parameterized complexity of happy coloring problems abstract in a vertex colored graph an edge is happy if its endpoints have the same color similarly a vertex is happy if all its incident edges are happy motivated by the computation of homophily in social networks we consider the algorithmic aspects of the following maximum happy edges k mhe problem given a partially k colored graph g and an integer l find an extended full k coloring of g making at least l edges happy when we want to make l vertices happy on the same input the problem is known as maximum happy vertices k mhv we perform an extensive study into the complexity of the problems particularly from a parameterized viewpoint for every k 3 we prove both problems can be solved in time 2 n n o 1 moreover by combining this result with a linear vertex kernel of size k l for k mhe we show that the edge variant can be solved in time 2 l n o 1 in contrast we prove that the vertex variant remains w 1 hard for the natural parameter l however the problem does admit a kernel with o k 2 l 2 vertices for the combined parameter k l from a structural perspective we show both problems are fixed parameter tractable for treewidth and neighborhood diversity which can both be seen as sparsity and density measures of a graph finally we extend the known completeness results of the problems by showing they remain hard on bipartite graphs and split graphs on the positive side we show the vertex variant can be solved optimally in polynomial time for cographs

global well posedness of incompressible inhomogeneous fluid systems with bounded density or non lipschitz velocity in this paper we first prove the global existence of weak solutions to the d dimensional incompressible inhomogeneous navier stokes equations with initial data a 0 in l infty mathbb r d u 0 u 0 h u 0 d in dot b 1 frac d p p r mathbb r d which satisfy mu a 0 l infty u 0 h dot b 1 frac d p p r rm exp c r mu 2r u 0 d dot b 1 frac d p p r 2r leqq c 0 mu for some positive constants c 0 c r and 1 p d 1 r the regularity of the initial velocity is critical to the scaling of this system and is general enough to generate non lipschitz velocity fields furthermore with additional regularity assumptions on the initial velocity or on the initial density we can also prove the uniqueness of such a solution we should mention that the classical maximal l p l q regularity theorem for the heat kernel plays an essential role in this context

lower bounds for the happy coloring problems in this paper we study the maximum happy vertices and the maximum happy edges problems mhv and mhe for short very recently the problems attracted a lot of attention and were studied in agrawal 17 aravind et al 16 choudhari and reddy 18 misra and reddy 17 main focus of our work is lower bounds on the computational complexity of these problems established lower bounds can be divided into the following groups np hardness of the above guarantee parameterization kernelization lower bounds answering questions of misra and reddy 17 exponential lower bounds under the set cover conjecture and the exponential time hypothesis and inapproximability results moreover we present an mathcal o ell k randomized algorithm for mhv and an mathcal o 2 k algorithm for mhe where ell is the number of colors used and k is the number of required happy vertices or edges these algorithms cannot be improved to subexponential taking proved lower bounds into account

local existence and uniqueness of strong solutions to the navier stokes equations with nonnegative density abstract in this paper we consider the initial boundary value problem to the nonhomogeneous incompressible navier stokes equations local strong solutions are established for any initial data 0 u 0 w 1 l h 0 1 with 1 and if 2 then the strong solution is unique the initial density is allowed to be nonnegative and in particular the initial vacuum is allowed the assumption on the initial data is weaker than the previous widely used one that 0 u 0 h 1 l h 0 1 h 2 and no compatibility condition is required

lemma for linear feedback shift registers and dfts applied to affine variety codes in this paper we establish a lemma in algebraic coding theory that frequently appears in the encoding and decoding of e g reed solomon codes algebraic geometry codes and affine variety codes our lemma corresponds to the nonsystematic encoding of affine variety codes and can be stated by giving a canonical linear map as the composition of an extension through linear feedback shift registers from a grobner basis and a generalized inverse discrete fourier transform we clarify that our lemma yields the error value estimation in the fast erasure and error decoding of a class of dual affine variety codes moreover we show that systematic encoding corresponds to a special case of erasure only decoding the lemma enables us to reduce the computational complexity of error evaluation from o n3 using gaussian elimination to o qn2 with some mild conditions on n and q where n is the code length and q is the finite field size

a lagrangian approach for the incompressible navier stokes equations with variable density we investigate the cauchy problem for the inhomogeneous navier stokes equations in the whole n dimensional space under some smallness assumption on the data we show the existence of global in time unique solutions in a critical functional framework the initial density is required to belong to the multiplier space of input amssym dot b n p 1 p 1 bbb r n in particular piecewise constant initial densities are admissible data provided the jump at the interface is small enough and generate global unique solutions with piecewise constant densities using lagrangian coordinates is the key to our results as it enables us to solve the system by means of the basic contraction mapping theorem as a consequence conditions for uniqueness are the same as for existence 2012 wiley periodicals inc

approximation and hardness results for the max k uncut problem abstract in the study of the homophily law of large scale complex networks we get a combinatorial optimization problem which we call the max k uncut problem given an n vertex undirected graph g v e with nonnegative weights w e e e defined on edges and a positive integer k the max k uncut problem asks to find a partition v 1 v 2 v k of v such that the total weight of edges that are not cut is maximized intuitively an edge that is not cut connects two vertices with the same or similar attributes since they are in the same part of the partition interestingly the max k uncut problem is just the complement of the classic min k cut problem for max k uncut we present a randomized 1 k n 2 approximation algorithm a greedy 1 2 k 1 n approximation algorithm and an 1 2 approximation algorithm by reducing it to densest k subgraph where is the approximation ratio of the densest k subgraph problem more importantly we show that max k uncut and densest k subgraph are in fact equivalent in approximability up to a factor of 2 we also prove an approximation hardness result for max k uncut under the assumption p np

geometric approach to higher weights the notion of higher or generalized weights of codes is just as natural as that of the classical hamming weight the authors adopt the geometric point of view and always treat the q ary case some results and proofs being new the main goal is to present a clear picture of what is known on the subject

rounding algorithms for a geometric embedding of minimum multiway cut given an undirected graph with edge costs and a subset ofk 3 nodes calledterminals a multiway ork way cut is a subset of the edges whose removal disconnects each terminal from the others the multiway cut problem is to find a minimum cost multiway cut this problem is max snp hard recently calinescu et al calinescu g h karloff y rabani 2000 an improved approximation algorithm for multiway cut j comput system sci 60 3 564 574 gave a novel geometric relaxation of the problem and a rounding scheme that produced a 3 2 1 k approximation algorithm in this paper we study their geometric relaxation in particular we study the worst case ratio between the value of the relaxation and the value of the minimum multicut the so called integrality gap of the relaxation fork 3 we show the integrality gap is 12 11 giving tight upper and lower bounds that is we exhibit a family of graphs with integrality gaps arbitrarily close to 12 11 and give an algorithm that finds a cut of value 12 11 times the relaxation value our lower bound shows that this is the best possible performance guarantee for any algorithm based purely on the value of the relaxation our upper bound meets the lower bound and improves the factor of 7 6 shown by calinescu et al for allk we show that there exists a rounding scheme with performance ratio equal to the integrality gap and we give explicit constructions of polynomial time rounding schemes that lead to improved upper bounds fork 4 and 5 our best upper bounds are based on computer constructed rounding schemes with computer proofs of correctness for generalk we give an algorithm with performance ratio 1 3438 e k our results were discovered with the help of computational experiments that we also describe here

on a lp estimate for the linearized compressible navier stokes equations with the dirichlet boundary conditions abstract in this paper a special l p estimate for the linearized compressible navier stokes in the lagrangian coordinates for the dirichlet boundary conditions is obtained the constant in the estimate does not depend on the length of time interval 0 t the result is essential to obtain an existence for regular solutions for the nonlinear problem with the lowest class of regularity in l p spaces

unique solvability for the density dependent navier stokes equations abstract in this paper we consider the incompressible navier stokes equations with a density dependent viscosity in a bounded domain of r n n 2 3 we prove the local existence of unique strong solutions for all initial data satisfying a natural compatibility condition this condition is also necessary for a very general initial data moreover we provide a blow up criterion for the regularity of the strong solution for these results the initial density need not be strictly positive it may vanish in an open subset of

greedy splitting algorithms for approximating multiway partition problems given a system v t f k where v is a finite set open image in new window is a submodular function and k 2 is an integer the general multiway partition problem mpp asks to find a k partition open image in new window v1 v2 vk of v that satisfiesopen image in new windowfor all i and minimizes f v1 f v2 f vk where open image in new window is a k partition of open image in new window hold mpp formulation captures a generalization in submodular systems of many np hard problems such as k way cut multiterminal cut target split and their generalizations in hypergraphs this paper presents a simple and unified framework for developing and analyzing approximation algorithms for various mpps

generalized bezout s theorem and its applications in coding theory this paper presents a generalized bezout theorem which can be used to determine a tighter lower bound of the number of distinct points of intersection of two or more plane curves a new approach to determine a lower bound on the minimum distance for algebraic geometric codes defined from a class of plane curves is introduced based on the generalized bezout theorem examples of more efficient linear codes are constructed using the generalized bezout theorem and the new approach for d 4 the linear codes constructed by the new construction are better than or equal to the known linear codes for d spl ges 5 these new codes are better than the known ag codes defined from whole spaces the klein codes 22 16 5 and 22 15 6 over gf 2 sup 3 and the improved hermitian code 64 56 6 over gf 2 sup 4 are also constructed

finding k cuts within twice the optimal two simple approximation algorithms for the minimum k cut problem are presented each algorithm finds a k cut having weight within a factor of 2 2 k of the optimal one of our algorithms is particularly efficient it requires a total of only n 1 maximum flow computations for finding a set of near optimal k cuts one for each value of k between 2 and n

algebraic geometric codes on minimal hirzebruch surfaces abstract we define a linear code c t x by evaluating polynomials of bidegree t x in the cox ring on f q rational points of a minimal hirzebruch surface over the finite field f q we give explicit parameters of the code notably using grobner bases the minimum distance provides an upper bound of the number of f q rational points of a non filling curve on a hirzebruch surface

footprints or generalized bezout s theorem in two previous papers the first by feng rao berg and zhu see ibid vol 43 p 1799 810 1997 and the second by feng zhu shi and rao see proc 35th afferton conf communication control and computing p 205 14 1997 the authors use a generalization of bezout s theorem to estimate the minimum distance and generalized hamming weights for a class of error correcting codes obtained by evaluation of polynomials in points of an algebraic curve the main aim of this article is to show that instead of using this rather complex method the same results and some improvements can be obtained by using the so called footprint from grobner basis theory we also develop the theory further such that the minimum distance and the generalized hamming weights not only can be estimated but also can actually be determined

generalized hamming weights for linear codes motivated by cryptographical applications the algebraic structure of linear codes from a new perspective is studied by viewing the minimum hamming weight as a certain minimum property of one dimensional subcodes a generalized notion of higher dimensional hamming weights is obtained these weights characterize the code performance on the wire tap channel of type ii basic properties of generalized weights are derived the values of these weights for well known classes of codes are determined and lower bounds on code parameters are obtained several open problems are also listed

a combinatorial strongly polynomial algorithm for minimizing submodular functions this paper presents a combinatorial polynomial time algorithm for minimizing submodular functions answering an open question posed in 1981 by grotschel lovasz and schrijver the algorithm employs a scaling scheme that uses a flow in the complete directed graph on the underlying set with each arc capacity equal to the scaled parameter the resulting algorithm runs in time bounded by a polynomial in the size of the underlying set and the length of the largest absolute function value the paper also presents a strongly polynomial version in which the number of steps is bounded by a polynomial in the size of the underlying set independent of the function values

algorithmic aspects of homophyly of networks abstract we investigate the algorithmic problems of the homophyly phenomenon in networks given an undirected graph g v e and a vertex coloring c v 1 2 k of g we say that a vertex v v is happy if v shares the same color with all its neighbors and unhappy otherwise and that an edge e e is happy if its two endpoints have the same color and unhappy otherwise supposing c is a partial vertex coloring of g we define the maximum happy vertices problem mhv for short as to color all the remaining vertices such that the number of happy vertices is maximized and the maximum happy edges problem mhe for short as to color all the remaining vertices such that the number of happy edges is maximized let k be the number of colors allowed in the problems we show that both mhv and mhe can be solved in polynomial time if k 2 and that both mhv and mhe are np hard if k 3 we devise a max 1 k 3 approximation algorithm for the mhv problem where is the maximum degree of vertices in the input graph and a 1 2 approximation algorithm for the mhe problem this is the first theoretical progress of these two natural and fundamental new problems

the complexity of multiterminal cuts in the multiterminal cut problem one is given an edge weighted graph and a subset of the vertices called terminals and is asked for a minimum weight set of edges that separates each terminal from all the others when the number k of terminals is two this is simply the mincut max flow problem and can be solved in polynomial time it is shown that the problem becomes np hard as soon as k 3 but can be solved in polynomial time for planar graphs for any fixed k the planar problem is np hard however if k is not fixed a simple approximation algorithm for arbitrary graphs that is guaranteed to come within a factor of 2 2 k of the optimal cut weight is also described

inhomogeneous navier stokes equations in the half space with only bounded density abstract in this paper we establish the global existence and uniqueness of solutions to the inhomogeneous navier stokes system in the half space the initial density only has to be bounded and close enough to a positive constant the initial velocity belongs to some critical besov space and the l norm of the inhomogeneity plus the critical norm to the horizontal components of the initial velocity has to be very small compared to the exponential of the norm to the vertical component of the initial velocity with a little bit more regularity for the initial velocity those solutions are proved to be unique in the last section of the paper our results are partially extended to the bounded domain case

approximation algorithms for classification problems with pairwise relationships metric labeling and markov random fields in a traditional classification problem we wish to assign one of k labels or classes to each of n objects in a way that is consistent with some observed data that we have about the problem an active line of research in this area is concerned with classification when one has information about pairwise relationships among the objects to be classified this issue is one of the principal motivations for the framework of markov random fields and it arises in areas such as image processing biometry and document analysis in its most basic form this style of analysis seeks to find a classification that optimizes a combinatorial function consisting of assignment costs based on the individual choice of label we make for each object and separation costs based on the pair of choices we make for two related objects we formulate a general classification problem of this type the metric labeling problem we show that it contains as special cases a number of standard classification frameworks including several arising from the theory of markov random fields from the perspective of combinatorial optimization our problem can be viewed as a substantial generalization of the multiway cut problem and equivalent to a type of uncapacitated quadratic assignment problem we provide the first nontrivial polynomial time approximation algorithms for a general family of classification problems of this type our main result is an o log k log log k approximation algorithm for the metric labeling problem with respect to an arbitrary metric on a set of k labels and an arbitrary weighted graph of relationships on a set of objects for the special case in which the labels are endowed with the uniform metric all distances are the same our methods provide a 2 approximation algorithm

incompressible flows with piecewise constant density we investigate the incompressible navier stokes equations with variable density the aim is to prove existence and uniqueness results in the case of discontinuous initial density in dimension n 2 3 assuming only that the initial density is bounded and bounded away from zero and that the initial velocity is smooth enough we get the local in time existence of unique solutions uniqueness holds in any dimension and for a wider class of velocity fields in particular all those results are true for piecewise constant densities with arbitrarily large jumps global results are established in dimension two if the density is close enough to a positive constant and in n dimensions if in addition the initial velocity is small the lagrangian formulation for describing the flow plays a key role in the analysis that is proposed in the present paper

the incompressible navier stokes equations in vacuum we are concerned with the existence and uniqueness issue for the inhomogeneous incompressible navier stokes equations supplemented with h 1 initial velocity and only bounded nonnegative density in contrast with all the previous works on that topics we do not require regularity or positive lower bound for the initial density or compatibility conditions for the initial velocity and still obtain unique solutions those solutions are global in the two dimensional case for general data and in the three dimensional case if the velocity satisfies a suitable scaling invariant smallness condition as a straightforward application we provide a complete answer to lions question in 25 page 34 concerning the evolution of a drop of incompressible viscous fluid in the vacuum