the entropy theorem via coding bounds the entropy theorem also known as the shannon mcmillan breiman theory or the asymptotic equipartition theorem asserts that for a stationary ergodic finite alphabet process the sequence 1 n log p x sub 1 sup n converges almost surely to the entropy rate h of the process the entropy theorem has been used to establish asymptotic bounds on the performance of noiseless codes here the coding theorems are established without using the entropy theorem and the coding theorems are then used to prove the entropy theorem the principle feature is the direct use of coding ideas to obtain the entropy theorem

time variant channel estimation using discrete prolate spheroidal sequences we propose and analyze a low complexity channel estimator for a multiuser multicarrier code division multiple access mc cdma downlink in a time variant frequency selective channel mc cdma is based on orthogonal frequency division multiplexing ofdm the time variant channel is estimated individually for every flat fading subcarrier assuming small intercarrier interference the temporal variation of every subcarrier over the duration of a data block is upper bounded by the doppler bandwidth determined by the maximum velocity of the users slepian showed that time limited snapshots of bandlimited sequences span a low dimensional subspace this subspace is also spanned by discrete prolate spheroidal dps sequences we expand the time variant subcarrier coefficients in terms of orthogonal dps sequences we call slepian basis expansion this enables a time variant channel description that avoids the frequency leakage effect of the fourier basis expansion the square bias of the slepian basis expansion per subcarrier is three magnitudes smaller than the square bias of the fourier basis expansion we show simulation results for a fully loaded mc cdma downlink with classic linear minimum mean square error multiuser detection the users are moving with 19 4 m s using the slepian basis expansion channel estimator and a pilot ratio of only 2 we achieve a bit error rate performance as with perfect channel knowledge

edge deletion problems if pi is a property on graphs or digraphs the edge deletion problem can be stated as follows find the minimum number of edges whose deletion results in a subgraph or subdigraph satisfying property pi several well studied graph problems can be formulated as edge deletion problems in this paper we show that the edge deletion problem is np complete for the following properties 1 without cycles of specified length l or of any length leqq l 2 connected and degree constrained 3 outerplanar 4 transitive digraph 5 line invertible 6 bipartite 7 transitively orientable for problems 5 6 7 we determine the best possible bounds on the node degrees for which the problems remain np complete

time varying channel estimation using two dimensional channel orthogonalization and superimposed training in this correspondence a method is presented for estimating double selective channels using superimposed training st the estimator is based on a subspace projection of the time varying channel onto a set of two dimensional orthogonal functions these functions are formed via the outer product of the discrete prolate spheroidal basis vectors and the universal basis vectors this approach allows the channel to be expanded in both the time delay and time dimensions with the fewest parameters when incomplete channel statistics are given this correspondence also provides a theoretical performance analysis of the estimation algorithm and its corroboration via simulations it is shown that this new method provides an enhancement in channel estimation when compared with state of the art approaches

probe order backtracking the algorithm for constraint satisfaction problems probe order backtracking has an average running time much faster than any previously analyzed algorithm under conditions where solutions are common the algorithm uses a probing assignment a preselected test assignment to unset variables to help guide the search for a solution if the problem is not satisfied when the unset variables are temporarily set to the probing assignment the algorithm selects one of the relations which is not satisfied by the probing assignment and selects an unset variable which affects the value of that relation it then does a backtracking splitting step where it generates subproblems by setting the selected variable each possible way each subproblem is simplified and then solved recursively for random problems with v variables t clauses and probability p that a literal appears in a clause the average time for probe order backtracking is no more than v n when p geq ln t v plus lower order terms the best previous result was p geq sqrt ln t v when the algorithm is combined with an algorithm of franco that makes selective use of resolution the average time for solving random problems is no more than v n for all values of p when t leq o n 1 3 v ln v 2 3 the best previous result was t leq o n 1 3 v ln v 1 6 probe order backtracking also runs in polynomial average time when p leq 1 v compared with the best previous result of p leq 1 2v with probe order backtracking the range of p that leads to more than polynomial time is much smaller than that for previously analyzed algorithms

approximation and intractability results for the maximum cut problem and its variants the maximum cut problem is known to be an important np complete problem with many applications the authors investigate this problem which they call the normal maximum cut problem and a variant of it which is referred to as the connected maximum cut problem they show that any n vertex e edge graph admits a cut with at least the fraction 1 2 1 2n of its edges thus improving the ratio 1 2 2 e known before it is shown that it is np complete to decide if a given graph has a normal maximum cut with at least a fraction 1 2 epsilon of its edges where the positive constant epsilon can be taken smaller than any value chosen the authors present an approximation algorithm for the normal maximum cut problem on any graph that runs in o e log e n log n p log p log n parallel time using p 1

hardness of edge modification problems for a graph property p consider the following computational problem given an input graph g what is the minimum number of edge modifications additions and or deletions that one has to apply to g in order to turn it into a graph that satisfies p namely what is the edit distance d g p of a graph g from satisfying p clearly the computational complexity of such a problem strongly depends on p for over 30 years this family of computational problems has been studied in several contexts and various algorithms as well as hardness results were obtained for specific graph properties alon shapira and sudakov studied in n alon a shapira b sudakov additive approximation for edge deletion problems in proc of the 46th ieee focs 2005 419 428 also annals of mathematics in press the approximability of the computational problem for the family of monotone graph properties namely properties that are closed under removal of edges and vertices they describe an efficient algorithm that achieves an o n 2 additive approximation to d g p for any monotone property p where g is an n vertex input graph and show that the problem of achieving an o n 2 e additive approximation is np hard for most monotone properties the methods in n alon a shapira b sudakov additive approximation for edge deletion problems in proc of the 46th ieee focs 2005 419 428 also annals of mathematics in press also provide a polynomial time approximation algorithm which computes d g p o n 2 for the broader family of hereditary graph properties which are closed under removal of vertices in this work we introduce two approaches for showing that improving upon the additive approximation achieved by this algorithm is np hard for several sub families of hereditary properties in addition we state a conjecture on the hardness of computing the edit distance from being induced h free for any forbidden graph h

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

complexity of partial satisfaction a conjunctive normal form expression cnf is said to be 2 satisfiable if and only if any two of its clauses are simultaneously satisfiable it is shown that every 2 satisfiable cnf has a truth assignment that satisfies at least the fraction h of its clauses where h sqrt 5 1 2 sim 0 618 the reciprocal of the golden ratio the proof is constructive in that it provides a polynomial time algorithm that will find for any 2 satisfiable cnf a truth assignment satisfying at least the fraction h of its clauses furthermore this result is optimal in that the constant h is as large as possible it is shown that for any rational h h the set of all 2 satisfiable cnfs that have truth assignments satisfying at least the fraction h of their clauses is an np complete set

a new bound for 3 satisfiable maxsat and its algorithmic application let f be a cnf formula with n variables and m clauses f is 3 satisfiable if for any 3 clauses in f there is a truth assignment which satisfies all of them lieberherr and specker 1982 and later yannakakis 1994 proved that in each 3 satisfiable cnf formula at least 23 of its clauses can be satisfied by a truth assignment we improve this result by showing that every 3 satisfiable cnf formula f contains a subset of variables u such that some truth assignment t will satisfy at least 23m 13m u rn clauses where m is the number of clauses of f m u is the number of clauses of f containing a variable from u n is the total number of variables in clauses not containing a variable in u and r is a positive absolute constant both u and t can be found in polynomial time we use our result to show that the following parameterized problem is fixed parameter tractable and moreover has a kernel with a linear number of variables in 3 s maxsat ae we are given a 3 satisfiable cnf formula f with m clauses and asked to determine whether there is an assignment which satisfies at least 23m k clauses where k is the parameter

the ergodic and entropy theorems revisited a new proof of birkhoff s ergodic theorem is given using a sample path covering idea an idea created by ornstein and weiss in their extension of the information convergence theorem to random fields a sketch of the ornstein weiss proof for processes is included in the appendix

efficient stabilization of cooperative matching games abstract cooperative matching games have drawn much interest partly because of the connection with bargaining solutions in the networking environment however it is not always guaranteed that a network under investigation gives rise to a stable bargaining outcome to address this issue we consider a modification process called stabilization that yields a network with stable outcomes where the modification should be as small as possible therefore the problem is cast to a combinatorial optimization problem in a graph recently the stabilization by edge removal was shown to be np hard on the contrary in this paper we show that other possible ways of stabilization namely edge addition vertex removal and vertex addition are all polynomial time solvable thus we obtain a complete complexity theoretic classification of the natural four variants of the network stabilization problem we further study weighted variants and prove that the variants for edge addition and vertex removal are np hard

on doubly selective channel estimation using superimposed training and discrete prolate spheroidal sequences channel estimation and data detection for frequency selective time varying channels are considered using superimposed training we employ a discrete prolate spheroidal basis expansion model dps bem to describe the time varying channel a periodic nonrandom training sequence is arithmetically added superimposed at low power to the information sequence at the transmitter before modulation and transmission therefore there is no loss in data transmission rate compared to time multiplexed tm training we first estimate the channel using dps bem and only the first order statistics of the observations in this estimator the unknown information sequence acts as interference resulting in a poor signal to noise and interference ratio snir for channel estimation we then apply a data dependent superimposed training sequence to either totally or partially cancel out the effects of the unknown information sequence at the receiver on channel estimation in total cancellation at certain frequencies the information bearing components are nulled to compensate for this information loss we investigate a partially data dependent pdd superimposed training scheme where a tradeoff is made between interference cancellation and frequency integrity design of certain parameters for pdd superimposed training is also investigated finally a deterministic maximum likelihood dml approach is used iteratively to enhance channel estimation and data detection computer simulation examples show that the proposed approaches are competitive with the conventional tm training without incurring data rate loss

node deletion problems on bipartite graphs a set of problems which has attracted considerable interest recently is the set of node deletion problems the general node deletion problem can be stated as follows given a graph find the minimum number of nodes whose deletion results in a subgraph satisfying property r in ly this problem was shown to be np complete for a large class of properties the class of properties that are hereditary on induced subgraphs using a small number of reduction schemes from the node cover problem since the node cover problem becomes polynomial on bipartite graphs it might be hoped that this is the case with other node deletion problems too in this paper we characterize those properties for which the bipartite restriction of the node deletion problem is polynomial and those for which it remains np complete similar results follow for analogous problems on other structures such as families of sets hypergraphs and 0 1 matrices for example in the case of matrices our result states that ifm is a class of 0 1 matrices which is closed under permutation and deletion of rows and columns then finding the largest submatrix inm of a matrix is polynomial if the matrices ofm have bounded rank and np complete otherwise 1 introduction a common approach when faced with an np complete problem c k gj is to restrict its input domain with the hope that the problem may become solvable in polynomial time under the restriction in fact it turns out that sometimes this is the case for example the node cover problem np complete on general graphs k can be solved efficiently when restricted to bipartite graphs this follows from k6nig s theorem that the node covering number of a bipartite graph equals the number of edges in a maximum matching and moreover a minimum node cover can be found efficiently from a maximum matching when the graph is bipartite see for example la in other cases however the node cover problem continues to be np complete under restriction for example the node cover problem on planar graphs gjs much work has been done recently kd ly ldl on a class of problems called the node deletion or maximum subgraph problems the general node deletion problem can be stated as follows given a graph or digraph g find a set of nodes of minimum cardinality whose deletion results in a graph or digraph satisfying a property r several of the known np complete problems such as the node cover the max clique the feedback node set k as well as some polynomial problems such as the connectivity of a graph e it can be formulated in an obvious way as node deletion problems by specifying appropriately the property r

deterministic blind identification of iir systems with output switching operations in this paper a deterministic blind identification approach is proposed for linear output switching systems which are modeled by multiple infinite impulse response iir dynamic functions by adopting a new over sampling strategy the concerned single input single output siso output switching system is equivalently transformed into a time invariant multi input multi output mimo system further by exploring the mutual relations among the multiple inputs the time invariant mimo system model and subsequently the output switching system model are identified uniquely up to a scalar constant using the proposed identification approach sufficient identifiability conditions are provided for output switching systems and numerical simulations are carried out to validate the proposed approach

sample converses in source coding theory the rate and distortion performance of a sequence of codes along a sample sequence of symbols generated by a stationary ergodic information source are studied two results are obtained 1 the source sample sequence is encoded by an arbitrary sequence of block codes which operate at a fixed rate level r and a sample converse is obtained which states that with probability one the lower limit of the code sample distortions is lower bounded by d r the value of the distortion rate function at r 2 the source sample sequence is encoded by an arbitrary sequence of variable rate codes which operate at a fixed distortion level d and a sample converse is obtained which states that with probability one the lower limit of the code sample rates is lower bounded by r d the value of the rate distortion function at d a novel ergodic theorem is used to obtain both sample converses

implications of forbidden structures for extremal algorithmic problems abstract we develop and analyze p optimal approximation algorithms for various generalized satisfiability problems and determine the corresponding p optimal thresholds the most novel aspect of the paper is that we develop new p optimal algorithms for restricted generalized satisfiability problems which are defined by forbidden subformulas all our algorithms run in linear time on a ram

the interactions between ergodic theory and information theory information theorists frequently use the ergodic theorem likewise entropy concepts are often used in information theory recently the two subjects have become partially intertwined as deeper results from each discipline find use in the other a brief history of this interaction is presented in this paper together with a more detailed look at three areas of connection namely recurrence theory blowing up bounds and direct sample path methods

a unified approach to approximation algorithms for bottleneck problems in this paper a powerful and yet simple technique for devising approximation algorithms for a wide variety of np complete problems in routing location and communication network design is investigated each of the algorithms presented here delivers an approximate solution guaranteed to be within a constant factor of the optimal solution in addition for several of these problems we can show that unless p np there does not exist a polynomial time algorithm that has a better performance guarantee

restrictions of minimum spanner problems abstract a t spanner of a graph g is a spanning subgraph h such that the distance between any two vertices in h is at most t times their distance in g spanners arise in the context of approximating the original graph with a sparse subgraph peleg d and schaffer a a 1989 j graph theory 13 1 99 116 the minimum t spanner problem seeks to find a t spanner with the minimum number of edges for the given graph in this paper we completely settle the complexity status of this problem for various values of t on chordal graphs split graphs bipartite graphs and convex bipartite graphs our results settle an open question raised by l cai 1994 discrete appl math 48 187 194 and also greatly simplify some of the proofs presented by cai and by l cai and m keil 1994 networks 24 233 249 we also give a factor 2 approximation algorithm for the minimum 2 spanner problem on interval graphs finally we provide approximation algorithms for the bandwidth minimization problem on convex bipartite graphs and split graphs using the notion of tree spanners