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

corrigendum to parameterized tractability of the maximum duo preservation string mapping problem theoret comput sci 646 2016 16 25 this is a corrigendum for our paper 1 as we have found that the first fpt algorithm for the maximum duo preservation string mapping problem we presented is incorrect however we show that by slightly modifying the color coding technique on which the algorithm is based we can fix the error thus giving a correct fpt algorithm for maximum duo preservation string mapping problem

solving the maximum duo preservation string mapping problem with linear programming abstract in this paper we introduce the maximum duo preservation string mapping problem mpsm which is complementary to the minimum common string partition problem mcsp when each letter occurs at most k times in any input string the version of mpsm is called k mpsm in order to design approximation algorithms for mpsm we also introduce the constrained maximum induced subgraph problem cmis and the constrained minimum induced subgraph cnis problem we show that both cmis and cnis are np complete we also study the approximation algorithms for the restricted version of cmis which is called k cmis k 2 using linear programming method we propose an approximation algorithm for 2 cmis with approximation ratio 2 and an approximation algorithm for k cmis k 3 with approximation ratio k 2 based on approximation algorithms for k cmis we get approximation algorithms for k mpsm with the same approximation ratio

scheduling and fixed parameter tractability fixed parameter tractability analysis and scheduling are two core domains of combinatorial optimization which led to deep understanding of many important algorithmic questions however even though fixed parameter algorithms are appealing for many reasons no such algorithms are known for many fundamental scheduling problems in this paper we present the first fixed parameter algorithms for classical scheduling problems such as makespan minimization scheduling with job dependent cost functions one important example being weighted flow time and scheduling with rejection to this end we identify crucial parameters that determine the problems complexity in particular we manage to cope with the problem complexity stemming from numeric input values such as job processing times which is usually a core bottleneck in the design of fixed parameter algorithms we complement our algorithms with mathsf w 1 w 1 hardness results showing that for smaller sets of parameters the respective problems do not allow fixed parameter algorithms in particular our positive and negative results for scheduling with rejection explore a research direction proposed by daniel marx

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

faster deterministic parameterized algorithm for k path abstract in the k path problem the input is a directed graph g and an integer k 1 and the goal is to decide whether there is a simple directed path in g with exactly k vertices we give a deterministic algorithm for k path with time complexity o 2 554 k this improves the previously best deterministic algorithm for this problem of zehavi esa 2015 whose time complexity is o 2 597 k the technique used by our algorithm can also be used to obtain faster deterministic algorithms for k tree r dimensional k matching graph motif and partial cover

on the spectrum of the zhang zagier height from recent work of zhang and of zagier we know that their height n is bounded away from 1 for every algebraic number a different from 0 1 1 2 3 2 the study of the related spectrum is especially interesting for it is linked to lehmer s problem and to a conjecture of bogomolov after recalling some definitions we show an improvement of the so called zhang zagier inequality to achieve this we need some algebraic numbers of small height so in the third section we describe an algorithm able to find them and we give an algebraic number with height 1 2875274z discovered in this way this search up to degree 64 suggests that the spectrum of n may have a limit point less than 1 292 we prove this fact in the fourth part

parameterized and approximation algorithms for finding two disjoint matchings we first present a randomized fixed parameter algorithm for the np hard problem of deciding if there are two matchings m 1 and m 2 in a given graph g such that m 1 m 2 is a given number k the algorithm runs in o 2 k k m n expected time and can be derandomized to run in o 2 2 k 12 log 2 2 k k n m n time where n respectively m is the number of vertices respectively edges in g we then extend the algorithm to the weighted version of the problem we further present a combinatorial approximation algorithm for the np hard problem of finding two disjoint matchings in a given edge weighted graph g so that their total weight is maximized the algorithm achieves an approximation ratio close to 0 76 and runs in o m n 3 n time where is the inverse ackermann function

the parameterized complexity of the k biclique problem given a graph g and an integer k the k b iclique problem asks whether g contains a complete bipartite subgraph with k vertices on each side whether there is an f k vgvo 1 time algorithm solving k b iclique for some computable function f has been a longstanding open problem we show that k b iclique is w 1 hard which implies that such an f k vgvo 1 time algorithm does not exist under the hypothesis w 1 fpt from parameterized complexity theory to prove this result we give a reduction which for every n vertex graph g and small integer k constructs a bipartite graph h e lc r e in time polynomial in n such that if g contains a clique with k vertices then there are k k 1 2 vertices in l with n 1 k common neighbors otherwise any k k 1 2 vertices in l have at most k 1 e common neighbors an additional feature of this reduction is that it creates a gap on the right side of the biclique such a gap might have further applications in proving hardness of approximation results assuming a randomized version of exponential time hypothesis we establish an f k vgvo sk time lower bound for k biclique for any computable function f combining our result with the work of bulatov and marx 2014 we obtain a dichotomy classification of the parameterized complexity of cardinality constraint satisfaction problems

bounds on the unstable eigenvalue for the asymmetric renormalization operator for period doubling we establish rigorous bounds for the unstable eigenvalue of the period doubling renormalization operator for asymmetric unimodal maps herglotz function techniques and cone invariance ideas are used our result generalizes an established result for conventional period doubling

parameterized maximum path coloring we study the well known max path coloring problem from a parameterized point of view focusing on trees and low treewidth networks we observe the existence of a variety of reasonable parameters for the problem such as the maximum degree and treewidth of the network graph the number of available colors and the number of requests one seeks to satisfy or reject in an effort to understand the impact of each of these parameters on the problem s complexity we study various parameterized versions of the problem deriving fixed parameter tractability and hardness results both for undirected and bi directed graphs

sums of heights of algebraic numbers for at x f x tg x we consider the set at 0 h t q the polynomials f x g x are in z x with only mild restrictions and h is the weil height of we show that this set is dense in d for some effectively computable limit point d

parameterized approximation algorithms for packing problems parameterized approximation is a topic of considerable interest in the field of parameterized complexity in the past decade new color coding related techniques including the breakthrough representative sets technique have been proven extremely powerful in the design of fast parameterized algorithms furthermore packing problems which are often solved via color coding related techniques have enjoyed a race towards obtaining the fastest parameterized algorithms that solve them therefore it is natural to ask whether packing problems admit efficient parameterized approximation algorithms in this paper we answer this question affirmatively we present tradeoffs that improve the running times of algorithms for well known special cases of the 3 setk packing problem at the cost of their accuracy consider a packing problem for which there is no known algorithm with approximation ratio and a parameter k if the value of an optimal solution is at least k we seek a solution of value at least k otherwise we seek an arbitrary solution clearly if the best known parameterized algorithm that finds a solution of value t runs in time o f t for some function f we are interested in running times better than o f k our main contribution lies in the adaptation of notions fundamental to the representative sets technique to the design of approximation algorithms we introduce the definition of approximate lopsided universal sets combine partial executions of representative sets based algorithms with approximation algorithms and adapt the iterative expansion framework in the context of representative sets to the design of parameterized approximation algorithms our ideas are intuitive and may be relevant to the design of other parameterized approximation algorithms

probabilistic checking of proofs a new characterization of np we give a new characterization of np the class np contains exactly those languages l for which membership proofs a proof that an input x is in l can be verified probabilistically in polynomial time using logarithmic number of random bits and by reading sublogarithmic number of bits from the proof we discuss implications of this characterization specifically we show that approximating clique and independent set even in a very weak sense is np hard

edge disjoint packing of stars and cycles abstract we study the parameterized complexity of two graph packing problems edge disjoint k packing of s stars and edge disjoint k packing of s cycles with respect to the choice of parameters we show that although the two problems are fpt with both k and s as parameters they are unlikely to be fixed parameter tractable when parameterized by only k or only s in terms of kernelization complexity we show that edge disjoint k packing of s stars has a kernel with size polynomial in both k and s but in contrast unless np conp poly edge disjoint k packing of s cycles does not have a kernel with size polynomial in both k and s and moreover does not have a kernel with size polynomial in s for any fixed k we also show that edge disjoint k packing of 4 cycles admits a 96 k 2 kernel in general graphs and a 96 k kernel in planar graphs

existence and properties of p tupling fixed points we prove the existence of fixed points of p tupling renormalization operators for interval and circle mappings having a critical point of arbitrary real degree r 1 some properties of the resulting maps are studied analyticity univalence behavior as r tends to infinity

parameterized tractability of the maximum duo preservation string mapping problem in this paper we investigate the parameterized complexity of the maximum duo preservation string mapping problem the complementary of the minimum common string partition problem we show that this problem is fixed parameter tractable when parameterized by the number k of conserved duos by first giving a parameterized algorithm based on the color coding technique and then presenting a reduction to a kernel of size o k 6

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

new proofs of the existence of the feigenbaum functions a new proof of the existence of analytic unimodal solutions of the cvitanovic feigenbaum functional equation g x g g x g x 1 const x r at 0 valid for all in 0 1 is given and the existence of the eckmann wittwer functions 8 is recovered the method also provides the existence of solutions for certain given values ofr and in particular forr 2 a proof requiring no computer

on the parameterized complexity of vertex cover and edge cover with connectivity constraints abstract we investigate the effect of certain natural connectivity constraints on the parameterized complexity of two fundamental graph covering problems namely vertex cover and edge cover specifically we impose the additional requirement that each connected component of a solution have at least t vertices resp edges from the solution for a fixed positive integer t and call the problem t total vertex cover resp t total edge cover in both cases the parameter k is the size of the solution we show that both problems remain fixed parameter tractable with these restrictions with running times of the form o c k for some constant c 0 in each case where the o notation hides polynomial factors for each fixed t 2 t total vertex cover has no polynomial kernel unless conp np poly for each fixed t 2 t total edge cover has a linear vertex kernel of size t 1 t k these results significantly improve earlier work on these problems we illustrate the utility of the technique used to solve t total vertex cover by applying it to derive an o c k time fpt algorithm for the t total edge dominating set problem our no poly kernel result for t total vertex cover and the known np hardness result for t total edge cover are in stark contrast to the fact that vertex cover has a 2 k vertex kernel and that edge cover is solvable in polynomial time this illustrates how even the slightest connectivity requirement results in a drastic change in the tractability of problems the curse of connectivity

parameterized counting matching and packing a family of hard problems that admit fptras abstract in the field of parameterized counting complexity the problems that are w 1 hard and admit fixed parameter tractable randomized approximation scheme fptras have attracted much attention in recent years in this paper we focus on the problems on parameterized counting matching and packing these problems include counting set packing counting matching and counting subgraph packing including both vertex disjoint and edge disjoint versions we study the parameterized complexity on these problems on the basis of some results for counting graph matchings we show that a series of problems are w 1 hard furthermore by extending the previous algorithm for counting 3 d matching we obtain fptras for each considered problem respectively our results indicate that the problems on parameterized counting matching and packing form a large family of problems that are w 1 hard and admit fptras

feigenbaum theory for unimodal maps with asymmetric critical point rigorous results we apply the methods of h epstein to prove the existence of a line of period 2 solutions of the feigenbaum period doubling renormalisation transformation these solutions govern the universal behaviour of families of unimodal maps with asymmetric critical points of degree d for which the d th derivative has differing left and right limits

improved kernel results for some fpt problems based on simple observations abstract in this paper we study the kernelization algorithms for several parameterized problems including parameterized co path set problem parameterized path contractibility problem and parameterized connected dominating set on g 7 graphs problem based on simple observations we give simple kernelization algorithms with kernel sizes 4 k 3 k 4 and o k 2 respectively which improves the previous best results 6 k 5 k 3 and o k 3 respectively

analyticity properties of the feigenbaum function analyticity properties of the feigenbaum function a solution ofg x 1g g x withg 0 1 g 0 0 g 0 0 are investigated by studying its inverse function which turns out to be herglotz or anti herglotz on all its sheets it is found thatg is analytic and uniform in a domain with a natural boundary

orbits of algebraic numbers with low heights we prove that the two smallest values of h h 1 1 h 1 1 are 0 and 0 4218 for any algebraic integer

on the parameterized complexity of multiple interval graph problems multiple interval graphs are a natural generalization of interval graphs where each vertex may have more than one interval associated with it many applications of interval graphs also generalize to multiple interval graphs often allowing for more robustness in the modeling of the specific application with this motivation in mind a recent systematic study of optimization problems in multiple interval graphs was initiated in this sequel we study multiple interval graph problems from the perspective of parameterized complexity the problems under consideration are k independent set k dominating set and k clique which are all known to be w 1 hard for general graphs and np complete for multiple interval graphs we prove that k clique is in fpt while k independent set and k dominating set are both w 1 hard we also prove that k independent dominating set a hybrid of the two above problems is also w 1 hard our hardness results hold even when each vertex is associated with at most two intervals and all intervals have unit length furthermore as an interesting byproduct of our hardness results we develop a useful technique for showing w 1 hardness via a reduction from the k multicolored clique problem a variant of k clique we believe this technique has interest in its own right as it should help in simplifying w 1 hardness results which are notoriously hard to construct and technically tedious

parameterized top k algorithms we study algorithmic techniques that produce the best k solutions to an instance of a parameterized np hard problem whose solutions are associated with a scoring function our parameterized top k algorithms proceed in two stages the first stage is a structure algorithm that on a problem instance constructs a structure of feasible size and the second stage is an enumerating algorithm that produces the k best solutions to the instance based on the structure we show that many algorithm design techniques for parameterized algorithms such as branch and search color coding and bounded treewidth can be adopted for designing efficient structure algorithms we then develop new techniques that support efficient enumerating algorithms in particular we show that for a large class of well known np optimization problems there are parameterized top k algorithms that produce the best k solutions for the problems in feasible amount of average time per solution when the parameter value is small finally we investigate the relation between fixed parameter tractability and parameterized top k algorithms

finding approximate and constrained motifs in graphs one of the most relevant topics in the analysis of biological networks is the identification of functional motifs inside a network a recent approach introduced in literature called graph motif represents the network as a vertex colored graph and the motif m as a multiset of colors an occurrence of a motif m in a vertex colored graph g is a connected induced subgraph of g whose vertex set is colored exactly as m in this paper we investigate three different variants of the graph motif problem the first two variants minimum adding motif min add graph motif and minimum substitution motif min sub graph motif deal with approximate occurrences of a motif in the graph while the third variant constrained graph motif cgm constrains the motif to contain a given set of vertices we investigate the computational and parameterized complexity of the three problems we show that min add graph motifand min sub graph motifare both np hard even when m is a set and the graph is a tree with maximum degree 4 in which each color appears at most twice then we show that min sub graph motifis fixed parameter tractable when parameterized by the size of m finally we consider the parameterized complexity of the cgmproblem we give a fixed parameter algorithm for graphs of bounded treewidth and show that the problem is w 2 hard when parameterized by m even if the input graph has diameter 2

revisiting the parameterized complexity of maximum duo preservation string mapping abstract in the maximum duo preservation string mapping max duo psm problem the input consists of two related strings a and b of length n and a nonnegative integer k the objective is to determine whether there exists a mapping m from the set of positions of a to the set of positions of b that maps only to positions with the same character and preserves at least k duos which are pairs of adjacent positions we develop a randomized algorithm that solves max duo psm in 4 k n o 1 time and a deterministic algorithm that solves this problem in 6 855 k n o 1 time the previous best known deterministic algorithm for this problem has 8 e 2 k o k n o 1 running time beretta et al 2016 1 2 we also show that max duo psm admits a problem kernel of size o k 3 improving upon the previous best known problem kernel of size o k 6

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

determinant sums for undirected hamiltonicity we present a monte carlo algorithm for hamiltonicity detection in an n vertex undirected graph running in o 1 657 n time to the best of our knowledge this is the first superpolynomial improvement on the worst case runtime for the problem since the o 2 n bound established for the traveling salesman problem tsp over 50 years ago r bellman j assoc comput mach 9 1962 pp 61 63 m held and r m karp j soc indust appl math 10 1962 pp 196 210 o f n suppresses polylogarithmic functions in f n it answers in part the first open problem in woeginger s 2003 survey on exact algorithms for np hard problems for bipartite graphs we improve the bound to o sqrt 2 n subset o 1 415 n time both the bipartite and the general algorithm can be implemented to use space polynomial in n we combine several recently resurrected ideas to get the results our main technical contribution is a new algebraic characterization of hamiltonian graphs we introduce an ex

approximating the maximum clique minor and some subgraph homeomorphism problems we consider the minor and homeomorphic analogues of the maximum clique problem i e the problems of determining the largest h such that the input graph on n vertices has a minor isomorphic to k h or a subgraph homeomorphic to k h respectively as well as the problem of finding the corresponding subgraphs we term them as the maximum clique minor problem and the maximum homeomorphic clique problem respectively we observe that a known result of kostochka and thomason supplies an o n bound on the approximation factor for the maximum clique minor problem achievable in polynomial time we also provide an independent proof of nearly the same approximation factor with explicit polynomial time estimation by exploiting the minor separator theorem of plotkin et al next we show that another known result of bollobas and thomason and of komlos and szemeredi provides an o n bound on the approximation factor for the maximum homeomorphic clique achievable in polynomial time on the other hand we show an w n 1 2 o 1 l o g n c lower bound for some constant c unless email protected zptime 2 l o g n o 1 on the best approximation factor achievable efficiently for the maximum homeomorphic clique problem nearly matching our upper bound finally we derive an interesting trade off between approximability and subexponential time for the problem of subgraph homeomorphism where the guest graph has maximum degree not exceeding three and low treewidth

editing to a graph of given degrees we consider the editing to a graph of given degrees problem that asks for a graph g non negative integers d k and a function v g 1 d whether it is possible to obtain a graph g from g such that the degree of v is v for any vertex v by at most k vertex or edge deletions or edge additions we construct an fpt algorithm for editing to a graph of given degrees parameterized by d k we complement this result by showing that the problem has no polynomial kernel unless np conp poly

arboricity and subgraph listing algorithms in this paper we introduce a new simple strategy into edge searching of a graph which is useful to the various subgraph listing problems applying the strategy we obtain the following four algorithms the first one lists all the triangles in a graph g in o a g m time where m is the number of edges of g and a g the arboricity of g the second finds all the quadrangles in o a g m time since a g is at most three for a planar graph g both run in linear time for a planar graph the third lists all the complete subgraphs k l of order l in o la g l 2 m time the fourth lists all the cliques in o a g m time per clique all the algorithms require linear space we also establish an upper bound on a g for a graph g a g leqq lceil 2m n 1 2 rceil where n is the number of vertices in g

calculus on arithmetic surfaces in a2 and a3 arakelov introduces an intersection calculus for arithmetic surfaces that is for stable models of curves over a number field in this paper we intend to show that his intersection product has a lot of useful properties more precisely we show that the following properties from the theory of algebraic surfaces have an analogue in our situation

comparing incomplete sequences via longest common subsequence abstract inspired by scaffold filling a recent approach for genome reconstruction from incomplete data we consider a variant of the well known longest common subsequence problem for the comparison of two sequences the new problem called longest filled common subsequence aims to compare a complete sequence with an incomplete one i e with some missing elements longest filled common subsequence lfcs given a complete sequence a an incomplete sequence b and a multiset m of symbols missing in b asks for a sequence b obtained by inserting the symbols of m into b so that b induces a common subsequence with a of maximum length we investigate the computational and approximation complexity of the problem and we show that it is np hard and apx hard when a contains at most two occurrences of each symbol and we give a polynomial time algorithm when the input sequences are over a constant size alphabet we give a 3 5 approximation algorithm for the longest filled common subsequence problem finally we present a fixed parameter algorithm for the problem when it is parameterized by the number of symbols inserted in b that match symbols of a

looking at the stars the problem of packing k vertex disjoint copies of a graph h into another graph g is np complete if h has more than two vertices in some connected component in the framework of parameterized complexity we analyze a particular family of instances of this problem namely the packing of stars we give a quadratic kernel for packing k copies of h k1 s when we consider the special case of s 2 i e h being a star with two leaves we give a linear kernel and an algorithm running in time o 25 301kk2 5 n3

the k distinct language parameterized automata constructions abstract in this paper we pioneer a study of parameterized automata constructions for languages related to the design of parameterized algorithms we focus on the k distinct language l k k defined as the set of words of length k over an alphabet whose symbols are all distinct this language is implicitly related to several breakthrough techniques developed during the last two decades to design parameterized algorithms for fundamental problems such as k path and r dimensional k matching building upon the color coding divide and color and narrow sieves techniques we obtain the following automata constructions for l k we develop non deterministic automata nfas of sizes 4 k o k n o 1 and 2 e k o k n o 1 where the latter satisfies a bounded ambiguity property relevant to approximate counting as well as a non deterministic xor automaton nxa of size 2 k n o 1 where n we show that our constructions can be used to develop both deterministic and randomized algorithms for k path r dimensional k matching and module motif in a natural manner considering also their approximate counting variants our framework is modular and consists of two parts designing an automaton for k distinct and designing a problem specific automaton as well as an algorithm for deciding whether the intersection automaton s language is empty or for counting the number of accepting paths in it

the number of irreducible factors of a polynomial i let f x be a polynomial with coefficients in an algebraic number field k we estimate the number of irreducible cyclotomic factors of f in k x the number of irreducible noncyclotomic factors of f the number of nth roots of unity among the roots of f and the number of primitive nth roots of unity among the roots of f all of these quantities are counted with multiplicity and estimated by expressions which depend explicitly on k on the degree of f and height of f and when appropriate on n we show by constructing examples that some of our results are essentially sharp

on the tractability of finding disjoint clubs in a network abstract we study a variant of the problem of finding a collection of disjoint s clubs in a given network given a graph the problem asks whether there exists a collection of at most r disjoint s clubs that covers at least k vertices of the network an s club is a connected graph that has diameter bounded by s for a positive integer s we demand that each club is non trivial that is it has order at least t 2 for some positive integer t we prove that the problem is apx hard even when the input graph has bounded degree s 2 t 3 and r v moreover we show that the problem is polynomial time solvable when s 4 t 3 and r v and when s 3 t 2 and r v finally for s 2 we present a fixed parameter algorithm for the problem when parameterized by the number of covered vertices

two edge disjoint paths with length constraints abstract we consider the problem of finding for two pairs s 1 t 1 and s 2 t 2 of vertices in an undirected graph an s 1 t 1 path p 1 and an s 2 t 2 path p 2 such that p 1 and p 2 share no edges and the length of each p i satisfies constraint l i where l i k i k i k i with l i indicating no length constraint on p i we regard k 1 and k 2 as parameters and investigate the parameterized complexity of the above problem when at least one of p 1 and p 2 has a length constraint for the 9 different cases of l 1 l 2 we obtain fpt algorithms for 7 of them by using random partition backed by some structural results on the other hand we prove that the problem admits no polynomial kernel for all 9 cases unless n p c o n p p o l y

efficient computation of representative families with applications in parameterized and exact algorithms let m e i be a matroid and let s ls1 str be a family of subsets of e of size p a subfamily s s is q representative for s if for every set y e of size at most q if there is a set x s disjoint from y with x y i then there is a set x s disjoint from y with x y i by the classic result of bollobas in a uniform matroid every family of sets of size p has a q representative family with at most p qp sets in his famous two families theorem from 1977 lovasz proved that the same bound also holds for any matroid representable over a field f we give an efficient construction of a q representative family of size at most p qp in time bounded by a polynomial in p qp t and the time required for field operations we demonstrate how the efficient construction of representative families can be a powerful tool for designing single exponential parameterized and exact exponential time algorithms the applications of our approach include the following in the l ong d irected c ycle problem the input is a directed n vertex graph g and the positive integer k the task is to find a directed cycle of length at least k in g if such a cycle exists as a consequence of our 6 75k o k no 1 time algorithm we have that a directed cycle of length at least log n if such a cycle exists can be found in polynomial time in the m inimum e quivalent g raph meg problem we are seeking a spanning subdigraph d of a given n vertex digraph d with as few arcs as possible in which the reachability relation is the same as in the original digraph d we provide an alternative proof of the recent results for algorithms on graphs of bounded treewidth showing that many connectivity problems such as h amiltonian c ycle or s teiner t ree can be solved in time 2o t n on n vertex graphs of treewidth at most t for the special case of uniform matroids on n elements we give a faster algorithm to compute a representative family we use this algorithm to provide the fastest known deterministic parameterized algorithms for k p ath k t ree and more generally k s ubgraph i somorphism where the k vertex pattern graph is of constant treewidth

kernel bounds for path and cycle problems connectivity problems like k path and k disjoint paths relate to many important milestones in parameterized complexity namely the graph minors project color coding and the recent development of techniques for obtaining kernelization lower bounds this work explores the existence of polynomial kernels for various path and cycle problems by considering nonstandard parameterizations we show polynomial kernels when the parameters are a given vertex cover a modulator to a cluster graph or a promised max leaf number we obtain lower bounds via cross composition e g for hamiltonian cycle and related problems when parameterized by a modulator to an outerplanar graph

beyond bidimensionality parameterized subexponential algorithms on directed graphs in this paper we make the first step beyond bidimensionality by obtaining subexponential time algorithms for problems on directed graphs we develop two different methods to achieve subexponential time parameterized algorithms for problems on sparse directed graphs we exemplify our approaches with two well studied problems for the first problem k leaf out branching which is to find an oriented spanning tree with at least k leaves we obtain an algorithm solving the problem in time 2 o k l o g k n n o 1 on directed graphs whose underlying undirected graph excludes some fixed graph h as a minor for the special case when the input directed graph is planar the running time can be improved to 2 o k n n o 1 the second example is a generalization of the directed hamiltonian path problem namely k internal out branching which is to find an oriented spanning tree with at least k internal vertices we obtain an algorithm solving the problem in time 2 o k l o g k n o 1 on directed graphs whose underlying undirected graph excludes some fixed apex graph h as a minor finally we observe that on these classes of graphs the k directed path problem is solvable in time o email protected kn f e for any e 0 where f is some function of e our methods are based on non trivial combinations of obstruction theorems for undirected graphs kernelization problem specific combinatorial structures and a layering technique similar to the one employed by baker to obtain ptas for planar graphs

kernel and fast algorithm for dense triplet inconsistency we study the parameterized complexity of inferring supertrees from sets of rooted triplets an important problem in phylogenetics for a set l of labels and a dense set t of triplets distinctly leaf labeled by 3 subsets of l we seek a tree distinctly leaf labeled by l and containing all but at most k triplets from t as homeomorphic subtree our results are the first polynomial kernel for this problem with o k 2 labels and a subexponential fixed parameter algorithm running in time o l 4 2 o k 1 3 l o g k

on finding rainbow and colorful paths in the colorful path problem we are given a graph g v e and an arbitrary vertex coloring function c v k the goal is to find a colorful path i e a path on k vertices that visits each color this problem has been introduced in the classical work of alon et al 1995 1 and the authors proposed a dynamic programming algorithm that runs in time 2 k n o 1 and uses o 2 k space since then the only progress obtained is reducing the space size to a polynomial at the cost of using randomization in this work we show that a progress in time complexity is unlikely if colorful path can be solved in time 2 e k n o 1 then set cover admits a 2 e n n m o 1 time algorithm the same applies to other versions of the problem when edges are colored instead of vertices or we ask for a walk instead of a path or when the requested path walk has specified endpoints we study also a second very related problem in rainbowst connectivity we are given a k edge colored graph and two vertices s and t the goal is to decide whether there is a rainbow path between s and t that is a path on which no color repeats in its vertex variant rainbow vertexst connectivity the input graph is k vertex colored and a rainbow path is defined analogously uchizawa et al 2011 14 show that both variants can be solved in 2 k n o 1 time and exponential space we show that the space size can be reduced to a polynomial while keeping the same running time in contrast to the polynomial space algorithm for colorful path our algorithm for finding rainbow paths is deterministic

on functional equations associated with the renormalization of non commuting circle mappings we investigate the functional equations associated with critical non commuting circle homeomorphisms using epstein s herglotz function methods we show that there are analytic solutions of a class of functional equations closely related to period two points of the circle map renormalization transformations for rotation number with continued fraction

polynomial fixed parameter algorithms a case study for longest path on interval graphs we study the design of fixed parameter algorithms for problems already known to be solvable in polynomial time the main motivation is to get more efficient algorithms for problems with unattractive polynomial running times here we focus on a fundamental graph problem longest path that is given an undirected graph find a maximum length path in g longest path is np hard in general but known to be solvable in o n4 time on n vertex interval graphs we show how to solve longest path on interval graphs parameterized by vertex deletion number k to proper interval graphs in o k9n time notably longest path is trivially solvable in linear time on proper interval graphs and the parameter value k can be approximated up to a factor of 4 in linear time from a more general perspective we believe that using parameterized complexity analysis may enable a refined understanding of efficiency aspects for polynomial time solvable problems similarly to what classical parameterized complexity analysis does for np hard problems

a 2 o k n algorithm for k cycle in minor closed graph families abstract let c be a proper minor closed family of graphs we present a randomized algorithm that given a graph g c with n vertices finds a simple cycle of size k in g if exists in 2 o k n time the algorithm applies to both directed and undirected graphs in previous linear time algorithms for this problem the runtime dependence on k is super exponential the algorithm can be derandomized yielding a 2 o k n log n time algorithm

fixed parameter tractability definability and model checking in this article we study parameterized complexity theory from the perspective of logic or more specifically descriptive complexity theory we propose to consider parameterized model checking problems for various fragments of first order logic as generic parameterized problems and show how this approach can be useful in studying both fixed parameter tractability and intractability for example we establish the equivalence between the model checking for existential first order logic the homomorphism problem for relational structures and the substructure isomorphism problem our main tractability result shows that model checking for first order formulas is fixed parameter tractable when restricted to a class of input structures with an excluded minor on the intractability side for every t ge 0 we prove an equivalence between model checking for first order formulas with t quantifier alternations and the parameterized halting problem for alternating turing machines with t alternations we discuss the close connection between this alternation hierarchy and downey and fellows w hierarchy on a more abstract level we consider two forms of definability called fagin definability and slicewise definability that are appropriate for describing parameterized problems we give a characterization of the class fpt of all fixed parameter tractable problems in terms of slicewise definability in finite variable least fixed point logic which is reminiscent of the immerman vardi theorem characterizing the class ptime in terms of definability in least fixed point logic

finding a path of superlogarithmic length we consider the problem of finding a long simple path in an undirected graph we present a polynomial time algorithm that finds a path of length omega bigl log l log log l 2 bigr where l denotes the length of the longest simple path in the graph this establishes the performance ratio o n log log n log n 2 for the longest path problem where n denotes the number of vertices in the graph

on the compressibility of mathcal np instances and cryptographic applications we study compression that preserves the solution to an instance of a problem rather than preserving the instance itself our focus is on the compressibility of mathcal np decision problems we consider mathcal np problems that have long instances but relatively short witnesses the question is whether one can efficiently compress an instance and store a shorter representation that maintains the information of whether the original input is in the language or not we want the length of the compressed instance to be polynomial in the length of the witness and polylog in the length of original input such compression enables succinctly storing instances until a future setting will allow solving them either via a technological or algorithmic breakthrough or simply until enough time has elapsed in this paper we first develop the basic complexity theory of compression including reducibility completeness and a stratification of mathcal np with respect to compression we then show that compressibility say of sat would have vast implications for cryptography including constructions of one way functions and collision resistant hash functions from any hard on average problem in mathcal np and cryptanalysis of key agreement protocols in the bounded storage model when mixed with time complexity based cryptography