algorithmic complexity of weakly semiregular partitioning and the representation number abstract a graph g is weakly semiregular if there are two numbers a b such that the degree of every vertex is a or b the weakly semiregular number of a graph g denoted by w r g is the minimum number of subsets into which the edge set of g can be partitioned so that the subgraph induced by each subset is a weakly semiregular graph we present a polynomial time algorithm to determine whether the weakly semiregular number of a given tree is two on the other hand we show that determining whether w r g 2 for a given bipartite graph g with at most three numbers in its degree set is np complete among other results for every tree t we show that w r t 2 log 2 t o 1 where t denotes the maximum degree of t a graph g is a d d s graph if the degree of every vertex of g lies in the interval d d s a d d 1 graph is said to be semiregular the semiregular number of a graph g denoted by s r g is the minimum number of subsets into which the edge set of g can be partitioned so that the subgraph induced by each subset is a semiregular graph we prove that the semiregular number of a tree t is t 2 on the other hand we show that determining whether s r g 2 for a given bipartite graph g with g 6 is np complete in the second part of the work we consider the representation number a graph g has a representation modulo r if there exists an injective map l v g z r such that vertices v and u are adjacent if and only if l u l v is relatively prime to r the representation number denoted by r e p g is the smallest r such that g has a representation modulo r narayan and urick conjectured that the determination of r e p g for an arbitrary graph g is a difficult problem 38 in this work we confirm this conjecture and show that if n p p then for any 0 there is no polynomial time 1 n 2 approximation algorithm for the computation of representation number of regular graphs with n vertices

chromatic index of graphs with no cycle with a unique chord the class c of graphs that do not contain a cycle with a unique chord was recently studied by trotignon and vuskovic in press 23 who proved for these graphs strong structure results which led to solving the recognition and vertex colouring problems in polynomial time in the present paper we investigate how these structure results can be applied to solve the edge colouring problem in the class we give computational complexity results for the edge colouring problem restricted to c and to the subclass c composed of the graphs of c that do not have a 4 hole we show that it is np complete to determine whether the chromatic index of a graph is equal to its maximum degree when the input is restricted to regular graphs of c with fixed degree d 3 for the subclass c we establish a dichotomy if the maximum degree is d 3 the edge colouring problem is np complete whereas if d 3 the only graphs for which the chromatic index exceeds the maximum degree are the odd holes and the odd order complete graphs a characterization that solves edge colouring problem in polynomial time we determine two subclasses of graphs in c of maximum degree 3 for which edge colouring is polynomial finally we remark that a consequence of one of our proofs is that edge colouring in np complete for r regular tripartite graphs of degree d 3 for r 3

a remark on computing distance functions we propose a new method for the reconstruction of the signed distance function in the context of level set methods the new method is a modification of the algorithm which makes use of the pde equation for the distance function introduced by m sussman p smereka and s osher 1994 j comput phys 119 146 it is based mainly on the use of a truly upwind discretization near the interface comparison with the previous algorithm shows a definite improvement when used with a first order upwind scheme the method provides first order accuracy for the signed distance function in the whole computational domain and second order accuracy in the location of the interface a second order version of the method is also presented

parameterized complexity of max lifetime target coverage in wireless sensor networks max lifetime target coverage can be viewed as a family of problems where the task is to partition the sensors into groups and assign their time slots such that the coverage lifetime is maximized while satisfying some coverage requirement unfortunately these problems are np hard to gain insight into the source of the complexity we initiate a systematic parameterized complexity study of two types of max lifetime target coverage max min target coverage and max individual target coverage we first prove that both problems remain np hard even in the special cases where each target is covered by at most two sensors or each sensor can cover at most two targets by contrast restricting the number of targets reduces the complexity of the considered problems in other words they are both fixed parameter tractable fpt with respect to the parameter number of targets moreover we extend our studies to the structural parameter number k of sensors covering at least two targets positively both problems are in fpt with respect to k finally we show that max min target coverage is in fpt with respect to the combined parameters number of groups and number of targets covered by each group

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

the parameterised complexity of list problems on graphs of bounded treewidth we consider the parameterised complexity of several list problems on graphs with parameter treewidth or pathwidth in particular we show that list edge chromatic number and list total chromatic number are fixed parameter tractable parameterised by treewidth whereas list hamilton path is w1 hard even parameterised by pathwidth these results resolve two open questions of fellows fomin lokshtanov rosamond saurabh szeider and thomassen 2011

the complexity of lsh feasibility in this paper we study the complexity of the following feasibility problem given an n n similarity matrix s as input is there a locality sensitive hash lsh for s we show that the lsh feasibility problem is np hard even in the following strong promise version either s admits an lsh or s is at l1 distance at least n2 from every similarity that admits an lsh we complement this hardness result by providing an o 3n algorithm for the lsh feasibility problem which improves upon the naive n n time algorithm we prove that this running time is tight modulo constants under the exponential time hypothesis

np hard graph problems and boundary classes of graphs any graph problem which is np hard in general graphs becomes polynomial time solvable when restricted to graphs in special classes when does a difficult problem become easy to answer this question we study the notion of boundary classes in the present paper we define this notion in its most general form describe several approaches to identify boundary classes and apply them to various graph problems

weighted and locally bounded list colorings in split graphs cographs and partial k trees abstract for a fixed number of colors we show that in node weighted split graphs cographs and graphs of bounded tree width one can determine in polynomial time whether a proper list coloring of the vertices of a graph such that the total weight of vertices of each color equals a given value in each part of a fixed partition of the vertices exists we also show that this result is tight in some sense by providing hardness results for the cases where any one of the assumptions does not hold the edge coloring variant is also studied as well as special cases of cographs and split graphs

free products of compact quantum groups we construct and study compact quantum groups from free products ofc algebras in this connection we discover two mysterious classes of natural compact quantum groups a u m anda o m thea u m s respectivelya o m s are non isomorphic to each other for differentm s and are not obtainable by the ordinary quantization method we also clarify some basic concepts in the theory of compact quantum groups

the defect of generalized fourier matrices the n n complex hadamard matrices form a real algebraic manifold cn we have cn mn t nun and following tadej and yczkowski we investigate here the computation of the enveloping tangent space t hcn thmn t thnun and notably of its dimension d h dim t hcn called undephased defect of h our main result is an explicit formula for the defect of the fourier matrix fg associated to an arbitrary finite abelian group g zn1 znr we also comment on the general question does the associated quantum permutation group see the defect with a probabilistic speculation involving diaconis shahshahani type variables

polyhedron of triangle free simple 2 matchings in subcubic graphs a simple 2 matching in an edge weighted graph is a subgraph all of whose nodes have degree 0 1 or 2 we consider the problem of finding a maximum weight simple 2 matching that contains no triangles which is closely related to a class of relaxations of the traveling salesman problem tsp our main results are for graphs with maximum degree 3 a complete description of the convex hull of incidence vectors of triangle free simple 2 matchings and a strongly polynomial time algorithm for the above problem our system requires the use of a type of comb inequality introduced by grotschel and padberg for the tsp polytope that has 0 1 2 coefficients and hence is more general than the well known blossom inequality used in edmonds characterization of the simple 2 matching polytope

tannaka krein duality for compact matrix pseudogroups twisted su n groups the notion of concrete monoidalw category is introduced and investigated a generalization of the tannaka krein duality theorem is proved it leads to new examples of compact matrix pseudogroups among them we have twistedsu n groups denoted bys u n it is shown that the representation theory fors u n is similar to that ofsu n irreducible representations are labeled by young diagrams and formulae for dimensions and multiplicity are the same as in the classical case

p k coloring problems in line graphs the p k coloring problems generalize the usual coloring problem by replacing stable sets by cliques and stable sets complexities of some variations of p k coloring problems split coloring and cocoloring are studied in line graphs polynomial algorithms or proofs of np completeness are given according to the complexity status we show that the most general p k coloring problems are more difficult than the cocoloring and the split coloring problems while there is no such relation between the last two problems we also give complexity results for the problem of finding a maximum p k colorable subgraph in line graphs finally upper bounds on the optimal values are derived in general graphs by sequential algorithms based on welsh powell and matula orderings

integer round up property for the chromatic number of some h perfect graphs we prove that every h perfect line graph and every t perfect claw free graph g has the integer round up property for the chromatic number for every non negative integral weight function c on the vertices of g the weighted chromatic number of g c can be obtained by rounding up its fractional relaxation as a corollary we obtain that the weighted chromatic number can be computed in polynomial time for these graphs finally we show a new example of a graph operation which preserves the integer round up property for the chromatic number and use it to provide a first example of a t perfect 3 colorable graph which does not have this property

jump number maximization for proper interval graphs and series parallel graphs in this paper the order theoretic problem of finding linear extensions of a partially ordered set poset with certain properties is mapped to the corresponding graphtheorectic problem given a poset a jump exists between two consecutive symbols in a topological sequence if there is no relation between these two symbols in the poset we want to find a topological sequence that minimizes or maximizes the total number of jumps in graph theoretic terms define the jump number optimization problem as finding a linear ordering of vertex sequence of a graph such that the number of pairs of consecutive vertices that are adjacent is maximized jump number minimization problem or minimized jump number maximization problem the complexities of both problems with respect to some special graphs are investigated first in this paper we then study the jump maximization problem for proper interval graphs and series parallel graphs from this study we hope that more researches can be derived from these interdisciplinary knowledge domains

stable laws and domains of attraction in free probability theory in this paper we determine the distributional behavior of sums of free in the sense of voiculescu identically distributed inflnitesimal random variables the theory is shown to parallel the classical theory of independent random variables though the limit laws are usually quite difierent our work subsumes all previously known instances of weak convergence of sums of free identically distributed random variables in particular we determine the domains of attraction of stable distributions in the free theory these freely stable distributions are studied in detail in the appendix where their unimodality and duality properties are demonstrated

space efficient euler partition and bipartite edge coloring abstract we describe space efficient algorithms for two problems on undirected multigraphs euler partition partitioning the edges into a minimum number of trails and bipartite edge coloring coloring the edges of a bipartite multigraph with the minimum number of colors let n m and 1 be the numbers of vertices and of edges and the maximum degree respectively of the input multigraph for euler partition we reduce the amount of working memory needed by a logarithmic factor from n m log n m bits to o n m bits while preserving a running time of o n m for bipartite edge coloring still using o n m bits of working memory we achieve a running time of o n m min log log log m log this is o m log log if m n log n log log n log n to be compared with o m log for the fastest known algorithm instrumental in obtaining the latter result is a new data structure presented here the subgraph stack that allows graph algorithms centered around recursive calls on substantially smaller subgraphs to be implemented in a space efficient manner

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

multivariate spline approximation of the signed distance function the signed distance function can effectively support many geometry processing tasks such as decimates smoothing and shape reconstruction since it provides efficient access to distance estimates in this paper we present an adaptive method to approximate the signed distance function of a smooth curve by using polynomial splines over type 2 triangulation the trimmed offsets are also studied

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

the full classification of orthogonal easy quantum groups we study easy quantum groups a combinatorial class of orthogonal quantum groups introduced by banica speicher in 2009 we show that there is a countable descending chain of easy quantum groups interpolating between bichon s free wreath product with the permutation group s n and a semi direct product of a permutation action of s n on a free product this reveals a series of new commutation relations interpolating between a free product construction and the tensor product furthermore we prove a dichotomy result saying that every hyperoctahedral easy quantum group is either part of our new interpolating series of quantum groups or belongs to a class of semi direct product quantum groups recently studied by the authors this completes the classification of easy quantum groups we also study combinatorial and operator algebraic aspects of the new interpolating series

structure in approximation classes the study of the approximability properties of np hard optimization problems has recently made great advances mainly due to the results obtained in the field of proof checking the last important breakthrough proves the apx completeness of several important optimization problems and thus reconciles two distinct views of approximation classes syntactic and computational s khanna et al in proc 35th ieee symp on foundations of computer science ieee computer society press los alamitos ca 1994 pp 819 830 in this paper we obtain new results on the structure of several computationally defined approximation classes in particular after defining a new approximation preserving reducibility to be used for as many approximation classes as possible we give the first examples of natural npo complete problems and the first examples of natural apx intermediate problems moreover we state new connections between the approximability properties and the query complexity of npo problems

quantum symmetry groups of finite spaces we determine the quantum automorphism groups of finite spaces these are compact matrix quantum groups in the sense of woronowicz

decompositions for the edge colouring of reduced indifference graphs the chromatic index problem finding the minimum number of colours required for colouring the edges of a graph is still unsolved for indifference graphs whose vertices can be linearly ordered so that the vertices contained in the same maximal clique are consecutive in this order we present new positive evidence for the conjecture every non neighbourhood overfull indifference graph can be edge coloured with maximum degree colours two adjacent vertices are twins if they belong to the same maximal cliques a graph is reduced if it contains no pair of twin vertices a graph is overfull if the total number of edges is greater than the product of the maximum degree by n 2 where n is the number of vertices we give a structural characterization for neighbourhood overfull indifference graphs proving that a reduced indifference graph cannot be neighbourhood overfull we show that the chromatic index for all reduced indifference graphs is the maximum degree we present two decomposition methods for edge colouring reduced indifference graphs with maximum degree colours

an upper bound for the choice number of star edge coloring of graphs abstract the star chromatic index of a multigraph g denoted s g is the minimum number of colors needed to properly color the edges of g such that no path or cycle of length four is bi colored a multigraph g is star k edge colorable if s g k dvo ak et al 2013 proved that every subcubic multigraph is star 7 edge colorable they conjectured in the same paper that every subcubic multigraph should be star 6 edge colorable in this paper we consider this problem in a more general setting we investigate star list edge coloring of general graph g and obtain an upper bound for the choice number of star edge coloring of graphs namely we proved that s l 2 3 2 1 2 1 2 2

a monolithic smoothing gap algorithm for contact impact based on the signed distance function a new algorithm has been developed for smoothing the surfaces in finite element formulations of contact impact a key feature of this method is that the smoothing is done implicitly by constructing smooth signed distance functions for the bodies these functions are then employed for the computation of the gap and other variables needed for implementation of contact impact the smoothed signed distance functions are constructed by a moving least squares approximation with a polynomial basis results show that when nodes are placed on a surface the surface can be reproduced with an error of about one per cent or less with either a quadratic or a linear basis with a quadratic basis the method exactly reproduces a circle or a sphere even for coarse meshes results are presented for contact problems involving the contact of circular bodies copyright c 2002 john wiley sons ltd

a noncommutative de finetti theorem invariance under quantum permutations is equivalent to freeness with amalgamation we show that the classical de finetti theorem has a canonical noncommutative counterpart if we strengthen exchangeability i e invariance of the joint distribution of the random variables under the action of the permutation group to invariance under the action of the quantum permutation group more precisely for an infinite sequence of noncommutative random variables x i i in mathbb n we prove that invariance of the joint distribution of the xi s under quantum permutations is equivalent to the fact that the xi s are identically distributed and free with respect to the conditional expectation onto the tail algebra of the xi s

two parameter families of quantum symmetry groups we introduce and study natural two parameter families of quantum groups motivated on one hand by the liberations of classical orthogonal groups and on the other by quantum isometry groups of the duals of the free groups specifically for each pair p q of non negative integers we define and investigate quantum groups o p q b p q s p q and h p q corresponding to respectively orthogonal groups bistochastic groups symmetric groups and hyperoctahedral groups in the first three cases the new quantum groups turn out to be related to the dual free products of free quantum groups studied earlier for h p q the situation is different and we show that h p 0 qiso fp where the latter can be viewed as a liberation of the classical isometry group of the p dimensional torus

when hamming meets euclid the approximability of geometric tsp and steiner tree we prove that the traveling salesman problem sc min tsp is sf max snp hard and thus sf np hard to approximate within some constant r 1 even if all cities lie in a euclidean space of dimension log n n is the number of cities and distances are computed with respect to any lp norm the running time of recent approximation schemes for geometric sc min tsp is doubly exponential in the number of dimensions our result implies that this dependence is necessary unless np has subexponential algorithms as an intermediate step we also prove the hardness of approximating sc min tsp in hamming spaces finally we prove a similar but weaker inapproximability result for the steiner minimal tree problem sc min st the reduction for sc min tsp uses error correcting codes the reduction for sc min st uses the integrality property of sc min cut linear programming relaxations the only previous inapproximability results for metric sc min tsp involved metrics where all distances are 1 or 2

bipartite edge coloring in o delta m time we show that a minimum edge coloring of a bipartite graph can be found in o delta m time where delta and m denote the maximum degree and the number of edges of g respectively it is equivalent to finding a perfect matching in a k regular bipartite graph in o km time by sharpening the methods a minimum edge coloring of a bipartite graph can be found in o p max delta log delta m time where p max delta is the largest prime factor of delta moreover a perfect matching in a k regular bipartite graph can be found in o pmax k m time

the 3 colorability problem on graphs with maximum degree four the 3 colorability problem is known to be np complete in the class of graphs with maximum degree four on the other hand due to the celebrated theorem of brooks the problem has a polynomial time solution for graphs with maximum degree three to make the complexity gap more precise we study a family of intermediate graph classes between these two extremes and classify all of them according to the computational complexity of the problem in particular we generalize brooks s theorem in the case of 3 colorability to a larger class by showing that every connected graph in that class is 3 colorable unless it is a complete graph on four vertices

approximating the max edge coloring problem the max edge coloring problem is a natural weighted generalization of the classical edge coloring problem arising in the domain of communication systems in this problem each color class is assigned the weight of the heaviest edge in this class and the objective is to find a proper edge coloring of the input graph minimizing the sum of all color classes weights we present new approximation results that improve substantially the known ones for several variants of the problem with respect to the class of the underlying graph in particular we deal with variants which are known to be np hard general and bipartite graphs or are proven to be np hard in this paper complete graphs with bi valued edge weights or whose complexity question still remains open trees

new algorithms for maximum disjoint paths based on tree likeness we study the classical mathsf np hard problems of finding maximum size subsets from given sets of k terminal pairs that can be routed via edge disjoint paths maxedp or node disjoint paths maxndp in a given graph the approximability of maxedp maxndp is currently not well understood the best known lower bound is 2 varomega sqrt log n assuming mathsf np not subseteq mathsf dtime n mathcal o log n this constitutes a significant gap to the best known approximation upper bound of mathcal o sqrt n due to chekuri et al theory comput 2 137 146 2006 and closing this gap is currently one of the big open problems in approximation algorithms in their seminal paper raghavan and thompson combinatorica 7 4 365 374 1987 introduce the technique of randomized rounding for lps their technique gives an mathcal o 1 approximation when edges or nodes may be used by mathcal o left log n log log n right paths in this paper we strengthen the fundamental results above we provide new bounds formulated in terms of the feedback vertex set number r of a graph which measures its vertex deletion distance to a forest in particular we obtain the following results for maxedp we give an mathcal o sqrt r log k r approximation algorithm up to a logarithmic factor our result strengthens the best known ratio mathcal o sqrt n due to chekuri et al as r le n further we show how to route varomega text opt pairs with congestion bounded by mathcal o log kr log log kr strengthening the bound obtained by the classic approach of raghavan and thompson for maxndp we give an algorithm that gives the optimal answer in time k r mathcal o r cdot n this is a substantial improvement on the run time of 2 kr mathcal o r cdot n which can be obtained via an algorithm by scheffler we complement these positive results by proving that maxedp is mathsf np hard even for r 1 and maxndp is mathsf w 1 hard when r is the parameter this shows that neither problem is fixed parameter tractable in r unless mathsf fpt mathsf w 1 and that our approximability results are relevant even for very small constant values of r

classical and free infinitely divisible distributions and random matrices we construct a random matrix model for the bijection between classical and free infinitely divisible distributions for every d 1 we associate in a quite natural way to each infinitely divisible distribution a distribution p d on the space of d d hermitian matrices such that p d p d p d the spectral distribution of a random matrix with distribution p d converges in probability to when d tends to it gives among other things a new proof of the almost sure convergence of the spectral distribution of a matrix of the gue and a projection model for the marchenko pastur distribution in an analogous way for every d 1 we associate to each infinitely divisible distribution a distribution l d on the space of complex non hermitian d x d random matrices if is symmetric the symmetrization of the spectral distribution of m d when m d is l d distributed converges in probability to

maximum cardinality simple 2 matchings in subcubic graphs a simple 2 matching in a graph is a subset of edges m such that every node is incident with at most two edges in m a simple 2 matching is called ck restricted k 2 if it contains no cycles of length k or less we begin by considering the problem of finding maximum cardinality simple 2 matchings in subcubic graphs i e graphs whose nodes have degree at most 3 for this problem we present a min max theorem a simple polynomial time algorithm and some other results then by generalizing an approach due to vornberger we show the following for subcubic graphs the maximum cardinality c3 and c4 restricted simple 2 matching problems are polynomial time solvable there exist min max theorems for these problems and for k 5 the maximum cardinality ck restricted simple 2 matching problems are np hard

integration with respect to the haar measure on unitary orthogonal and symplectic group we revisit the work of the first named author and using simpler algebraic arguments we calculate integrals of polynomial functions with respect to the haar measure on the unitary group u d the previous result provided exact formulas only for 2d bigger than the degree of the integrated polynomial and we show that these formulas remain valid for all values of d also we consider the integrals of polynomial functions on the orthogonal group o d and the symplectic group sp d we obtain an exact character expansion and the asymptotic behavior for large d thus we can show the asymptotic freeness of haar distributed orthogonal and symplectic random matrices as well as the convergence of integrals of the itzykson zuber type

the complexity of controlled selection we identify a class of problems called controlled selection problems and study their complexity a controlled selection problem involves a random selection of a sample from a large set subject to certain structural constraints the structural constraints model some desired relationships among the elements of the set which we want to preserve under random selection for example the set may be divided into equivalence classes and we may want to ensure that not too many elements are chosen from the same class or the set may correspond to vertices of a graph and we may want to ensure that the sample does not induce a cycle controlled selection problems arise in many disciplines including statistics mathematical programming combinatorial algorithms and resource allocation this is the first attempt to unify them we develop techniques for determining the complexity of controlled selection problems given the structural constraints we use these techniques to determine the complexity of many types of controlled selection problems we show that a controlled selection problem is polynomial time reducible to the problem of finding a maximum weight sample satisfying the structural constraints we also show that a general reduction in the opposite direction would imply that one way functions do not exist in contrast we show that most natural controlled selection problems have the same complexity within polynomial time as the problem of finding a maximum weight sample satisfying the structural constraints we also present efficient algorithms for several controlled selection problems

free wreath product quantum groups the monoidal category approximation properties and free probability in this paper we find the fusion rules of the free wreath product quantum groups g sn for all compact matrix quantum groups of kac type g and n 4 this is based on a combinatorial description of the intertwiner spaces between certain generating representations of g sn the combinatorial properties of the intertwiner spaces in g sn allow us to obtain several probabilistic applications we prove also the monoidal equivalence between g sn and a compact quantum group whose dual is a discrete quantum subgroup of the free product g suq 2 for some 0 q 1 we obtain as a corollary certain stability results for the operator algebras associated with the free wreath products of quantum groups such as the acpap property and exactness

the tournament scheduling problem with absences abstract we study time scheduling problems with allowed absences as a new kind of graph coloring problem one may think of a sport tournament where each player each team is permitted a certain number t of absences we then examine how many rounds are needed to schedule the whole tournament in the worst case this upper limit depends on t and on the structure of the graph g whose edges represent the games that have to be played but also on whether or not the absences are announced before the tournament starts therefore we actually have two upper limits for the number of required rounds we have t g for pre scheduling if all absences are pre fixed and we have ol t g for on line scheduling if we have to stay flexible and deal with absences when they occur we conjecture that t g g 2 t and that ol t g g 2 t the first conjecture is stronger than the total coloring conjecture while the second is weaker than the on line list edge coloring conjecture our conjectures hold for all bipartite graphs for complete graphs we prove them partially lower and upper bounds to t g and ol t g for general multigraphs g are established too

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

bounding the maximum size of a packet radio network the author addresses the problem arising in the topology design of packet radio networks prns which use time division multiplexing and have a diameter constraint of what is the maximum number n sub c f k of users which can be contained in a diameter k prn with f time slots per frame the author assumed that users cannot transmit and receive simultaneously and cannot transmit receive more than one packet at a time this assumption implies that no two channels accessed by the same user may be assigned the same time slot it is shown that the problem of determining n sub c f k is identical to the problem of determining the largest number of vertices which can be contained in an f edge colorable directed graph with diameter k lower bounds on n sub c f k for f 2 k 3 4 5 are obtained by generating large graphs of this type the graphs are constructed and colored by simple and fast procedures which are similar for different values of f and k an extensive bibliography on the edge coloring problem is included

link scheduling in polynomial time two polynomial time algorithms are given for scheduling conversations in a spread spectrum radio network the constraint on conversations is that each station can converse with only one other station at a time the first algorithm is strongly polynomial and finds a schedule of minimum length that allows each pair of neighboring stations to converse directly for a prescribed length of time the second algorithm is designed for the situation in which messages must be relayed multiple hops the algorithm produces in polynomial time a routing vector and compatible link schedule that jointly meet a prespecified end to end demand so that the schedule has the smallest possible length

the complexity of path coloring and call scheduling abstract modern high performance communication networks pose a number of challenging problems concerning the efficient allocation of resources to connection requests in all optical networks with wavelength division multiplexing connection requests must be assigned paths and colors wavelengths such that intersecting paths receive different colors and the goal is to minimize the number of colors used this path coloring problem is proved np hard for undirected and bidirected ring networks path coloring in undirected tree networks is shown to be equivalent to edge coloring of multigraphs which implies a polynomial time optimal algorithm for trees of constant degree as well as np hardness and an approximation algorithm with absolute approximation ratio 4 3 and asymptotic approximation ratio 1 1 for trees of arbitrary degree for bidirected trees path coloring is shown to be np hard even in the binary case a polynomial time optimal algorithm is given for path coloring in undirected or bidirected trees with n nodes under the assumption that the number of paths touching every single node of the tree is o log n 1 e call scheduling is the problem of assigning paths and starting times to calls in a network with bandwidth reservation such that the maximum completion time is minimized in the case of unit bandwidth requirements unit edge capacities and unit call durations call scheduling is equivalent to path coloring if either the bandwidth requirements or the call durations can be arbitrary call scheduling is shown np hard for virtually every network topology

quantum bernstein s theorem and the hyperoctahedral quantum group we study an extension of bernstein theorem to the setting of quantum groups for a d tuple of free identically distributed random variables we consider a problem of preservation of freeness under the transformations from the free orthogonal quantum group for a subset not contained in the hyperoctahedral quantum group we prove that preservation of freeness characterizes wigner s semicircle law we show that freeness is always preserved if the quantum subset is contained in the hyperoctahedral quantum group we provide examples of quantum subsets which show that our result is an extension of results known in the literature we study an extension of bernstein theorem to the setting of quantum groups for a d tuple of free identically distributed random variables we consider a problem of preservation of freeness under the transformations from the free orthogonal quantum group for a subset not contained in the hyperoctahedral quantum group we prove that preservation of freeness characterizes wigner s semicircle law we show that freeness is always preserved if the quantum subset is contained in the hyperoctahedral quantum group we provide examples of quantum subsets which show that our result is an extension of results known in the literature

stochastic aspects of easy quantum groups we consider several orthogonal quantum groups satisfying the easiness assumption axiomatized in our previous paper for each of them we discuss the computation of the asymptotic law of tr uk with respect to the haar measure u being the fundamental representation for the classical groups on sn we recover in this way some well known results of diaconis and shahshahani

the np completeness of edge coloring we show that it is np complete to determine the chromatic index of an arbitrary graph the problem remains np complete even for cubic graphs

liberation of orthogonal lie groups abstract we show that under suitable assumptions we have a one to one correspondence between classical groups and free quantum groups in the compact orthogonal case we classify the groups under correspondence with the result that there are exactly 6 of them o n s n h n b n s n b n we investigate the representation theory aspects of the correspondence with the result that for o n s n h n b n this is compatible with the bercovici pata bijection finally we discuss some more general classification problems in the compact orthogonal case notably with the construction of a new quantum group

on excessive index of certain networks matching is one of the most extensively studied areas in computer science and is interesting from the combinatorial point of view as well a matching in a graph g v e is a subset m of edges no two of which have a vertex in common a matching m is said to be perfect if every vertex in g is an endpoint of one of the edges in m the excessive index of a graph g is the minimum number of perfect matchings to cover the edge set of g the study of excessive index has a number of applications particularly in scheduling theory in this paper we determine the excessive index for certain classes of graphs including augmented butterfly network and honeycomb network we also prove that the excessive index does not exist for butterfly and benes networks

minimizing makespan in no wait job shops in this paper we study polynomial time approximation schemes ptases for the no wait job shop scheduling problem with the makespan objective function it is known that the problem is maxsnp hard in the case when each job is allowed to have three operations or more we show that if each job has at most two operations the problem admits a ptas if the number of machines is a constant i e not part of the input if the number of machines is not a constant we show that the problem is hard to approximate within a factor better than 5 4

on the classification of easy quantum groups abstract in 2009 banica and speicher began to study the compact quantum groups g with s n g o n whose intertwiner spaces are induced by some partitions these so called easy quantum groups have a deep connection to combinatorics we continue their work on classifying these objects by introducing some new examples of easy quantum groups in particular we show that the six easy groups o n s n h n b n s n and b n split into seven cases o n s n h n b n s n b n and b n on the side of free easy quantum groups also we give a complete classification in the half liberated and in the nonhyperoctahedral case

algorithmic complexity of proper labeling problems a proper labeling of a graph is an assignment of integers to some elements of a graph which may be the vertices the edges or both of them such that we obtain a proper vertex coloring via the labeling subject to some conditions the problem of proper labeling offers many variants and received a great interest during recent years we consider the algorithmic complexity of some variants of the proper labeling problems we present some polynomial time algorithms and np completeness results for them

chromatic gallai identities operating on lovasz number if g is a triangle free graph then two gallai identities can be written as alpha g overline chi l g v g alpha l g overline chi g where alpha and overline chi denote the stability number and the clique partition number and l g is the line graph of g we show that surprisingly both equalities can be preserved for any graph g by deleting the edges of the line graph corresponding to simplicial pairs of adjacent arcs according to any acyclic orientation of g as a consequence one obtains an operator phi which associates to any graph parameter beta such that alpha g le beta g le overline chi g for all graph g a graph parameter phi beta such that alpha g le phi beta g le overline chi g for all graph g we prove that vartheta g le phi vartheta g and that phi overline chi f g le overline chi f g for all graph g where vartheta is lovasz theta function and overline chi f is the fractional clique partition number moreover overline chi f g le phi vartheta g for triangle free g comparing to the previous strengthenings psi vartheta and vartheta triangle of vartheta numerical experiments show that phi vartheta is a significant better lower bound for overline chi than vartheta

densities matchings and fractional edge colorings given a multigraph g v e with a positive rational weight w e on each edge e the weighted density problem wdp is to find a subset u of v with u ge 3 and odd that maximizes 2w

edge colouring of regular graphs of large degree we consider the following conjecture letgbe ak regular simple graph with an even numbernof vertices ifk n 2thengisk edge colourable we show that this conjecture is true for graphs that are join of two graphs and we provide a polynomial time algorithm for finding a k edge colouring of these graphs

computational complexity of covering three vertex multigraphs a covering projection from a graph g onto a graph h is a mapping of the vertices of g onto the vertices of h such that for every vertex v of g the neighborhood of v is mapped bijectively onto the neighborhood of its image moreover if g and h are multigraphs then this local bijection has to preserve multiplicities of the neighbors as well the notion of covering projection stems from topology but has found applications in areas such as the theory of local computation and construction of highly symmetric graphs it provides a restrictive variant of the constraint satisfaction problem with additional symmetry constraints on the behavior of the homomorphisms of the structures involved we investigate the computational complexity of the problem of deciding the existence of a covering projection from an input graph g to a fixed target graph h among other partial results this problem has been shown np hard for simple regular graphs h of valency greater than 2 and a full characterization of computational complexity has been shown for target multigraphs with 2 vertices we extend the previously known results to the ternary case i e we give a full characterization of the computational complexity in the case of multigraphs with 3 vertices we show that even in this case a p np completeness dichotomy holds

algorithmic problems in right angled artin groups complexity and applications abstract in this paper we consider several classical and novel algorithmic problems for right angled artin groups some of which are closely related to graph theoretic problems and study their computational complexity we study these problems with a view towards applications to cryptography

computing clique and chromatic number of circular perfect graphs in polynomial time a main result of combinatorial optimization is that clique and chromatic number of a perfect graph are computable in polynomial time grotschel et al in combinatorica 1 2 169 197 1981 perfect graphs have the key property that clique and chromatic number coincide for all induced subgraphs we address the question whether the algorithmic results for perfect graphs can be extended to graph classes where the chromatic number of all members is bounded by the clique number plus one we consider a well studied superclass of perfect graphs satisfying this property the circular perfect graphs and show that for such graphs both clique and chromatic number are computable in polynomial time as well in addition we discuss the polynomial time computability of further graph parameters for certain subclasses of circular perfect graphs all the results strongly rely upon lovasz s theta function

an approximation algorithm for diagnostic test scheduling in multicomputer systems the problem of diagnostic test scheduling dts is to assign to each edge e of a diagnostic graph g a time interval of length l e so that intervals corresponding to edges at any given vertex do not overlap and the overall finishing time is minimum in this correspondence we show that the dts problem is np complete then we present a longest first sequential scheduling algorithm which runs in worst case time o dm log n and uses o m space to produce a solution of length less than four times optimal then we show that the general performance bound can be strengthened to 3 opt g for low degree graphs and to 2 opt g in some special cases of binomial diagnostic graphs

satisfactory graph partition variants and generalizations the satisfactory partition problem asks for deciding if a given graph has a partition of its vertex set into two nonempty parts such that each vertex has at least as many neighbors in its part as in the other part this problem was introduced by gerber and kobler m gerber d kobler algorithmic approach to the satisfactory graph partitioning problem european journal of operational research 125 2000 283 291 and studied further by other authors in this paper we first review some applications and related problems then we survey structural complexity and approximation results obtained for satisfactory partition and for some of its variants and generalizations a list of open questions concludes this survey

the np completeness of chromatic index in triangle free graphs with maximum vertex of degree 3 we restrict the known case about the complexity of edge coloring for cubic graphs in the case of triangle free graphs with maximum vertex degree 3

a variational level set approach to multiphase motion a coupled level set method for the motion of multiple junctions of e g solid liquid and grain boundaries which follows the gradient flow for an energy functional consisting of surface tension proportional to length and bulk energies proportional to area is developed the approach combines the level set method of s osher and j a sethian with a theoretical variational formulation of the motion by f reitich and h m soner the resulting method uses as many level set functions as there are regions and the energy functional is evaluated entirely in terms of level set functions the gradient projection method leads to a coupled system of perturbed by curvature terms hamilton jacobi equations the coupling is enforced using a single lagrange multiplier associated with a constraint which essentially prevents a regions from overlapping and b the development of a vacuum the numerical implementation is relatively simple and the results agree with and go beyond the theory as given in 12 other applications of this methodology including the decomposition of a domain into subregions with minimal interface length are discussed finally some new techniques and results in level set methodology are presented

h coloring degree bounded acyclic digraphs an np complete coloring or homomorphism problem may become polynomial time solvable when restricted to graphs with degrees bounded by a small number but remain np complete if the bound is higher we investigate an analogous phenomenon for digraphs focusing on the three smallest digraphs h with np complete h colorability problems it turns out that in all three cases the h coloring problem is polynomial time solvable for digraphs with in degrees at most 1 regardless of the out degree bound respectively with out degrees at most 1 regardless of the in degree bound on the other hand as soon as both in and out degrees are bounded by constants greater than or equal to 2 all three problems are again np complete interestingly the situation changes when we further restrict the inputs to be acyclic digraphs in two of the three cases the np complete problems remain np complete in the third case the h coloring problem turns out to be polynomial time solvable for acyclic digraphs with in degrees at most 2 regardless of the out degree bound respectively with out degrees at most 2 regardless of the in degree bound we also show that in this case as soon as both in and out degrees are bounded by constants greater than or equal to 3 the h coloring problem is once again np complete even for acyclic digraphs a conjecture proposed for graphs h by feder hell and huang states that any variant of the h coloring problem which is np complete without degree constraints is also np complete with degree constraints provided the degree bounds are high enough it has been verified for h coloring of digraphs but the degree bounds are quite high and are stated in terms of the sum of in and out degrees our results confirm the conjecture with the best possible bounds that are needed for np completeness moreover they underscore the fact that the bound must apply separately to both the in degrees and the out degrees

on edge colouring indifference graphs copyright c 1997 elsevier science b v all rights reserved vizingrs theorem states that the chromatic index g of a graph g is either the maximum degree d g or d g 1 a graph g is called overfull if ve g v d g vv g v 2 a sufficient condition for g d g 1 is that g contains an overfull subgraph h with d h d g plantholt proved that this condition is necessary for graphs with a universal vertex in this paper we conjecture that for indifference graphs this is also true as supporting evidence we prove this conjecture for general graphs with three maximal cliques and with no universal vertex and for indifference graphs with odd maximum degree for the latter subclass we prove that d

de finetti theorems for easy quantum groups we study sequences of noncommutative random variables which are invariant under quantum transformations coming from an orthogonal quantum group satisfying the easiness condition axiomatized in our previous paper for 10 easy quantum groups we obtain de finetti type theorems characterizing the joint distribution of any infinite quantum invariant sequence in particular we give a new and unified proof of the classical results of de finetti and freedman for the easy groups sn on which is based on the combinatorial theory of cumulants we also recover the free de finetti theorem of kostler and speicher and the characterization of operator valued free semicircular families due to curran we consider also finite sequences and prove an approximation result in the spirit of diaconis and freedman

the dimension and basis of spaces of multivariate splines abstract the purpose of this paper is to study the recent development of certain aspects of multivariate spline spaces more specifically we will survey the existence of bivariate spline the dimension and basis of the bivariate spline spaces s k with various partitions furthermore we will also introduce some results on the higher dimensional splines

multiplicative functions on the lattice of non crossing partitions and free convolution the lattice of non crossing partitions was introduced by kreweras kre in 1972 and examined further by poupard pou and edelman edel ede2 in the last time there has been an increased interest in this lattice on one hand from the purely combinatorial point of view siu eds sire bss and on the other hand from a quantum probabilistic point of view gss spel spe2 spe3 we became interested in this lattice when we noticed that it is connected with some new form of convolution for probability measures the free convolution introduced by voiculescu voil voi2 voi3 this connection between the free convolution and the lattice of non crossing partitions is exactly analogous to the connection between the usual convolution and the lattice of all partitions we shall work out this connection in the last part of this work the intention of this article is to show that both the purely combinatorial and the probabilistic point of view can benefit from each other motivated by voiculescu s formula for the r series of free convolution we examine multiplicative functions on the lattice of non crossing partitions and prove our main theorem which is much in the spirit of voiculescu s formula we shall then use this main theorem for deriving some known results on non crossing partitions in a unified way this part of our work is purely combinatorial the free convolution serves only as a motivation for our theorem in the last section we shall switch to the probabilistie side and show how the lattice of non crossing partitions determines the structure of the free convolution voiculescu s formula will then follow as an easy corollary of our main theorem in this respect we shall get a purely combinatorial proof of this formula without any operator algebraic or functional analytic tools