hand we also show that every efficient distributed approximation algorithm for this problem must violate the capacity constraints our result is achieved in two steps we first develop a 2 approximate sequential primal dual algorithm that violates the capacity constraints by a factor of 2 subsequently we present a distributed version of this algorithm we demonstrate that the sequential algorithm has an inherent need for synchronization which forces any naive distributed implementation to use a linear number of communication rounds the challenge in this step is therefore to achieve a reduction of the communication complexity to a polylogarithmic number of rounds without worsening the approximation guarantee

randomized distributed edge coloring via an extension of the chernoff hoeffding bounds certain types of routing scheduling and resource allocation problems in a distributed setting can be modeled as edge coloring problems we present fast and simple randomized algorithms for edge coloring a graph in the synchronous distributed point to point model of computation our algorithms compute an edge coloring of a graph g with n nodes and maximum degree delta with at most 1 6 delta o log 1 delta n colors with high probability arbitrarily close to 1 for any fixed delta 0 they run in polylogarithmic time the upper bound on the number of colors improves upon the 2 delta 1 coloring achievable by a simple reduction to vertex coloring to analyze the performance of our algorithms we introduce new techniques for proving upper bounds on the tail probabilities of certain random variables the chernoff hoeffding bounds are fundamental tools that are used very frequently in estimating tail probabilities however they assume stochastic independence among certain random variables which may not always hold our results extend the chernoff hoeffding bounds to certain types of random variables which are not stochastically independent we believe that these results are of independent interest and merit further study

self self dual spaces of polynomials abstract a space of polynomials v of dimension 7 is called self dual if the divided wronskian of any 6 subspace is in v a self dual space v has a natural inner product the divided wronskian of any isotropic 3 subspace of v is a square of a polynomial we call v self self dual if the square root of the divided wronskian of any isotropic 3 subspace is again in v we show that the self self dual spaces have a natural non degenerate skew symmetric 3 form defined in terms of wronskians we show that the self self dual spaces correspond to g2 populations related to the bethe ansatz of the gaudin model of type g2 and prove that a g2 population is isomorphic to the g2 flag variety

a simple parallel algorithm for the maximal independent set problem two basic design strategies are used to develop a very simple and fast parallel algorithms for the maximal independent set mis problem the first strategy consists of assigning identical copies o

a note on appointment scheduling with piecewise linear cost functions we consider the problem of determining the optimal schedules for a given sequence of jobs on a single processor the objective is to minimize the expected total cost incurred by job waiting and processor idling where the job processing times are random variables it is known in the prior literature that if the processing times are integers and the costs are linear functions satisfying a mild condition then the problem can be solved in a polynomial number of expected cost evaluations in this work we extend the result to piecewise linear cost functions which include many useful objective functions in practice our analysis explores the hidden dual network flow structure of the appointment scheduling problem and thus greatly simplifies that of prior work we also find the number of samples needed to compute a near optimal solution when only some independent samples of processing times are known

randomized obdd based graph algorithms abstract implicit graph algorithms deal with the characteristic function e of the edge set e of a graph g v e encoding the nodes by binary vectors e can be represented by an ordered binary decision diagram obdd which is a well known data structure for boolean functions obdd based graph algorithms solve graph optimization problems by mainly using functional operations and are a heuristic approach to cope with massive graphs these algorithms heavily rely on a compact representation of the underlying boolean functions which is why all previously known obdd based algorithms are deterministic since random functions are not compressible in general here the first randomized obdd based algorithms are presented where random functions with limited independence are used to overcome the large representation size on the theoretical part we investigate the size of obdds representing k wise independent random functions in addition we give a construction of almost k wise independent random functions by means of a random obdd generation and show an almost matching lower bound of the obdd size representing such random functions on the algorithmic part randomization often facilitates the design of simple algorithms which in the context of obdd based algorithms means a small number of functional operations and as few input variables of the used boolean functions as possible we present two randomized implicit algorithms a randomized minimum spanning tree algorithm and a maximal matching algorithm the matching algorithm needs o log 3 v functional operations in expectation and uses functions with at most 3 log v variables which is both better than the best known deterministic algorithms w r t functional operations and variables the algorithm may be of independent interest the experimental evaluation of the algorithms shows that they outperform known obdd based algorithms for the maximal matching problem and the minimum spanning tree problem

local computation lower and upper bounds the question of what can be computed and how efficiently is at the core of computer science not surprisingly in distributed systems and networking research an equally fundamental question is what can be computed in a distributed fashion more precisely if nodes of a network must base their decision on information in their local neighborhood only how well can they compute or approximate a global optimization problem in this paper we give the first polylogarithmic lower bound on such local computation for optimization problems including minimum vertex cover minimum connected dominating set maximum matching maximal independent set and maximal matching in addition we present a new distributed algorithm for solving general covering and packing linear programs for some problems this algorithm is tight with the lower bounds whereas for others it is a distributed approximation scheme together our lower and upper bounds establish the local computability and approximability of a large class of problems characterizing how much local information is required to solve these tasks

improved distributed approximate matching we present distributed network algorithms to compute weighted and unweighted matchings with improved approximation ratios and running times the computational model is a network of processors exchanging o log n bit messages the congest model for unweighted graphs we give an algorithm providing 1 e approximation in o log n time for any constant e 0 improving on the classical approximation in olog n time of israeli and itai 1986 the time complexity of the algorithm depends on 1he exponentially in the general case and polynomially in bipartite graphs for weighted graphs we present another algorithm which provides e approximation in general graphs in o loge 1log n time improving on the previously known algorithms which attain e approximation in o log n time or approximation in o n time all our algorithms are randomized the complexity bounds hold both with high probability and for the expected running time

local approximation of the maximum cut in regular graphs abstract this paper is devoted to the distributed complexity of finding an approximation of the maximum cut maxcut in graphs a classical algorithm consists in letting each vertex choose its side of the cut uniformly at random this does not require any communication and achieves an approximation ratio of at least 1 2 in expectation when the graph is d regular and triangle free a slightly better approximation ratio can be achieved with a randomized algorithm running in a single round here we investigate the round complexity of deterministic distributed algorithms for maxcut in regular graphs we first prove that if g is d regular with d even and fixed no deterministic algorithm running in a constant number of rounds can achieve a constant approximation ratio we then give a simple one round deterministic algorithm achieving an approximation ratio of 1 d for d regular graphs when d is odd we show that this is best possible in several ways and in particular no deterministic algorithm with approximation ratio 1 d with 0 can run in a constant number of rounds we also prove results of a similar flavor for the maxdicut problem in regular oriented graphs where we want to maximize the number of arcs oriented from the left part to the right part of the cut

efficient quantum algorithms for simulating sparse hamiltonians we present an efficient quantum algorithm for simulating the evolution of a quantum state for a sparse hamiltonian h over a given time t in terms of a procedure for computing the matrix entries of h in particular when h acts on n qubits has at most a constant number of nonzero entries in each row column and h is bounded by a constant we may select any positive integer k such that the simulation requires o log n t1 1 2k accesses to matrix entries of h we also show that the temporal scaling cannot be significantly improved beyond this because sublinear time scaling is not possible

symmetry breaking depending on the chromatic number or the neighborhood growth we deterministically compute a d 1 coloring and a maximal independent set mis in time o d 1 2 q 1 h log n for d 1 email protected email protected 1 i h where d j is defined as the maximal number of nodes within distance j for a node and d email protected 1 our greedy coloring and mis algorithms improve the state of the art algorithms running in o d log n for a large class of graphs i e graphs of moderate neighborhood growth with h 36 we also state and analyze a randomized coloring algorithm in terms of the chromatic number the run time and the used colors our algorithm runs in time o email protected log n for email protected email protected log 1 1 l o g nn and email protected o d log 1 1 l o g nn for graphs of polylogarithmic chromatic number the analysis reveals an exponential gap compared to the fastest d 1 coloring algorithm running in time o email protected logn the algorithm works without knowledge of g and uses less than d colors i e 1 1 o g d with high probability to the best of our knowledge this is the first distributed algorithm for such general graphs taking the chromatic number g into account we also improve on the state of the art deterministic computation of 2 c ruling sets

lower bounds for local approximation in the study of deterministic distributed algorithms it is commonly assumed that each node has a unique o log n bit identifier we prove that for a general class of graph problems local algorithms constant time distributed algorithms do not need such identifiers a port numbering and orientation is sufficient our result holds for so called simple po checkable graph optimisation problems this includes many classical packing and covering problems such as vertex covers edge covers matchings independent sets dominating sets and edge dominating sets we focus on the case of bounded degree graphs and show that if a local algorithm finds a constant factor approximation of a simple po checkable graph problem with the help of unique identifiers then the same approximation ratio can be achieved on anonymous networks as a corollary of our result we derive a tight lower bound on the local approximability of the minimum edge dominating set problem by prior work there is a deterministic local algorithm that achieves the approximation factor of 4 1 2 in graphs of maximum degree this approximation ratio is known to be optimal in the port numbering model our main theorem implies that it is optimal also in the standard model in which each node has a unique identifier our main technical tool is an algebraic construction of homogeneously ordered graphs we say that a graph is r homogeneous if its nodes are linearly ordered so that an fraction of nodes have pairwise isomorphic radius r neighbourhoods we show that there exists a finite r homogeneous 2k regular graph of girth at least g for any

discrete miura opers and solutions of the bethe ansatz equations solutions of the bethe ansatz equations associated to the xxx model of a simple lie algebra open image in new window come in families called the populations we prove that a population is isomorphic to the flag variety of the langlands dual lie algebra open image in new window the proof is based on the correspondence between the solutions of the bethe ansatz equations and special difference operators which we call the discrete miura opers the notion of a discrete miura oper is one of the main results of the paper