a polynomial time approximation algorithm for the permanent of a matrix with nonnegative entries we present a polynomial time randomized algorithm for estimating the permanent of an arbitrary n n matrix with nonnegative entries this algorithm technically a fully polynomial randomized approximation scheme computes an approximation that is with high probability within arbitrarily small specified relative error of the true value of the permanent

average case intractability vs worst case intractability we show that not all sets in np or other levels of the polynomial time hierarchy have efficient average case algorithms unless the arthur merlin classes ma and am can be derandomized to np and various subclasses of p poly collapse to p furthermore other complexity classes like p pp and pspace are shown to be intractable on average unless they are easy in the worst case

interactive proofs and the hardness of approximating cliques the contribution of this paper is two fold first a connection is established between approximating the size of the largest clique in a graph and multi prover interactive proofs second an efficient multi prover interactive proof for np languages is constructed where the verifier uses very few random bits and communication bits last the connection between cliques and efficient multi prover interaction proofs is shown to yield hardness results on the complexity of approximating the size of the largest clique in a graph of independent interest is our proof of correctness for the multilinearity test of functions

gap definability as a closure property gap definability and the gap closure operator were defined by s fenner l fortnow and s kurth j comput system sci 48 116 148 1994 few complexity classes were known at that time to be gap definable in this paper we give simple characterizations of both gap definability and the gap closure operator and we show that many complexity classes are gap definable includingp p p p1 pspace exp nexp mp middle bitp andbp p if a class is closed under union and intersection and contains and then it is gap definable if and only if it containsspp its gap closure is the closure of this class together withsppunder union and intersection on the other hand we give some examples of classes which are reasonable and gap definable but not closed under union resp intersection complement finally we show that a complexity class such aspspaceorpp if it is not equal tospp contains a maximal gap definable many one reduction closed subclass which is properly betweensppand the class of allpspace incomplete pp incomplete sets with respect to containment the gap closure of the class of all incomplete sets inpspace resp pp ispspace resp pp

proof verification and the hardness of approximation problems we show that every language in np has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a constant number of bits in the proof if a string is in the language then there exists a proof such that the verifier accepts with probability 1 i e for every choice of its random string for strings not in the language the verifier rejects every provided proof with probability at least 1 2 our result builds upon and improves a recent result of arora and safra 1998 whose verifiers examine a nonconstant number of bits in the proof though this number is a very slowly growing function of the input length as a consequence we prove that no max snp hard problem has a polynomial time approximation scheme unless np p the class max snp was defined by papadimitriou and yannakakis 1991 and hard problems for this class include vertex cover maximum satisfiability maximum cut metric tsp steiner trees and shortest superstring we also improve upon the clique hardness results of feige et al 1996 and arora and safra 1998 and show that there exists a positive e such that approximating the maximum clique size in an n vertex graph to within a factor of n e is np hard

the complexity of counting cuts and of computing the probability that a graph is connected several enumeration and reliability problems are shown to be p complete and hence at least as hard as np complete problems included are important problems in network reliability analysis namely computing the probability that a graph is connected and counting the number of minimum cardinality s t cuts or directed network cuts also shown to be p complete are counting vertex covers in a bipartite graph counting antichains in a partial order and approximating the probability that a graph is connected and the probability that a pair of vertices is connected

fuzzy vector quantization algorithms and their application in image compression this paper presents the development and evaluation of fuzzy vector quantization algorithms these algorithms are designed to achieve the quality of vector quantizers provided by sophisticated but computationally demanding approaches while capturing the advantages of the frequently used in practice k means algorithm such as speed simplicity and conceptual appeal the uncertainty typically associated with clustering tasks is formulated in this approach by allowing the assignment of each training vector to multiple clusters in the early stages of the iterative codebook design process a training vector assignment strategy is also proposed for the transition from the fuzzy mode where each training vector can be assigned to multiple clusters to the crisp mode where each training vector can be assigned to only one cluster such a strategy reduces the dependence of the resulting codebook on the random initial codebook selection the resulting algorithms are used in image compression based on vector quantization this application provides the basis for evaluating the computational efficiency of the proposed algorithms and comparing the quality of the resulting codebook design with that provided by competing techniques

block interpolation a framework for tight exponential time counting complexity we devise a framework for proving tight lower bounds under the counting exponential time hypothesis mathsf eth introduced by dell et al our framework allows to convert many known mathsf p hardness results for counting problems into results of the following type if the given problem admits an algorithm with running time 2 o n on graphs with n vertices and mathcal o n edges then mathsf eth fails as exemplary applications of this framework we obtain such tight lower bounds for the evaluation of the zero one permanent the matching polynomial and the tutte polynomial on all non easy points except for two lines

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

holomorphic solutions to functional differential equations abstract the pantograph equation is perhaps one of the most heavily studied class of functional differential equations owing to its numerous applications in mathematical physics biology and problems arising in industry this equation is characterized by a linear functional argument heard 1973 10 considered a generalization of this equation that included a nonlinear functional argument his work focussed on the asymptotic behaviour of solutions for a real variable x as x in this paper we revisit heard s equation but study it in the complex plane using results from complex dynamics we show that any nonconstant solution that is holomorphic at the origin must have the unit circle as a natural boundary we consider solutions that are holomorphic on the julia set of the nonlinear argument we show that the solutions are either constant or have a singularity at the origin there is a special case of heard s equation that includes only the derivative and the functional term for this case we construct solutions to the equation and illustrate the general results using classical complex analysis

the np completeness of some edge partition problems we show that for each fixed n geqq 3 it is np complete to determine whether an arbitrary graph can be edge partitioned into subgraphs isomorphic to the complete graph k n the np completeness of a number of other edge partition problems follows immediately

analyzing glauber dynamics by comparison of markov chains a popular technique for studying random properties of a combinatorial set is to design a markov chain monte carlo algorithm for many problems there are natural markov chains connecting the set of allowable configurations which are based on local moves or glauber dynamics typically these single site update algorithms are difficult to analyze so often the markov chain is modified to update several sites simultaneously recently there has been progress in analyzing these more complicated algorithms for several important combinatorial problems in this work we use the comparison technique of diaconis and saloff coste to show that several of the natural single point update algorithms are efficient the strategy is to relate the mixing rate of these algorithms to the corresponding nonlocal algorithms which have already been analyzed this allows us to give polynomial time bounds for single point update algorithms for problems such as generating planar tilings and random triangulations of convex polygons we also survey several other comparison techniques along with specific applications which have been used in the context of estimating mixing rates of markov chains

relations among mod classes abstract mod classes are those subclasses of p p 1 where the deterministic polynomial time computation consists only of checking whether the oracle answer for the input lies in a certain residue class modulo a fixed natural number k for the so defined classes modk p we prove that modk p modm p if k and m have the same prime divisors and as a consequence of this modk p is closed under union for all k

the complexity of bayesian networks specified by propositional and relational languages abstract we examine the complexity of inference in bayesian networks specified by logical languages we consider representations that range from fragments of propositional logic to function free first order logic with equality in doing so we cover a variety of plate models and of probabilistic relational models we study the complexity of inferences when network query and domain are the input the inferential and the combined complexity when the network is fixed and query and domain are the input the query data complexity and when the network and query are fixed and the domain is the input the domain complexity we draw connections with probabilistic databases and liftability results and obtain complexity classes that range from polynomial to exponential levels we identify new languages with tractable inference and we relate our results to languages based on plates and probabilistic relational models

approximate counting for complex weighted boolean constraint satisfaction problems constraint satisfaction problems or csps have been extensively studied in for instance artificial intelligence database theory graph theory and statistical physics from a practical viewpoint it is beneficial to approximately solve those csps when one tries to approximate the total number of truth assignments that satisfy all boolean valued constraints for unweighted boolean csps there is a known trichotomy theorem by which all such counting problems are neatly classified into exactly three categories under polynomial time randomized approximation preserving reductions in contrast we obtain a dichotomy theorem of approximate counting for complex weighted boolean csps provided that all complex valued unary constraints are freely available to use it is the expressive power of free unary constraints that enables us to prove such a stronger complete classification theorem this discovery makes a step forward in the quest for the approximation complexity classification of all counting csps to deal with complex weights we employ proof techniques of factorization and arity reduction along the line of solving holant problems moreover we introduce a novel notion of t constructibility that naturally induces approximation preserving reducibility our result also gives an approximation analogue of the dichotomy theorem on the complexity of exact counting for complex weighted boolean csps

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

maximum independent set on diluted triangular lattices core percolation and maximum independent set on random graphs have recently been characterized using the methods of statistical physics here we present a statistical physics study of these problems on bond diluted triangular lattices core percolation critical behavior is found to be consistent with the standard percolation values though there are strong finite size effects a transfer matrix method is developed and applied to find accurate values of the density and degeneracy of the maximum independent set on lattices of limited width but large length an extrapolation of these results to the infinite lattice limit yields high precision results which are tabulated these results are compared to results found using both vertex based and edge based local probability recursion algorithms which have proven useful in the analysis of hard computational problems such as the satisfiability problem

partition functions on k regular graphs with 0 1 vertex assignments and real edge functions we prove a complexity dichotomy theorem for a class of partition functions over k regular graphs for any fixed k these problems can be viewed as graph homomorphisms from an arbitrary k regular input graph g to the weighted two vertex graph on 0 1 defined by a real valued symmetric function h we completely classify the computational complexity of this problem we show that there are exactly the following alternatives for any given h depending on h over k regular graphs either 1 the problem is p hard even for planar graphs 2 the problem is p hard for general non planar graphs but solvable in polynomial time for planar graphs or 3 the problem is solvable in polynomial time for general graphs the dependence on h is an explicit criterion furthermore we show that in case 2 the problem is solvable in polynomial time over k regular planar graphs by exactly the theory of holographic algorithms using matchgates

quantum and classical complexity classes separations collapses and closure properties we study the complexity of quantum complexity classes such as eqp bqp and nqp quantum analogs of p bpp and np respectively using classical complexity classes such as zpp wpp and c p the contributions of this paper are threefold first via oracle constructions we show that no relativizable proof technique can improve the best known classical upper bound for bqp bqp awpp journal of computer and system sciences 59 2 1999 240 to bqp wpp and the best known classical lower bound for eqp p eqp to zpp eqp second we prove that there are oracles a and b such that relative to a corp is immune to nqp and relative to b bqp is immune to pc p extending a result of de graaf and valiant technical report quant ph 0211179 quantum physics 2002 we construct a relativized world where eqp is immune to modpkp third motivated by the fact that counting classes e g lwpp awpp etc are the best known classical upper bounds on quantum complexity classes we study properties of these counting classes we prove that wpp is closed under polynomial time truth table reductions while we construct an oracle relative to which wpp is not closed under polynomial time turing reductions the latter result implies that proving the equality of the similar appearing classes lwpp and wpp would require nonrelativizable proof techniques we also prove that both awpp and app are closed under tup reductions we use closure properties of wpp and awpp to prove interesting consequences in terms of the complexity of the polynomial hierarchy of the following hypotheses nqp bqp and eqp nqp

spin systems on k regular graphs with complex edge functions let k 1 be an integer and let h h 0 0 h 0 1 h 1 0 h 1 1 be a complex valued symmetric function on domain 0 1 i e where h 0 1 h 1 0 we introduce a new technique called a syzygy and prove a dichotomy theorem for the following class of problems specified by k and h given an arbitrary k regular graph g v e where the function h is attached to each edge compute z g email protected s v 0 1 u v eh s u s v z is known as the partition function of the spin system also known as counting graph homomorphisms on domain size two and is a special case of holant problems the dichotomy theorem gives a complete classification of the computational complexity of this problem depending on k and h the dependence on k and h is explicit

set partitioning via inclusion exclusion given a set n with n elements and a family mathcal f of subsets we show how to partition n into k such subsets in 2 n n o 1 time we also consider variations of this problem where the subsets may overlap or are weighted and we solve the decision counting summation and optimization versions of these problems our algorithms are based on the principle of inclusion exclusion and the zeta transform in effect we get exact algorithms in 2 n n o 1 time for several well studied partition problems including domatic number chromatic number maximum k cut bin packing list coloring and the chromatic polynomial we also have applications to bayesian learning with decision graphs and to model based data clustering if only polynomial space is available our algorithms run in time 3 n n o 1 if membership in mathcal f can be decided in polynomial time we solve chromatic number in o 2 2461 n time and domatic number in o 2 8718 n time finally we present a family of polynomial space approximation algorithms that find a number between chi g and lceil 1 epsilon chi g rceil in time o 1 2209 n 2 2461 e epsilon n

the complexity of the counting constraint satisfaction problem the counting constraint satisfaction problem csp h over a finite relational structure h can be expressed as follows given a relational structure g over the same vocabulary determine the number of homomorphisms from g to h in this article we characterize relational structures h for which csp h can be solved in polynomial time and prove that for all other structures the problem is p complete

vector quantization of image subbands a survey subband and wavelet decompositions are powerful tools in image coding because of their decorrelating effects on image pixels the concentration of energy in a few coefficients their multirate multiresolution framework and their frequency splitting which allows for efficient coding matched to the statistics of each frequency band and to the characteristics of the human visual system vector quantization vq provides a means of converting the decomposed signal into bits in a manner that takes advantage of remaining inter and intraband correlation as well as of the more flexible partitions of higher dimensional vector spaces since 1988 a growing body of research has examined the use of vq for subband wavelet transform coefficients we present a survey of these methods

approximating the permanent of graphs with large factors let g u v e be a bipartite graph with u v n the factor size of g f is the maximum number of edge disjoint perfect matchings in g we characterize the complexity of counting the number of perfect matchings in classes of graphs parameterized by factor size we describe the simple algorithm which is an approximation algorithm for the permanent that is a natural simplification of the algorithm suggested by broder 1986 and analyzed by jerrum and sinclair 1988a b compared to the algorithm by jerrum and sinclair 1988a b the simple algorithm achieves a polynomial speed up in the running time to compute the permanent a combinatorial lemma is used to prove that the simple algorithm runs in time no n f thus 1 for all constants 0 the simple algorithm runs in polynomial time for graphs with factor size at least n 2 for some constant c the simple algorithm is the fastest known approximation for graphs with factor size at least c log n compare with the approximation algorithms described in karmarkar 1988 we prove the following complementary hardness results for functions f such that 3 f n n 3 the exact counting problem for f n regular bipartite graphs is p complete for and e 0 for any function f such that 3 f n n1 e approximate counting for f n regular bipartite graphs is as hard as approximate counting for all bipartite graphs an announcement of these results appears in dagum 1988

an update on minc s survey of open problems involving permanents abstract we summarise the progress which has been made since 1986 on the conjectures and open problems listed in h minc s survey articles on the theory of permanents

combining fractal image compression and vector quantization in fractal image compression the code is an efficient binary representation of a contractive mapping whose unique fixed point approximates the original image the mapping is typically composed of affine transformations each approximating a block of the image by another block called domain block selected from the same image the search for a suitable domain block is time consuming moreover the rate distortion performance of most fractal image coders is not satisfactory we show how a few fixed vectors designed from a set of training images by a clustering algorithm accelerates the search for the domain blocks and improves both the rate distortion performance and the decoding speed of a pure fractal coder when they are used as a supplementary vector quantization codebook we implemented two quadtree based schemes a fast top down heuristic technique and one optimized with a lagrange multiplier method for the 8 bits per pixel bpp luminance part of the 512 spl kappa 512 lena image our best scheme achieved a peak signal to noise ratio of 32 50 db at 0 25 bpp

counting trees in a phylogenetic network is p complete answering a problem posed by nakhleh we prove that counting the number of phylogenetic trees inferred by a binary phylogenetic network is p complete an immediate consequence of this result is that counting the number of phylogenetic trees commonly inferred by two binary phylogenetic networks is also p complete

the computational complexity of tutte invariants for planar graphs for each pair of algebraic numbers x y the complexity of computing the tutte polynomial t g x y of a planar graph g is determined this computation is found to be overline rm p complete except when x 1 y 1 1 2 or when x y is one of 1 1 1 1 j j 2 or j 2 j where j e 2 pi i 3 in which case it is polynomial time computable a corollary gives the computational complexity of various enumeration problems for planar graphs

design of vector quantizer for image compression using self organizing feature map and surface fitting we propose a new scheme of designing a vector quantizer for image compression first a set of codevectors is generated using the self organizing feature map algorithm then the set of blocks associated with each code vector is modeled by a cubic surface for better perceptual fidelity of the reconstructed images mean removed vectors from a set of training images is used for the construction of a generic codebook further huffman coding of the indices generated by the encoder and the difference coded mean values of the blocks are used to achieve better compression ratio we proposed two indices for quantitative assessment of the psychovisual quality blocking effect of the reconstructed image our experiments on several training and test images demonstrate that the proposed scheme can produce reconstructed images of good quality while achieving compression at low bit rates

approximating the partition function of the ferromagnetic potts model we provide evidence that it is computationally difficult to approximate the partition function of the ferromagnetic q state potts model when q 2 specifically we show that the partition function is hard for the complexity class rhpi under approximation preserving reducibility thus it is as hard to approximate the partition function as it is to find approximate solutions to a wide range of counting problems including that of determining the number of independent sets in a bipartite graph our proof exploits the first order phase transition of the random cluster model which is a probability distribution on graphs that is closely related to the q state potts model

the complexity of reliability computations in planar and acyclic graphs we show that the problem of computing source sink reliability is np hard in fact p complete even for undirected and acyclic directed source sink planar graphs having vertex degree at most three thus the source sink reliability problem is unlikely to have an efficient algorithm even when the graph can be laid out on a rectilinear grid

on the hardness of approximate reasoning abstract many ai problems when formalized reduce to evaluating the probability that a propositional expression is true in this paper we show that this problem is computationally intractable even in surprisingly restricted cases and even if we settle for an approximation to this probability we consider various methods used in approximate reasoning such as computing degree of belief and bayesian belief networks as well as reasoning techniques such as constraint satisfaction and knowledge compilation that use approximation to avoid computational difficulties and reduce them to model counting problems over a propositional domain we prove that counting satisfying assignments of propositional languages is intractable even for horn and monotone formulae and even when the size of clauses and number of occurrences of the variables are extremely limited this should be contrasted with the case of deductive reasoning where horn theories and theories with binary clauses are distinguished by the existence of linear time satisfiability algorithms what is even more surprising is that as we show even approximating the number of satisfying assignments i e approximating approximate reasoning is intractable for most of these restricted theories we also identify some restricted classes of propositional formulae for which efficient algorithms for counting satisfying assignments can be given

dempster s rule of combination is p complete research note abstract we consider the complexity of combining bodies of evidence according to the rules of the dempster shafer theory of evidence we prove that given as input a set of tables representing basic probability assignments m 1 m n over a frame of discernment and a set a the problem of computing the combined basic probability value m 1 m n a is p complete as a corollary we obtain that while the simple belief plausibility and commonality values bel a pl a and q a can be computed in polynomial time the problems of computing the combinations bel 1 bel n a pl 1 pl n a and q 1 q n a are p complete

bound entanglement in tree graphs in this paper we discuss the entanglement properties of graph diagonal states with particular emphasis on calculating the threshold for the transition between the presence and absence of entanglement i e the separability point special consideration is made of the thermal states of trees including the linear cluster state we characterize the type of entanglement present and describe the optimal entanglement witnesses and their implementation on a quantum computer up to an additive approximation in the case of general graphs we invoke a relation with the partition function of the classical ising model thereby intimating a connection to computational complexity theoretic tasks finally we show that the entanglement is robust to some classes of local perturbations

computational aspects of mining maximal frequent patterns in this paper we study the complexity theoretic aspects of mining maximal frequent patterns from the perspective of counting the number of all distinct solutions we present the first formal proof that the problem of counting the number of maximal frequent itemsets in a database of transactions given an arbitrary support threshold is p complete thereby providing theoretical evidence that the problem of mining maximal frequent itemsets is np hard we also extend our complexity analysis to other similar data mining problems that deal with complex data structures such as sequences trees and graphs we investigate several variants of these mining problems in which the patterns of interest are subsequences subtrees or subgraphs and show that the associated problems of counting the number of maximal frequent patterns are all either p complete or p hard

tally np sets and easy census functions abstract we study the question of whether every p set has an easy i e polynomial time computable census function we characterize this question in terms of unlikely collapses of language and function classes such as p1 fp where p1 is the class of functions that count the witnesses for tally np sets we prove that every p1ph function can be computed in fp p1 p1 consequently every p set has an easy census function if and only if every set in the polynomial hierarchy does we show that the assumption p1 fp implies p bpp and ph modk p for each k 2 we also relate a set s property of having an easy census function to other well studied properties of sets such as rankability and scalability the closure of the rankable sets under p isomorphisms finally we prove that it is no more likely that the census function of any set in p can be approximated more precisely can be n enumerated in time n for fixed and than that it can be precisely computed in polynomial time

exponential decay for the fragmentation or cell division equation abstract we consider a classical integro differential equation that arises in various applications as a model for cell division or fragmentation in biology it describes the evolution of the density of cells that grow and divide we prove the existence of a stable steady distribution first positive eigenvector under general assumptions in the variable coefficients case we also prove the exponential convergence for large times of solutions toward such a steady state

the complexity of counting self avoiding walks in subgraphs of two dimensional grids and hypercubes valiant siam j comput 8 1979 410 421 showed that the problem of computing the number of simple s t paths in graphs is p complete both in the case of directed graphs and in the case of undirected graphs welsh complexity knots colourings and counting cambridge university press cambridge 1993 p 17 asked whether the problem of computing the number of self avoiding walks of a given length in the complete two dimensional grid is complete for p1 the tally version of p this paper offers a partial answer to the question of welsh it is p complete to compute the number of self avoiding walks of a given length in a subgraph of a two dimensional grid several variations of the problem are also studied and shown to be p complete this paper also studies the problem of computing the number of self avoiding walks in a subgraph of a hypercube similar completeness results are shown for the problem by scaling the computation time to exponential it is shown that computing the number of self avoiding walks in hypercubes is a complete problem for exp in the case when a subgraph of a hypercube is specified by its dimension and a boolean circuit that accepts the nodes finally this paper studies the complexity of testing whether a given word over the four letter alphabet u d l r represents a self avoiding walk in a two dimensional grid a linear space lower bound is shown for nondeterministic turing machines with a 1 way input head to make this test

algorithms for four variants of the exact satisfiability problem we present four polynomial space and exponential time algorithms for variants of the exact satisfiability problem first an o 1 1120n where n is the number of variables time algorithm for the np complete decision problem of exact 3 satisfiability and then an o 1 1907n time algorithm for the general decision problem of exact satisfiability the best previous algorithms run in o 1 1193n and o 1 2299n time respectively for the p complete problem of counting the number of models for exact 3 satisfiability we present an o 1 1487n time algorithm we also present an o 1 2190n time algorithm for the general problem of counting the number of models for exact satisfiability presenting a simple reduction we show how this algorithm can be used for computing the permanent of a 0 1 matrix

rank maximal matchings structure and algorithms abstract let g a p e be a bipartite graph where a denotes a set of applicants p denotes a set of posts and ranks on the edges denote preferences of the agents over posts a matching m in g is rank maximal if it matches the maximum number of applicants to their top rank post subject to this the maximum number of applicants to their second rank post and so on in this paper we develop a switching graph characterization of rank maximal matchings which is a useful tool that encodes all rank maximal matchings in an instance the characterization leads to simple and efficient algorithms for several interesting problems in particular we give an efficient algorithm to compute the set of rank maximal pairs in an instance we show that the problem of counting the number of rank maximal matchings is p complete and also give an fpras for the problem finally we consider the problem of deciding whether a rank maximal matching is popular among all the rank maximal matchings in a given instance and give an efficient algorithm for the problem

segmentation based compression for graylevel images abstract this paper presents a segmentation based lossy image compression slic algorithm the segmentation scheme biswas and pal pattern recog lett 21 2000 entropy based and hierarchical in nature provides sub images of homogeneous regions the compression algorithm encodes a graylevel image through global approximations of sub images by 2 d bezier bernstein polynomial along with corrections if needed over regions in sub images by local approximation contours by 1 d bezier bernstein polynomial and texture if present by huffman coding scheme using hilbert scan on texture blocks order of the 2 d polynomials has been computed with the help of an image quality index iqi the proposed compression algorithm also examines the compression result by encoding contours through their approximation based on stretching of discrete circular arcs stretching is done by affine transformation compression results in both the cases have been compared with jpeg results attempts have been made to evaluate the quality of reconstructed images through a fidelity vector whose components are different objective measures

inapproximability of the tutte polynomial the tutte polynomial of a graph g is a two variable polynomial t g x y that encodes many interesting properties of the graph we study the complexity of the following problem for rationals x and y take as input a graph g and output a value which is a good approximation to t g x y jaeger et al have completely mapped the complexity of exactly computing the tutte polynomial they have shown that this is p hard except along the hyperbola x 1 y 1 1 and at four special points we are interested in determining for which points x y there is a fully polynomial randomised approximation scheme fpras for t g x y under the assumption rp np we prove that there is no fpras at x y if x y is in one of the half planes x np there is no fpras at the point x y 0 email protected when l 2 is a positive integer thus there is no fpras for counting nowhere zero l flows for l 2 this is an interesting consequence of our work since the corresponding decision problem is in p for example for l 6 although our main concern is to distinguish regions of the tutte plane that admit an fpras from those that do not we also note that the latter regions exhibit different levels of intractability at certain points x y for example the integer points on the x axis or any point in the positive quadrant there is a randomised approximation scheme for t g x y that runs in polynomial time using an oracle for an np predicate on the other hand we identify a region of points x y at which even approximating t g x y is as hard as p

sublinear time algorithms for monomer dimer systems on bounded degree graphs for a graph g let z g be the partition function of the monomer dimer system defined by k m k g k where m k g is the number of matchings of size k in g we consider graphs of bounded degree and develop a sublinear time algorithm for estimating logz g at an arbitrary value 0 within additive error en with high probability the query complexity of our algorithm does not depend on the size of g and is polynomial in 1 e and we also provide a lower bound quadratic in 1 e for this problem this is the first analysis of a sublinear time approximation algorithm for a p complete problem our approach is based on the correlation decay of the gibbs distribution associated with z g we show that our algorithm approximates the probability for a vertex to be covered by a matching sampled according to this gibbs distribution in a near optimal sublinear time we extend our results to approximate the average size and the entropy of such a matching within an additive error with high probability where again the query complexity is polynomial in 1 e and the lower bound is quadratic in 1 e our algorithms are simple to implement and of practical use when dealing with massive datasets our results extend to other systems where the correlation decay is known to hold as for the independent set problem up to the critical activity

monte carlo methods for the ferromagnetic potts model using factor graph duality normal factor graph duality offers new possibilities for monte carlo algorithms in graphical models specifically we consider the problem of estimating the partition function of the ferromagnetic ising and potts models by monte carlo methods which are known to work well at high temperatures but to fail at low temperatures we propose monte carlo methods uniform sampling and importance sampling in the dual normal factor graph and demonstrate that they behave differently they work particularly well at low temperatures by comparing the relative error in estimating the partition function we show that the proposed importance sampling algorithm significantly outperforms the state of the art deterministic and monte carlo methods for the ferromagnetic ising model in an external field we show the equivalence between the valid configurations in the dual normal factor graph and the terms that appear in the high temperature series expansion of the partition function following this result we discuss connections with jerrum sinclair s polynomial randomized approximation scheme the subgraphs world process for evaluating the partition function of ferromagnetic ising models

predictive vector quantization with intrablock prediction support region a predictive vector quantization pvq structure is proposed where the encoder uses a predictor based on an intrablock support region followed by a modified vector quantizer stage simulation results show that a modification on a previously published pvq system led to an improvement of 1 db in psnr for lenna

counting models for 2 sat and 3 sat formulae we here present algorithms for counting models and max weight models for 2sat and 3sat formulae they use polynomial space and run in o 1 2561n and o 1 6737n time respectively where n is the number of variables this is faster than the previously best algorithms for counting nonweighted models for 2sat and 3sat which run in o 1 3247n and o 1 6894n time respectively in order to prove these time bounds we develop new measures of formula complexity allowing us to conveniently analyze the effects of certain factors with a large impact on the total running time we also provide an algorithm for the restricted case of separable 2sat formulae with fast running times for well studied input classes for all three algorithms we present interesting applications such as computing the permanent of sparse 0 1 matrices

approximating the permanent a randomised approximation scheme for the permanent of a 0 1s presented the task of estimating a permanent is reduced to that of almost uniformly generating perfect matchings in a graph the latter is accomplished by simulating a markov chain whose states are the matchings in the graph for a wide class of 0 1 matrices the approximation scheme is fully polynomial i e runs in time polynomial in the size of the matrix and a parameter that controls the accuracy of the output this class includes all dense matrices those that contain sufficiently many 1 s and almost all sparse matrices in some reasonable probabilistic model for 0 1 matrices of given density for the approach sketched above to be computationally efficient the markov chain must be rapidly mixing informally it must converge in a short time to its stationary distribution a major portion of the paper is devoted to demonstrating that the matchings chain is rapidly mixing apparently the first such result for a markov chain with genuinely c

holographic algorithms by fibonacci gates abstract we introduce fibonacci gates as a polynomial time computable primitive and develop a theory of holographic algorithms based on these gates the fibonacci gates play the role of matchgates in valiant s theory valiant 2008 19 they give rise to polynomial time computable counting problems on general graphs while matchgates mainly work over planar graphs only we develop a signature theory and characterize all realizable signatures for fibonacci gates for bases of arbitrary dimensions we prove a basis collapse theorem we apply this theory to give new polynomial time algorithms for certain counting problems we also use this framework to prove that some slight variations of these counting problems are p hard holographic algorithms with fibonacci gates prove to be useful as a general tool for classification results of counting problems dichotomy theorems cai et al 2009 7

the complexity of counting in sparse regular and planar graphs we show that a number of graph theoretic counting problems remain cal np hard indeed cal p complete in very restricted classes of graphs in particular we prove that the problems of counting matchings vertex covers independent sets and extremal variants of these all remain hard when restricted to planar bipartite graphs of bounded degree or regular graphs of constant degree we obtain corollaries about counting cliques in restricted classes of graphs and counting satisfying assignments to restricted classes of monotone 2 cnf formulae to achieve these results a new interpolation based reduction technique which preserves properties such as constant degree is introduced

geometric sets of low information content abstract in this paper we study language classes defined by nonuniform families of hyperplanes and halfspaces these are subclasses of p poly upto many one equivalence these classes are essentially the classes elt1 and lt1 of languages accepted by nonuniform families of polynomial size depth 1 circuits with a weighted exact threshold gate at the root respectively weighted threshold gate at the root we investigate the consequences of intractable sets being reducible to sets in elt1 and lt1 using the polynomial time algorithm for linear programming and ideas from the theory of convex polytopes we prove that every disjunctively self reducible bd cylinder that is many one reducible to a set in elt1 or lt1 is already in p as a consequence we derive that if a complete set for any class k np mod k p pp c p many one reduces to a set in elt1 or lt1 then k p for families of hyperplanes over finite fields we prove that every turing self reducible set that is many one reducible to a set defined by such a family is in p and every word decreasing self reducible set that is many one reducible to a set defined by such a family is in np co np furthermore as an interesting connection it turns out that several sparse and tally reduction classes are many one reducible to elt1 or lt1 thus our results concerning reductions to elt1 and lt1 subsume the strongest existing collapse results concerning reductions of any class k np mod k p pp c p to sparse sets for various reducibilities

solutions to an advanced functional partial differential equation of the pantograph type a model for cells structured by size undergoing growth and division leads to an initial boundary value problem that involves a first order linear partial differential equation with a functional term here size can be interpreted as dna content or mass it has been observed experimentally and shown analytically that solutions for arbitrary initial cell distributions are asymptotic as time goes to infinity to a certain solution called the steady size distribution the full solution to the problem for arbitrary initial distributions however is elusive owing to the presence of the functional term and the paucity of solution techniques for such problems in this paper we derive a solution to the problem for arbitrary initial cell distributions the method employed exploits the hyperbolic character of the underlying differential operator and the advanced nature of the functional argument to reduce the problem to a sequence of simple cauchy problems the existence of solutions for arbitrary initial distributions is established along with uniqueness the asymptotic relationship with the steady size distribution is established and because the solution is known explicitly higher order terms in the asymptotics can be readily obtained

random generation of combinatorial structures from a uniform distribution abstract the class of problems involving the random generation of combinatorial structures from a uniform distribution is considered uniform generation problems are in computational difficulty intermediate between classical existence and counting problems it is shown that exactly uniform generation of efficiently verifiable combinatorial structures is reducible to approximate counting and hence is within the third level of the polynomial hierarchy natural combinatorial problems are presented which exhibit complexity gaps between their existence and generation and between their generation and counting versions it is further shown that for self reducible problems almost uniform generation and randomized approximate counting are inter reducible and hence of similar complexity

conditional entropy constrained residual vq with application to image coding this paper introduces an extension of entropy constrained residual vector quantization vq where intervector dependencies are exploited the method which we call conditional entropy constrained residual vq employs a high order entropy conditioning strategy that captures local information in the neighboring vectors when applied to coding images the proposed method is shown to achieve better rate distortion performance than that of entropy constrained residual vector quantization with less computational complexity and lower memory requirements moreover it can be designed to support progressive transmission in a natural way it is also shown to outperform some of the best predictive and finite state vq techniques reported in the literature this is due partly to the joint optimization between the residual vector quantizer and a high order conditional entropy coder as well as the efficiency of the multistage residual vq structure and the dynamic nature of the prediction

polynomial time approximation algorithms for the ising model the paper presents a randomised algorithm which evaluates the partition function of an arbitrary ferromagnetic ising system to any specified degree of accuracy the running time of the algorithm in

two dimensional monomer dimer systems are computationally intractable the classic problem of counting monomer dimer arrangements on a two dimensional lattice is analyzed using techniques from theoretical computer science under a certain assumption made precise in the text it can be shown that the general problem is computationally intractable this negative result contrasts with the special case of a system with monomer density zero for which efficient solutions have been known for some time a second much easier result obtained under the same assumption is that the partition function of a three dimensional ising system is computationally intractable again the negative result contrasts with known efficient techniques for evaluating the partition function of a two dimensional system

the complexity of computing the sign of the tutte polynomial we study the complexity of computing the sign of the tutte polynomial of a graph as there are only three possible outcomes positive negative and zero this seems at first sight more like a decision problem than a counting problem surprisingly however there are large regions of the parameter space for which computing the sign of the tutte polynomial is actually p hard as a trivial consequence approximating the polynomial is also p hard in this case thus approximately evaluating the tutte polynomial in these regions is as hard as exactly counting the satisfying assignments to a cnf boolean formula for most other points in the parameter space we show that computing the sign of the polynomial is in fp whereas approximating the polynomial can be done in polynomial time with an np oracle as a special case we completely resolve the complexity of computing the sign of the chromatic polynomial this is easily computable at q 2 and when q leq 32 27 and is np hard to compute for all other va

predecessor existence problems for finite discrete dynamical systems we study the predecessor existence problem for finite discrete dynamical systems given a finite discrete dynamical system s and a configuration c the predecessor existence or pre problem is to determine whether there is a configuration c such that s has a transition from c to c in addition to the decision version we also study the following variants the predecessor existence or pre problem counting the number of predecessors the unique predecessor existence or upre problem deciding whether there is a unique predecessor and the ambiguous predecessor existence or apre problem given a configuration c and a predecessor c of c deciding whether there is a different predecessor c of c general techniques are presented for simultaneously characterizing the computational complexity of the pre problem and its three variants our hardness results are based on the concept of simultaneous reductions single transformations that can be used to simultaneously prove the hardness of the different variants of the pre problem for their respective complexity classes our easiness results are based on dynamic programming and they extend the previous results on pre problem for one dimensional cellular automata the hardness results together with the easiness results provide a tight separation between easy and hard instances further the results imply similar bounds for other classes of finite discrete dynamical systems including discrete hopfield and recurrent neural networks concurrent state machines systolic networks and one and two dimensional cellular automata our results extend the earlier results of green sutner and orponen on the complexity of the predecessor existence problem and its variants

gradient algorithms for designing predictive vector quantizers a predictive vector quantizer pvq is a vector extension of a predictive quantizer it consists of two parts a conventional memoryless vector quantizer vq and a vector predictor two gradient algorithms for designing a pvq are developed in this paper the steepest descent sd algorithm and the stochastic gradient sg algorithm both have the property of improving the quantizer and the predictor in the sense of minimizing the distortion as measured by the average mean squared error the differences between the two design approaches are the period and the step size used in each iteration to update the codebook and predictor the sg algorithm updates once for each input training vector and uses a small step size while the sd updates only once for a long period possibly one pass over the entire training sequence and uses a relatively large step size code designs and tests are simulated for both gauss markov sources and for sampled speech waveforms and the results are compared to codes designed using techniques that attempt to optimize only the quantizer for the predictor and not vice versa

on unapproximable versions of np complete problems we prove that all of karp s 21 original np complete problems have a version that is hard to approximate these versions are obtained from the original problems by adding essentially the same simple constraint we further show that these problems are absurdly hard to approximate in fact no polynomial time algorithm can even approximate log k of the magnitude of these problems to within any constant factor where logk denotes the logarithm iterated k times unless np is recognized by slightly superpolynomial randomized machines we use the same technique to improve the constant epsilon such that max clique is hard to approximate to within a factor of n epsilon finally we show that it is even harder to approximate two counting problems counting the number of satisfying assignments to a monotone 2sat formula and computing the permanent of 1 0 1 matrices

a complete dichotomy rises from the capture of vanishing signatures we prove a complexity dichotomy theorem for holant problems over an arbitrary set of complex valued symmetric constraint functions mathcal f on boolean variables this extends and unifies all previous dichotomies for holant problems on symmetric constraint functions taking values without a finite modulus we define and characterize all symmetric vanishing signatures they turn out to be essential to the complete classification of holant problems the dichotomy theorem has an explicit tractability criterion expressible in terms of holographic transformations a holant problem defined by a set of constraint functions mathcal f is solvable in polynomial time if it satisfies this tractability criterion and is p hard otherwise the tractability criterion can be intuitively stated as follows a set mathcal f is tractable if 1 every function in mathcal f has arity at most two or 2 mathcal f is transformable to an affine type or 3 mathcal f is transformable to a product type o

on approximation algorithms for p the theme of this paper is to investigate to what extent approximation possibly together with randomization can reduce the complexity of problems in valiant s class p in general any function in p can be approximated to within any constant factor by a function in the class delta 3 p of the polynomial time hierarchy relative to a particular oracle delta 3 p cannot be replaced by delta 2 p in this result another part of the paper introduces a model of random sampling where the size of a set x is estimated by checking for various sample sets s whether or not s intersects x for various classes of sample sets upper and lower bounds on the number of samples required to estimate the size of x are discussed this type of sampling is motivated by particular problems in p such as computing the size of a backtrack search tree in the case of backtrack search trees a sample amounts to checking whether a certain path exists in the tree one of the lower bounds suggests that such tests

image coding using vector quantization a review a review of vector quantization techniques used for encoding digital images is presented first the concept of vector quantization is introduced then its application to digital images is explained spatial predictive transform hybrid binary and subband vector quantizers are reviewed the emphasis is on the usefulness of the vector quantization when it is combined with conventional image coding techniques or when it is used in different domains

the complexity of planar counting problems we prove the p hardness of the counting problems associated with various satisfiability graph and combinatorial problems when restricted to planar instances these problems include 3sat 1 3sat 1 ex3sat minimum vertex cover minimum dominating set minimum feedback vertex set x3c partition into triangles and clique cover we also prove the np completeness of the ambiguous satisfiability problems j b saxe two papers on graph embedding problems tech report cmu cs 80 102 dept of computer science carnegie mellon univ pittsburgh pa 1980 and the dp completeness with respect to random polynomial reducibility of the unique satisfiability problems l g valiant and v v vazirani np is as easy as detecting unique solutions in proc 17th acm symp on theory of computing 1985 pp 458 463 associated with several of the above problems when restricted to planar instances previously very few p hardness results no sf np hardness results and no dp completeness results were known for counting problems ambiguous satisfiability problems and unique satisfiability problems respectively when restricted to planar instances assuming sf p neq np one corollary of the above results is that there are no epsilon approximation algorithms for the problems of maximizing or minimizing a linear objective function subject to a planar system of linear inequality constraints over the integers

natural boundaries for solutions to a certain class of functional differential equations this paper is concerned with a generalization of a functional differential equation known as the pantograph equation the pantograph equation contains a linear functional argument in this paper we generalize this functional argument to include nonlinear polynomials in contrast to the entire solutions generated by the pantograph equation for the retarded case we show that in the nonlinear case natural boundaries occur these boundaries are linked to the julia set of the polynomial functional argument

dichotomy results for fixed point counting in boolean dynamical systems we present dichotomy theorems regarding the computational complexity of counting fixed points in boolean discrete dynamical systems i e finite discrete dynamical systems over the domain 0 1 for a class f of boolean functions and a class g of graphs an f g system is a boolean dynamical system with local transitions functions lying in f and graphs in g we show that if local transition functions are given by lookup tables then the following complexity classification holds let f be a class of boolean functions closed under superposition and let g be a graph class closed under taking minors if f contains all min functions all max functions or all self dual and monotone functions and g contains all planar graphs then it is p complete to compute the number of fixed points in an f g system otherwise it is computable in polynomial time we also prove a dichotomy theorem for the case that local transition functions are given by formulas over logical bases this theorem has a significantly more complicated structure than the theorem for lookup tables a corresponding theorem for boolean circuits coincides with the theorem for formulas

a computational proof of complexity of some restricted counting problems we explore a computational approach to proving the intractability of certain counting problems these problems can be described in various ways and they include concrete problems such as counting the number of vertex covers or independent sets for 3 regular graphs the high level principle of our approach is algebraic which provides sufficient conditions for interpolation to succeed another algebraic component is holographic reductions we then analyze in detail polynomial maps on r 2 induced by some combinatorial constructions these maps define sufficiently complicated dynamics of r 2 that we can only analyze them computationally in this paper we use both numerical computation as intuitive guidance and symbolic computation as proof theoretic verification to derive that a certain collection of combinatorial constructions in myriad combinations fulfills the algebraic requirements of proving p hardness the final result is a dichotomy theorem for a class of counting problems this includes a class of generic holant problems with an arbitrary real valued edge signature over 2 3 regular undirected graphs in particular it includes all partition functions with 0 1 vertex assignments and an arbitrary real valued edge function over all 3 regular undirected graphs

free bits pcps and nonapproximability towards tight results this paper continues the investigation of the connection between probabilistically checkable proofs pcps and the approximability of np optimization problems the emphasis is on proving tight nonapproximability results via consideration of measures such as the free bit complexity and the amortized free bit complexity of proof systems the first part of the paper presents a collection of new proof systems based on a new error correcting code called the long code we provide a proof system that has amortized free bit complexity of 2 epsilon implying that approximating maxclique within n frac13 e and approximating the chromatic number within n frac15 e are hard assuming np neq corp for any e 0 we also derive the first explicit and reasonable constant hardness factors for min vertex cover msat 2 and max cut and we improve the hardness factor for msat 3 we note that our nonapproximability factors for maxsnp problems are appreciably close to the values known to be achievable by polynomial time algorithms finally we note a general approach to the derivation of strong nonapproximability results under which the problem reduces to the construction of certain gadgets the increasing strength of nonapproximability results obtained via the pcp connection motivates us to ask how far this can go and whether pcps are inherent in any way the second part of the paper addresses this the main result is a reversal of the connection due to feige et al fglss connection j acm 43 1996 pp 268 292 where the latter had shown how to translate proof systems for np into np hardness of approximation results for maxclique we show how any np hardness of approximation result for maxclique yields a proof system for np roughly our result says that for any constant f if maxclique is np hard to approximate within n 1 1 f then np subseteq overline fpcp log f the latter being the class of languages possessing proofs of logarithmic randomness and amortized free bit complexity f this suggests that pcps are inherent to obtaining nonapproximability results furthermore the tight relation suggests that reducing the amortized free bit complexity is necessary for improving the nonapproximability results for maxclique the third part of our paper initiates a systematic investigation of the properties of pcp and fpcp free pcp as a function of the following various parameters randomness query complexity free bit complexity amortized free bit complexity proof size etc we are particularly interested in triviality results which indicate which classes are not powerful enough to capture np we also distill the role of randomized reductions in this area and provide a variety of useful transformations between proof checking complexity classes

the pantograph equation in the complex plane abstract the subject matter of this paper focuses on two functional differential equations with complex lag functions we address ourselves to the existence and uniqueness of solutions and to their asymptotic behaviour