fptass for trimming weighted trees given a tree with nonnegative edge cost and nonnegative vertex weight and a number k 0 we consider the following four cut problems cutting vertices of weight at most or at least k from the tree by deleting some edges such that the remaining part of the graph is still a tree and the total cost of the edges being deleted is minimized or maximized the minmstcut problem cut vertices of weight at mostk and minimize the total cost of the edges being deleted can be solved in linear time and space and the other three problems are np hard in this paper we design an o nl e time o l 2 e n space algorithm for maxmstcut and o nl 1 e logn time o l 2 e n space algorithms for the other two problems minlstcut and maxlstcut where n is the number of vertices in the tree l the number of leaves and e 0 the prescribed error bound

improved parameterized and exact algorithms for cut problems on trees we study the multicut on trees and the generalized multiway cut on trees problems for the multicut on trees problem we present a parameterized algorithm that runs in time o k where 2 1 1 554 is the positive root of the polynomial x 4 2 x 2 1 this improves the current best algorithm of chen et al that runs in time o 1 619 k for the generalized multiway cut on trees problem we show that this problem is solvable in polynomial time if the number of terminal sets is fixed this answers an open question posed in a recent paper by liu and zhang by reducing the generalized multiway cut on trees problem to the multicut on trees problem our results give a parameterized algorithm that solves the generalized multiway cut on trees problem in time o k

the approximability of constraint satisfaction problems we study optimization problems that may be expressed as boolean constraint satisfaction problems an instance of a boolean constraint satisfaction problem is given by m constraints applied to n boolean variables different computational problems arise from constraint satisfaction problems depending on the nature of the underlying constraints as well as on the goal of the optimization task here we consider four possible goals max csp min csp is the class of problems where the goal is to find an assignment maximizing the number of satisfied constraints minimizing the number of unsatisfied constraints max ones min ones is the class of optimization problems where the goal is to find an assignment satisfying all constraints with maximum minimum number of variables set to 1 each class consists of infinitely many problems and a problem within a class is specified by a finite collection of finite boolean functions that describe the possible constraints that may be used tight bounds on the approximability of every problem in max csp were obtained by creignou j comput system sci 51 1995 pp 511 522 in this work we determine tight bounds on the approximability i e the ratio to within which each problem may be approximated in polynomial time of every problem in max ones min csp and min ones combined with the result of creignou this completely classifies all optimization problems derived from boolean constraint satisfaction our results capture a diverse collection of optimization problems such as max 3 sat max cut max clique min cut nearest codeword etc our results unify recent results on the in approximability of these optimization problems and yield a compact presentation of most known results moreover these results provide a formal basis to many statements on the behavior of natural optimization problems that have so far been observed only empirically

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

the multi multiway cut problem in this paper we define and study a natural generalization of the multicut and multiway cut problems the minimum multi multiway cut problem the input to the problem is a weighted undirected graph g v e and k sets s 1 s 2 s k of vertices the goal is to find a subset of edges of minimum total weight whose removal completely disconnects each one of the sets s 1 s 2 s k i e disconnects every pair of vertices u and v such that u email protected s i for some i this problem generalizes both the multicut problem when s i 2 for email protected email protected k and the multiway cut problem when k 1 we present an approximation algorithm for the multi multiway cut problem with an approximation ratio which matches that obtained by garg vazirani and yannakakis on the standard multicut problem namely our algorithm has an o logk approximation ratio moreover we consider instances of the minimum multi multiway cut problem which are known to have an optimal solution of light weight we show that our algorithm has an approximation ratio substantially better than o logk when restricted to such light instances specifically we obtain an o loglp approximation algorithm for the problem when all edge weights are at least 1 here lp denotes the value of a natural linear programming relaxation of the problem the latter improves the o loglplogloglp approximation ratio for the minimum multicut problem implied by the work of seymour and even et al

multicuts and integral multiflows in rings we show how to solve in polynomial time the multicut and the maximum integral multiflow problems in rings moreover we give linear time procedures to solve both problems in rings with uniform capacities

a polynomial algorithm for the k cut problem for fixed k the k cut problem is to find a partition of an edge weighted graph into k nonempty components such that the total edge weight between components is minimum this problem is np complete for an arbitrary k and its version involving fixing a vertex in each component is np hard even for k 3 we present a polynomial algorithm for k fixed that runs in onki 2 3k 2 4tn m steps where tn m is the running time required to find the minimum s t cut on a graph with n vertices and m edges

minimal multicut and maximal integer multiflow a survey we present a survey about the maximum integral multiflow and minimum multicut problems and their subproblems such as the multiterminal cut and the unsplittable flow problems we consider neither continuous multiflow nor minimum cost multiflow most of the results are very recent and some are new we recall the dual relationship between both problems give complexity results and algorithms firstly in unrestricted graphs and secondly in several special graphs trees bipartite or planar graphs a table summarizes the most important results

stabilizing network bargaining games by blocking players cooperative matching games shapley and shubik and network bargaining games kleinberg and tardos are games described by an undirected graph where the vertices represent players an important role in such games is played by stable graphs that are graphs whose set of inessential vertices those that are exposed by at least one maximum matching are pairwise non adjacent in fact stable graphs characterize instances of such games that admit the existence of stable outcomes in this paper we focus on stabilizing instances of the above games by blocking as few players as possible formally given a graph g we want to find a minimum cardinality set of vertices such that its removal from g yields a stable graph we give a combinatorial polynomial time algorithm for this problem and develop approximation algorithms for some np hard weighted variants where each vertex has an associated non negative weight our approximation algorithms are lp based and we show that our analysis are almost tight by giving suitable lower bounds on the integrality gap of the used lp relaxations

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

structure theorem and isomorphism test for graphs with excluded topological subgraphs we generalize the structure theorem of robertson and seymour for graphs excluding a fixed graph h as a minor to graphs excluding h as a topological subgraph we prove that for a fixed h every graph excluding h as a topological subgraph has a tree decomposition where each part is either almost embeddable to a fixed surface or has bounded degree with the exception of a bounded number of vertices furthermore we prove that such a decomposition is computable by an algorithm that is fixed parameter tractable with parameter h we present two algorithmic applications of our structure theorem to illustrate the mechanics of a typical application of the structure theorem we show that on graphs excluding h as a topological subgraph partial dominating set find k vertices whose closed neighborhood has maximum size can be solved in time f h k cdot n o 1 more significantly we show that on graphs excluding h as a topological subgraph graph isomorphism can be solved in time n f h this result unifies and generalizes two previously known important polynomial time solvable cases of graph isomorphism bounded degree graphs e m luks j comput system sci 25 1982 pp 42 65 and h minor free graphs i n ponomarenko zap nauchn sem leningrad otdel mat inst steklov lomi 174 1988 pp 147 177 182 the proof of this result needs a generalization of our structure theorem to the context of invariant treelike decomposition

partitioning a graph into small pieces with applications to path transversal given a graph g v e and an integer k in mathbb n we study k vertex separator resp k edge separator where the goal is to remove the minimum number of vertices resp edges such that each connected component in the resulting graph has less than k vertices we also study k path transversal where the goal is to remove the minimum number of vertices such that there is no simple path of length k our main results are the following improved approximation algorithms

using branch and price and cut to solve origin destination integer multicommodity flow problems we present a column generation model and branch and price and cut algorithm for origin destination integer multicommodity flow problems the origin destination integer multicommodity flow problem is a constrained version of the linear multicommodity flow problem in which flow of a commodity defined in this case by an origin destination pair may use only one path from origin to destination branch and price and cut is a variant of branch and bound with bounds provided by solving linear programs using column and cut generation at nodes of the branch and bound tree because our model contains one variable for each origin destination path for every commodity the linear programming relaxations at nodes of the branch and bound tree are solved using column generation i e implicit pricing of nonbasic variables to generate new columns or to prove lp optimality we devise a new branching rule that allows columns to be generated efficiently at each node of the branch and bound tree then we describe cuts cover inequalities that can be generated at each node of the branch and bound tree these cuts help to strengthen the linear programming relaxation and to mitigate the effects of problem symmetry we detail the implementation of our combined column and cut generation method and present computational results for a set of test problems arising from telecommunications applications we illustrate the value of our branching rule when used to find a heuristic solution and compare branch and price and branch and price and cut methods to find optimal solutions for highly capacitated problems

polynomial flow cut gaps and hardness of directed cut problems we study the multicut and the sparsest cut problems in directed graphs in the multicut problem we are a given an n vertex graph g along with k source sink pairs and the goal is to find the minimum cardinality subset of edges whose removal separates all source sink pairs the sparsest cut problem has the same input but the goal is to find a subset of edges to delete so as to minimize the ratio of the number of deleted edges to the number of source sink pairs that are separated by this deletion the natural linear programming relaxation for multicut corresponds by lp duality to the well studied maximum fractional multicommodity flow problem while the standard lp relaxation for sparsest cut corresponds to maximum concurrent flow therefore the integrality gap of the linear programming relaxation for multicut sparsest cut is also the flow cut gap the largest gap achievable for any graph between the maximum flow value and the minimum cost solution for the corresponding cut problem our first result is that the flow cut gap between maximum multicommodity flow and minimum multicut is n1 7 in directed graphs we show a similar result for the gap between maximum concurrent flow and sparsest cut in directed graphs these results improve upon a long standing lower bound of log n for both types of flow cut gaps we notice that these polynomially large flow cut gaps are in a sharp contrast to the undirected setting where both these flow cut gaps are known to be log n our second result is that both directed multicut and sparsest cut are hard to approximate to within a factor of 2 log1 n for any constant 0 unless np zpp this improves upon the recent log n log log n hardness result for these problems we also show that existence of pcp s for np with perfect completeness polynomially small soundness and constant number of queries would imply a polynomial factor hardness of approximation for both these problems all our results hold for directed acyclic graphs

on the complexity of computing the k restricted edge connectivity of a graph the k restricted edge connectivity of a graph g denoted by k g is defined as the minimum size of an edge set whose removal leaves exactly two connected components each containing at least k vertices this graph invariant which can be seen as a generalization of a minimum edge cut has been extensively studied from a combinatorial point of view however very little is known about the complexity of computing k g very recently in the parameterized complexity community the notion of good edge separation of a graph has been defined which happens to be essentially the same as the k restricted edge connectivity motivated by the relevance of this invariant from both combinatorial and algorithmic points of view in this article we initiate a systematic study of its computational complexity with special emphasis on its parameterized complexity for several choices of the parameters we provide a number of np hardness and w 1 hardness results as well as fpt algorithms

thresholded covering algorithms for robust and max min optimization in a two stage robust covering problem one of several possible scenarios will appear tomorrow and require to be covered but costs are higher tomorrow than today what should you anticipatorily buy today so that the worst case cost summed over both days is minimized we consider the k k robust model where the possible scenarios tomorrow are given by all demand subsets of size k k in this paper we give the following simple and intuitive template for k k robust covering problems having built some anticipatory solution if there exists a single demand whose augmentation cost is larger than some threshold augment the anticipatory solution to cover this demand as well and repeat we show that this template gives good approximation algorithms for k k robust versions of many standard covering problems set cover steiner tree steiner forest minimum cut and multicut our k k robust approximation ratios nearly match the best bounds known for their deterministic counterparts the main technical contribution lies in proving certain net type properties for these covering problems which are based on dual rounding and primal dual ideas these properties might be of some independent interest as a by product of our techniques we also get algorithms for max min problems of the form given a covering problem instance which k k of the elements are costliest to cover for the problems mentioned above we show that their k k max min versions have performance guarantees similar to those for the k k robust problems

planar formulae and their uses we define the set of planar boolean formulae and then show that the set of true quantified planar formulae is polynomial space complete and that the set of satisfiable planar formulae is np complete using these results we are able to provide simple and nearly uniform proofs of np completeness for planar node cover planar hamiltonian circuit and line geometric connected dominating set and of polynomial space completeness for planar generalized geography the np completeness of planar node cover and planar hamiltonian circuit and line were first proved elsewhere m r garey and d s johnson the rectilinear steiner tree is np complete siam j appl math 32 1977 pp 826 834 and m r garey d s johnson and r e tarjan the planar hamilton circuit problem is np complete siam j comp 5 1976 pp 704 714

an algorithm for the numerical evaluation of the associated legendre functions that runs in time independent of degree and order abstract we describe a method for the numerical evaluation of normalized versions of the associated legendre functions p and q of degrees 0 1 000 000 and orders for arguments in the interval 1 1 our algorithm which runs in time independent of and is based on the fact that while the associated legendre functions themselves are extremely expensive to represent via polynomial expansions the logarithms of certain solutions of the differential equation defining them are not we exploit this by numerically precomputing the logarithms of carefully chosen solutions of the associated legendre differential equation and representing them via piecewise trivariate chebyshev expansions these precomputed expansions which allow for the rapid evaluation of the associated legendre functions over a large swath of parameter domain mentioned above are supplemented with asymptotic and series expansions in order to cover it entirely the results of numerical experiments demonstrating the efficacy of our approach are presented and our code for evaluating the associated legendre functions is publicly available

the complexity of multicut and mixed multicut problems in di graphs the multicut problem with respect to the fixed pattern di graph q is the following given a di graph g an integer k and a 1 1 mapping p of the vertices of q to a subset x of v g determine whether there exists a set s of at most k edges arcs of g such that g s has no p q p q path for every edge arc qq of q we first prove a dichotomy result for the complexity they are all either polynomial or np complete for multicut problems in di graphs in terms of the pattern q then we consider a variant where the pattern is a mixed graph m the input is a digraph d an integer k and a 1 1 mapping p from v m to some subset of v d and the goal is to decide whether there exists a set s of at most k arcs of d that meets every p q p q path in d for every arc qq of m and every p p p p path in the underlying undirected graph of d for every edge pp of m again we prove a dichotomy result for the complexity of such mixed multicut problems in terms of the pattern m it turns out that the only polynomial cases are those where the pattern is either a polynomial graph pattern or a polynomial digraph pattern as soon as m contains both edges and arcs the m mixed cut problem becomes np complete finally we prove that the directed feedback arc set problem is np complete even in the class of digraphs which have a feedback vertex set of size 2

complexity and exact algorithms for vertex multicut in interval and bounded treewidth graphs multicut is a fundamental network communication and connectivity problem it is defined as given an undirected graph and a collection of pairs of terminal vertices find a minimum set of edges or vertices whose removal disconnects each pair we mainly focus on the case of removing vertices where we distinguish between allowing or disallowing the removal of terminal vertices complementing and refining previous results from the literature we provide several np completeness and fixedparameter tractability results for restricted classes of graphs such as trees interval graphs and graphs of bounded treewidth

clustering with local restrictions we study a family of graph clustering problems where each cluster has to satisfy a certain local requirement formally let m be a function on the subsets of vertices of a graph g in the m p q partition problem the task is to find a partition of the vertices into clusters where each cluster c satisfies the requirements that 1 at most q edges leave c and 2 m c

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

complexity of the multicut problem in its vanilla partial and generalized versions in graphs of bounded treewidth abstract in the multicut problem the input consists of a graph and a set of pairs of terminal vertices that have to be disconnected by the removal of a set of edges with minimum total weight the partial multicut problem generalizes multicut by only requiring a given number of pairs of terminals to be disconnected by the removal of edges it has been shown that multicut remains apx hard if the treewidth t w g of its input graph g is bounded but that it is fpt w r t t w g h where h is the demand graph whose vertices are the terminals and whose edges are the pairs of terminals we prove that this also holds for partial multicut furthermore it has been proved that multicut also becomes polynomial time solvable if t w g is bounded and h is complete this is the multiterminal cut problem we extend this result in two directions by proving that multicut remains polynomial time solvable if t w g h is bounded and that this remains true for partial multicut finally we show that if we further generalize the problem to allow non unitary profits for pairs of terminals then the problem is weakly np hard and has an fptas if t w g h is bounded and becomes apx hard if t w g h is bounded

correlation clustering in general weighted graphs we consider the following general correlation clustering problem n bansal a blum s chawla correlation clustering in proc 43rd annu ieee symp on foundations of computer science vancouver canada november 2002 pp 238 250 given a graph with real nonnegative edge weights and a edge labelling partition the vertices into clusters to minimize the total weight of cut edges and uncut edges thus edges with large weights representing strong correlations between endpoints encourage those endpoints to belong to a common cluster while edges with large weights encourage the endpoints to belong to different clusters in contrast to most clustering problems correlation clustering specifies neither the desired number of clusters nor a distance threshold for clustering both of these parameters are effectively chosen to be best possible by the problem definition correlation clustering was introduced by bansal et al correlation clustering in proc 43rd annu ieee syrup on foundations of computer science vancouver canada november 2002 pp 238 250 motivated by both document clustering and agnostic learning they proved np hardness and gave constant factor approximation algorithms for the special case in which the graph is complete full information and every edge has the same weight we give an o log n approximation algorithm for the general case based on a linear programming rounding and the region growing technique we also prove that this linear program has a gap of log n and therefore our approximation is tight under this approach we also give an o r3 approximation algorithm for kr r minor free graphs on the other hand we show that the problem is equivalent to minimum multicut and therefore apx hard and difficult to approximate better than log n

parameterized complexity of critical node cuts we consider the following graph cut problem called critical node cut cnc given a graph g on n vertices and two positive integers k and x determine whether g has a set of k vertices whose removal leaves g with at most x connected pairs of vertices we analyze this problem in the framework of parameterized complexity that is we are interested in whether or not this problem is solvable in f n o 1 time i e whether or not it is fixed parameter tractable for various natural parameters we consider four such parameters the size k of the required cut the upper bound x on the number of remaining connected pairs the lower bound y on the number of connected pairs to be removed the treewidth w of g we determine whether or not cnc is fixed parameter tractable for each of these parameters we determine this also for all possible aggregations of these four parameters apart from w k moreover we also determine whether or not cnc admits a polynomial kernel for all these parameterizations that is whether or not there is an algorithm that reduces each instance of cnc in polynomial time to an equivalent instance of size o 1 where is the given parameter

on weighted multiway cuts in trees a min max theorem is developed for the multiway cut problem of edge weighted trees we present a polynomial time algorithm to construct an optimal dual solution if edge weights come in unary representation applications to biology also require some more complex edge weights we describe a dynamic programming type algorithm for this more general problem from biology and show that our min max theorem does not apply to it

packing and covering with linear programming a survey this paper considers the polyhedral results and the min max results on packing and covering problems of the decade since the strong perfect graph theorem published in 2006 the main such results are available for the packing problem however there are still important polyhedral questions that remain open for the covering problem the main questions are still open although there has been important progress we survey some of the main results with emphasis on those where linear programming and graph theory come together they mainly concern the covering of cycles or dicycles in graphs or signed graphs either with vertices or edges this includes the multicut and integral multiflow problems

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

the checkpoint problem in this paper we consider the checkpoint problem the input consists of an undirected graph g a set of source destination pairs s 1 t 1 s 2 t 2 s k t k and a collection p of paths connecting the s i t i pairs a feasible solution is a multicut e namely a set of edges whose removal disconnects every source destination pair for each email protected p we define cp e p email protected e in the sum checkpoint scp problem the goal is to minimize p pcp e p while in the maximum checkpoint mcp problem the goal is to minimize max p pcp e p these problems have several natural applications e g in urban transportation and network security in a sense they combine the multicut problem and the minimum membership set cover problem for the sum objective we show that weighted scp is equivalent with respect to approximability to undirected multicut thus there exists an o logn approximation for scp in general graphs our current approximability results for the max objective have a wide gap we provide an approximation factor of o nlogn opt for mcp and a hardness of 2 under the assumption p np the hardness holds for trees this solves an open problem of nelson 2009 25 we complement the lower bound by an almost matching upper bound with an asymptotic approximation factor of 2 on trees with all s i t i having an ancestor descendant relation we give a combinatorial exact algorithm besides the algorithm being combinatorial its running time improves by many orders of magnitude the lp algorithm that follows from total unimodularity finally we show strong hardness for the well known problem of finding a path with minimum forbidden pairs which in a sense can be considered the dual to the checkpoint problem despite various works on this problem hardness of approximation was not known prior to this work we show that the problem cannot be approximated within cn for some constant c 0 unless p np this is the strongest type of hardness possible it carries over to directed acyclic graphs and is a huge improvement over the plain np hardness of gabow h n gabow finding paths and cycles of superpolylogarithmic length siam j comput 36 6 2007 1648 1671

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

interpolative butterfly factorization this paper introduces the interpolative butterfly factorization for nearly optimal implementation of several transforms in harmonic analysis when their explicit formulas satisfy certain analytic properties and the matrix representations of these transforms satisfy a complementary low rank property a preliminary interpolative butterfly factorization is constructed based on interpolative low rank approximations of the complementary low rank matrix a novel sweeping matrix compression technique further compresses the preliminary interpolative butterfly factorization via a sequence of structure preserving low rank approximations the sweeping procedure propagates the low rank property among neighboring matrix factors to compress dense submatrices in the preliminary butterfly factorization to obtain an optimal one in the butterfly scheme for an n times n matrix it takes o n log n operations and complexity to construct the factorization as a product of o log n sparse matrices each with o n nonz

divide and conquer approximation algorithms via spreading metrics we present a novel divide and conquer paradigm for approximating np hard graph optimization problems the paradigm models graph optimization problems that satisfy two properties first a divide and conquer approach is applicable second a fractional spreading metric is computable in polynomial time the spreading metric assigns lengths to either edges or vertices of the input graph such that all subgraphs for which the optimization problem is nontrivial have large diameters in addition the spreading metric provides a lower bound t on the cost of solving the optimization problem we present a polynomial time approximation algorithm for problems modeled by our paradigm whose approximation factor is o min log t log log t log k log log k where k denotes the number of interesting vertices in the problem instance and is at most the number of vertices we present seven problems that can be formulated to fit the paradigm for all these problems our algorithm improves previous results the problems are 1 linear arrangement 2 embedding a graph in a d dimensional mesh 3 interval graph completion 4 minimizing storage time product 5 subset feedback sets in directed graphs and multicuts in circular networks 6 symmetric multicuts in directed networks 7 balanced partitions and p separators for small values of p in directed graphs

parameterized graph separation problems we consider parameterized problems where some separation property has to be achieved by deleting as few vertices as possible the following five problems are studied delete k vertices such that a each of the given l terminals is separated from the others b each of the given l pairs of terminals is separated c exactly l vertices are cut away from the graph d exactly l connected vertices are cut away from the graph e the graph is separated into at least l components we show that if both k and l are parameters then a b and d are fixed parameter tractable while c and e are w 1 hard

a faster strongly polynomial minimum cost flow algorithm in this paper we present a new strongly polynomial time algorithm for the minimum cost flow problem based on a refinement of the edmonds karp scaling technique our algorithm solves the uncapacitated minimum cost flow problem as a sequence of on log n shortest path problems on networks with n nodes and m arcs and runs in on log nm n log n time using a standard transformation this approach yields an om log nm n log n algorithm for the capacitated minimum cost flow problem this algorithm improves the best previous strongly polynomial time algorithm due to z galil and e tardos by a factor of n2 m our algorithm for the capacitated minimum cost flow problem is even more efficient if the number of arcs with finite upper bounds say m is much less than m in this case the running time of the algorithm is om n log nm n log n

explicit solutions of the g heat equation for a class of initial conditions abstract we obtain the viscosity solution of the g heat equation with the initial condition x x n for each integer n 1 using the method of g brownian motion

parameterized complexity of the anchored k core problem for directed graphs we consider the directed anchoredk core problem where the task is for a given directed graph g and integers b k and p to find an induced subgraph h with at least p vertices the core such that all but at most b vertices the anchors of h have in degree at least k we undertake a systematic analysis of the computational complexity of the directed anchoredk core problem

on the complexity of the selective graph coloring problem in some special classes of graphs in this paper we consider the selective graph coloring problem given an integer k 1k 1 and a graph g v e g v e with a partition v1 vpv1 vp of vv it consists in deciding whether there exists a set v v in gg such that v vi 1 v vi 1 for all i 1 p i 1 p and such that the graph induced by v v is kk colorable we investigate the complexity status of this problem in various classes of graphs

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

on the parameterized complexity of separating certain sources from the target abstract multicut is proven to be fpt parameterized by the size of the cutset p we focus on its generalization partial multicut setting the number of pairs of terminals to be separated as a parameter r it asks for the minimum edge vertex cut whose removal would separate r sources s i from their target t i our main contributions concern its single target subproblem partial one target cut potc we show that potc when edges are to be cut is fpt parameterized by the cutset size p but is w 1 hard when parameterized only with r the vertex cutting version of potc is w 1 hard regardless of parameter configurations involving r and p

solving integer programs over monotone inequalities in three variables a framework for half integrality and good approximations abstract we define a class of monotone integer programs with constraints that involve up to three variables each a generic constraint in such integer program is of the form ax by z c where a and b are nonnegative and the variable z appears only in that constraint we devise an algorithm solving such problems in time polynomial in the length of the input and the range of variables u the solution is obtained from a minimum cut on a graph with o nu nodes and o mu arcs where n is the number of variables of the types x and y and m is the number of constraints our algorithm is also valid for nonlinear objective functions nonmonotone integer programs are optimization problems with constraints of the type ax by z c without restriction on the signs of a and b such problems are in general np hard we devise here an algorithm relying on a transformation to the monotone case that delivers half integral superoptimal solutions in polynomial time such solutions provide bounds on the optimum value that can only be superior to bounds provided by linear programming relaxation when the half integral solution can be rounded to an integer feasible solution this is a 2 approximate solution in that the technique is a unified 2 approximation technique for a large class of problems the results apply also for general integer programming problems with worse approximation factors that depend on a quantifier measuring how far the problem is from the class of problems we describe the algorithm described here has a wide array of problem applications an additional important consequence of our results is that nonmonotone problems in the framework are max snp hard and at least as hard to approximate as vertex cover problems that are amenable to the analysis provided here are easily recognized the analysis itself is entirely technical and involves manipulating the constraints and transforming them to a totally unimodular system while losing no more than a factor of 2 in the integrality

improving multicut in directed trees by upgrading nodes abstract in this paper we consider the network improvement problem for multicut by upgrading nodes in a directed tree t v e with multiple sources and multiple terminals in a node based upgrading model a node v can be upgraded at the expense of c v and such an upgrade reduces weights on all edges incident to v the objective is to upgrade a minimum cost subset s v of nodes such that the resulting network has a multicut in which no edge has weight larger than a given value d we first obtain a minimum cardinality node multicut v c for tree t then find the minimum cost upgrading set based on the upgrading sets for the subtrees rooted at the nodes in v c we show that our algorithm is polynomial when the number of source terminal pairs is upper bounded by a given value

subset feedback vertex set on graphs of bounded independent set size abstract the weighted subset feedback vertex set problem is a generalization of the classical feedback vertex set problem and asks for a vertex set of minimum weight size that intersects all cycles containing a vertex of a predescribed set of vertices although subset feedback vertex set and feedback vertex set exhibit different computational complexity on split graphs no similar characterization is known on other classes of graphs towards the understanding of the complexity difference between the two problems it is natural to study the importance of structural graph parameters here we consider graphs of bounded independent set number for which it is known that weighted feedback vertex set can be solved in polynomial time we provide a dichotomy result with respect to the size of a maximum independent set in particular we show that weighted subset feedback vertex set can be solved in polynomial time for graphs with 3 whereas we prove that the problem remains np hard for graphs with 4 moreover we show that the unweighted subset feedback vertex set problem can be solved in polynomial time on graphs of bounded independent set number by giving an algorithm with running time n o to complement our results we demonstrate how our ideas can be extended to other terminal set problems on graphs of bounded independent set size node multiway cut is a terminal set problem that asks for a vertex set of minimum size that intersects all paths connecting any two terminals based on our findings for subset feedback vertex set we settle the complexity of node multiway cut as well as its variants where nodes are weighted and or the terminals are deletable for every value of the given independent set number

the complexity of soft constraint satisfaction over the past few years there has been considerable progress in methods to systematically analyse the complexity of constraint satisfaction problems with specified constraint types one very powerful theoretical development in this area links the complexity of a set of constraints to a corresponding set of algebraic operations known as polymorphisms in this paper we extend the analysis of complexity to the more general framework of combinatorial optimisation problems expressed using various forms of soft constraints we launch a systematic investigation of the complexity of these problems by extending the notion of a polymorphism to a more general algebraic operation which we call a multimorphism we show that many tractable sets of soft constraints both established and novel can be characterised by the presence of particular multimorphisms we also show that a simple set of np hard constraints has very restricted multimorphisms finally we use the notion of multimorphism to give a complete classification of complexity for the boolean case which extends several earlier classification results for particular special cases

fast approximate graph partitioning algorithms we study graph partitioning problems on graphs with edge capacities and vertex weights the problems of b balanced cuts and k balanced partitions are unified into a new problem called minimum capacity rho separators a rho separator is a subset of edges whose removal partitions the vertex set into connected components such that the sum of the vertex weights in each component is at most rho times the weight of the graph we present a new and simple o log n approximation algorithm for minimum capacity rho separators which is based on spreading metrics yielding an o log n approximation algorithm both for b balanced cuts and k balanced partitions in particular this result improves the previous best known approximation factor for k balanced partitions in undirected graphs by a factor of o log k we enhance these results by presenting a version of the algorithm that obtains an o log opt approximation factor the algorithm is based on a technique called spreading metrics that enables us to formulate directly the minimum capacity rho separator problem as an integer program we also introduce a generalization called the simultaneous separator problem where the goal is to find a minimum capacity subset of edges that separates a given collection of subsets simultaneously we extend our results to directed graphs for values of rho geq 1 2 we conclude with an efficient algorithm for computing an optimal spreading metric for rho separators this yields more efficient algorithms for computing b balanced cuts than were previously known

the multi terminal maximum flow network interdiction problem abstract this paper defines and studies the multi terminal maximum flow network interdiction problem mtnip in which a network user attempts to maximize flow in a network among k 3 pre specified node groups while an interdictor uses limited resources to interdict network arcs to minimize this maximum flow the paper proposes an exact mtnip e and an approximating model mpnim to solve this np hard problem and presents computational results to compare the models mtnip e is obtained by first formulating mtnip as bi level min max program and then converting it into a mixed integer program where the flow is explicitly minimized mpnim is binary integer program that does not minimize the flow directly it partitions the node set into disjoint subsets such that each node group is in a different subset and minimizes the sum of the arc capacities crossing between different subsets computational results show that mpnim can solve all instances in a few seconds while mtnip e cannot solve about one third of the problems in 24 hour the optimal objective function values of both models are equal to each other for some problems while they differ from each other as much as 46 2 in the worst case however when the post interdiction flow capacity incurred by the solution of mpnim is computed and compared to the objective value of mtnip e the largest difference is only 7 90 implying that mpnim may be a very good approximation to mtnip e

representative sets and irrelevant vertices new tools for kernelization we continue the development of matroid based techniques for kernelization initiated by the present authors 47 we significantly extend the usefulness of matroid theory in kernelization by showing applications of a result on representative sets due to lovasz 51 and marx 53 as a first result we show how representative sets can be used to derive a polynomial kernel for the elusive almost 2 sat problem where the task is to remove at most k clauses to make a 2 cnf formula satisfiable solving a major open problem in kernelization this result also yields a new o log opt approximation for the problem improving on the o log n approximation of agarwal et al 3 and an implicit o log opt approximation due to even et al 24 we further apply the representative sets tool to the problem of finding irrelevant vertices in graph cut problems that is vertices that can be made undeletable without affecting the answer to the problem this gives the first significant progress towards a polynomial kernel for the multiway cut problem in particular we get a kernel of o ks 1 vertices for multiway cut instances with at most s terminals both these kernelization results have significant spin off effects producing the first polynomial kernels for a range of related problems more generally the irrelevant vertex results have implications for covering min cuts in graphs for a directed graph g v e and sets s t v let r be the size of a minimum s t vertex cut which may intersect s and t we can find a set z v of size o s t r that contains a minimum a b vertex cut for every a s b t similarly for an undirected graph g v e a set of terminals x v and a constant s we can find a set z v of size o x s 1 that contains a minimum multiway cut for every partition of x into at most s pairwise disjoint subsets both results are polynomial time we expect this to have further applications in particular we get direct reduction rule based kernelizations for all problems above in contrast to the indirect compression based kernel previously given for odd cycle transversal 47 all our results are randomized with failure probabilities that can be made exponentially small in n due to needing a representation of a matroid to apply the representative sets tool

min sum clustering with penalties abstract given a complete graph g v e a weight function w e n 0 on its edges and p n 0 a penalty function on its vertices the penalized k min sum problem is the problem of finding a partition of v to k 1 sets s 1 s k 1 minimizing i 1 k w s i p s k 1 where for s v w s e i j s w e and p s i s p i our main result is a randomized approximation scheme for the metric version of the penalized 1 min sum problem when the ratio between the minimal and maximal penalty is bounded for the metric penalized k min sum problem where k is a constant we offer a 2 approximation

correlation clustering with constrained cluster sizes and extended weights bounds we consider the problem of correlation clustering on graphs with constraints on both the cluster sizes and the positive and negative weights of edges our contributions are twofold first we introduce the problem of correlation clustering with bounded cluster sizes second we extend the regime of weight values for which the clustering may be performed with constant approximation guarantees in polynomial time and apply the results to the bounded cluster size problem

on the regularity theory of fully nonlinear parabolic equations ii recently m crandall and p l lions 3 developed a very successful method for proving the existence of solutions of nonlinear second order partial differential equations their method called the theory of viscosity solutions also applies to fully nonlinear equations in which even the second order derivatives can enter in nonlinear fashion solutions produced by the viscosity method are guaranteed to be continuous but not necessarily smooth here we announce smoothness results for viscosity solutions our methods extend those of 1 we obtain krylov safonov i e c estimates 8 c 1 schauder c and w estimates for viscosity solutions of uniformly parabolic equations in general form the results can be viewed as a priori estimates on the classical c solutions our method produces in particular regularity results for a broad new array of nonlinear heat equations including the bellman equation 6

reconstructing polyatomic structures from discrete x rays we address a discrete tomography problem that arises in the study of the atomic structure of crystal lattices a polyatomic structure t can be defined as an integer lattice in dimension d 2 whose points may be occupied by c distinct types of atoms to analyze t we conduct measurements that we call discrete x rays a discrete x ray in direction determines the number of atoms of each type on each line parallel to given such non parallel x rays we wish to reconstruct t the complexity of the problem for c 1 one atom type has been completely determined by gardner et al technical report 970 05012 techn univ munchen fak f math 1997 who proved that the problem is np complete for any dimension d 2 and 3 non parallel x rays and that it can be solved in polynomial time otherwise ryser mathematical association of america and quinn boden rahway new jersey 1963 the np completeness result above clearly extends to any c 2 and therefore when studying the polyatomic case we can assume that 2 as shown in another article by the same authors gardner et al theoret comput sci 1997 to appear this problem is also np complete for c 6 atoms even for dimension d 2 and axis parallel x rays gardner et al 1997 conjecture that the problem remains np complete for c 3 4 5 although as they point out the proof idea in gardner et al 1997 does not seem to extend to c 5 we resolve the conjecture from gardner et al 1997 by proving that the problem is indeed np complete for c 3 in 2d even for axis parallel x rays our construction relies heavily on some structure results for the realizations of 0 1 matrices with given row and column sums

butterfly factorization via randomized matrix vector multiplications this paper presents an adaptive randomized algorithm for computing the butterfly factorization of an m times n matrix with m approx n provided that both the matrix and its transpose can be rapi

partial multicuts in trees let t v e be an undirected tree in which each edge is associated with a non negative cost and let s1 t1 sk tk be a collection of k distinct pairs of vertices given a requirement parameter t k the partial multicut on a tree problem asks to find a minimum cost set of edges whose removal from t disconnects at least t out of these k pairs this problem generalizes the well known multicut on a tree problem in which we are required to disconnect all given pairs the main contribution of this paper is an 8 3 e approximation algorithm for partial multicut on a tree whose run time is strongly polynomial for any fixed e 0 this result is achieved by introducing problem specific insight to the general framework of using the lagrangian relaxation technique in approximation algorithms our algorithm utilizes a heuristic for the closely related prize collecting variant in which we are not required to disconnect all pairs but rather incur penalties for failing to do so we provide a lagrangian multiplier preserving algorithm for the latter problem with an approximation factor of 2 finally we present a new 2 approximation algorithm for multicut on a tree based on lp rounding

key generation algorithms for pairwise independent networks based on graphical models we consider two secret key generation problems under a pairwise independent network model and propose low complexity key generation schemes in a framework that connects our problems to network flow problems in graphs our schemes have two components 1 local key generation and 2 global key propagation in the local key generation we use point to point source coding with side information to establish pairwise keys from which we construct a graph with the capacity of each edge being the key rate of the corresponding point to point local key in the global key propagation depending on the particular problem secret keys are delivered to users in the network using various network flow algorithms in particular in the first problem in which one is required to generate a group key for a group of users in the network we propose a network coding based global key propagation approach this approach has a low complexity and has a better performance than the existing approach in the second problem in which one is required to generate multiple keys simultaneously for different pairs of users we propose a multicommodity flow based global key propagation approach we show that the proposed approach is optimal for the case of generating two keys for the general case of generating more than two keys we show that the sum rate of the proposed scheme is larger than an upper bound characterized in this paper divided by a constant

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

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

a new approach to the maximum flow problem all previously known efficient maximum flow algorithms work by finding augmenting paths either one path at a time as in the original ford and fulkerson algorithm or all shortest length augmenting paths at once using the layered network approach of dinic an alternative method based on the preflow concept of karzanov is introduced a preflow is like a flow except that the total amount flowing into a vertex is allowed to exceed the total amount flowing out the method maintains a preflow in the original network and pushes local flow excess toward the sink along what are estimated to be shortest paths the algorithm and its analysis are simple and intuitive yet the algorithm runs as fast as any other known method on dense graphs achieving an o n 3 time bound on an n vertex graph by incorporating the dynamic tree data structure of sleator and tarjan we obtain a version of the algorithm running in o nm log n 2 m time on an n vertex m edge graph this is as fast as any known method for any graph density and faster on graphs of moderate density the algorithm also admits efficient distributed and parallel implementations a parallel implementation running in o n 2 log n time using n processors and o m space is obtained this time bound matches that of the shiloach vishkin algorithm which also uses n processors but requires o n 2 space

fast algorithms for shortest paths in planar graphs with applications graph decomposition and data structures techniques are presented that exploit the structure of planar graphs to yield faster algorithms for a number of shortest path and related problems improved

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

on the hardness of finding near optimal multicuts in directed acyclic graphs abstract given an edge weighted di graph and a list of source sink pairs of vertices of this graph the minimum multicut problem consists in selecting a minimum weight set of edges or arcs whose removal leaves no path from each source to the corresponding sink this is a well known np hard problem and improving several previous results we show that it remains apx hard in unweighted directed acyclic graphs dag even with only two source sink pairs this is also true if we remove vertices instead of arcs

parameterized complexity in multiple interval graphs domination partition separation irredundancy we show that the problem k dominating set and its several variants including k connected dominating set k independent dominating set and k dominating clique when parameterized by the solution size k are w 1 hard in either multiple interval graphs or their complements or both on the other hand we show that these problems belong to w 1 when restricted to multiple interval graphs and their complements this answers an open question of fellows et al in sharp contrast we show that d distance k dominating set for d 2 is w 2 complete in multiple interval graphs and their complements we also show that k perfect code and d distance k perfect code for d 2 are w 1 complete even in unit 2 track interval graphs in addition we present various new results on the parameterized complexities of k vertex clique partition and k separating vertices in multiple interval graphs and their complements and present a very simple alternative proof of the w 1 hardness of k irredundant set in general graphs