complexity of graph self assembly in accretive systems and self destructible systems self assembly is a process in which small objects autonomously associate with each other to form larger complexes it is ubiquitous in biological constructions at the cellular and molecular scale and has also been identified by nanoscientists as a fundamental method for building nano scale structures recent years have seen convergent interest and efforts in studying self assembly from mathematicians computer scientists physicists chemists and biologists however most complexity theoretical studies of self assembly utilize mathematical models with two limitations 1 only attraction while no repulsion is studied 2 only assembled structures of two dimensional square grids are studied in this paper we study the complexity of the assemblies resulting from the cooperative effect of repulsion and attraction in a more general setting of graphs this allows for the study of a more general class of self assembled structures than the previous tiling model we define two novel assembly models namely the accretive graph assembly model and the self destructible graph assembly model and identify a fundamental problem in them the sequential construction of a given graph we refer to it as the accretive graph assembly problem agap and the self destructible graph assembly problem dgap in the respective models our main results are i agap is np complete even if the maximum degree of the graph is restricted to 4 or the graph is restricted to be planar with maximum degree 5 ii counting the number of sequential assembly orderings that result in a target graph agap is p complete and iii dgap is pspace complete even if the maximum degree of the graph is restricted to 6 this is the first pspace complete result in self assembly we also extend the accretive graph assembly model to a stochastic model and prove that determining the probability of a given assembly in this model is p complete

the within strip discrete unit disk cover problem we present a study of the within strip discrete unit disk cover wsdudc problem which is a restricted version of the discrete unit disk cover dudc problem for the wsdudc problem there exists a set of points and a set of unit disks in the plane and the points and disk centres are conned to a strip of xed height an optimal solution to the wsdudc problem is a set of disks of minimum cardinality that covers all points in the input set we describe a range of approximation algorithms for the problem including 4 and 3 approximate algorithms which apply for strips of height 2 p 2 3 0 94 and 0 8 units respectively as well as a general scheme for any strip with less than unit height we prove that the wsdudc problem is np complete on strips of any xed height which is our most interesting result from a theoretical standpoint the result is also quite surprising since a number of similar problems are tractable on strips of xed height finally we discuss how these results may be applied to known dudc approximation algorithms

some np hard polygon decomposition problems the inherent computational complexity of polygon decomposition problems is of theoretical interest to researchers in the field of computational geometry and of practical interest to those working in syntactic pattern recognition three polygon decomposition problems are shown to be np hard and thus unlikely to admit efficient algorithms the problems are to find minimum decompositions of a polygonal region into perhaps overlapping convex star shaped or spiral subsets we permit the polygonal region to contain holes the proofs are by transformation from boolean three satisfiability a known np complete problem several open problems are discussed

berge s theorem for noncompact image sets abstract for an upper semi continuous set valued mapping from one topological space to another and for a lower semi continuous function defined on the product of these spaces berge s theorem states lower semi continuity of the minimum of this function taken over the image sets it assumes that the image sets are compact for hausdorff topological spaces this paper extends berge s theorem to set valued mappings with possible noncompact image sets and studies relevant properties of minima

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

the complexity of coloring games on perfect graphs abstract in this paper we consider the following type of game two players must color the vertices of a given graph g v e in a prescribed order in such a way that no two adjacent vertices are colored with the same color in one variant the first player which is unable to move loses the game in another variant player 1 wins the game if and only if the game ends with all vertices colored in this paper we obtain several results on the complexity of the problem to decide whether there is a winning strategy for player 1 in a given game instance when g is restricted to split graphs interval graphs or bipartite graphs

the planar k means problem is np hard in the k means problem we are given a finite set s of points in m and integer k 1 and we want to find k points centers so as to minimize the sum of the square of the euclidean distance of each point in s to its nearest center we show that this well known problem is np hard even for instances in the plane answering an open question posed by dasgupta 2007 7

the othello game on an n n board is pspace complete abstract given an arbitrary position of the othello game played on an n n board the problem of determining the winner is shown to be pspace complete it can be reduced from generalized geography played on bipartite graphs with maximum degree 3

the computational complexity of probabilistic inference using bayesian belief networks research note abstract bayesian belief networks provide a natural efficient method for representing probabilistic dependencies among a set of variables for these reasons numerous researchers are exploring the use of belief networks as a knowledge representation in artificial intelligence algorithms have been developed previously for efficient probabilistic inference using special classes of belief networks more general classes of belief networks however have eluded efforts to develop efficient inference algorithms we show that probabilistic inference using belief networks is np hard therefore it seems unlikely that an exact algorithm can be developed to perform probabilistic inference efficiently over all classes of belief networks this result suggests that research should be directed away from the search for a general efficient probabilistic inference algorithm and toward the design of efficient special case average case and approximation algorithms

the tractability of segmentation and scene analysis one of the fundamental problems in computer vision is the segmentation of an image into semantically meaningful regions based only on image characteristics a single segmentation can be determined using a linear number of evaluations of a uniformity predicate however minimising the number of regions is shown to be an np complete problem we also show that the variational approach to segmentation based on minimising a criterion combining the overall variance of regions and the number of regions also gives rise to an np complete problem when a library of object models is available segmenting the image becomes a problem of scene analysis a sufficient condition for the reconstruction of a 3d scene from a 2d image to be solvable in polynomial time is that the scene contains no cycles of mutually occluding objects and that no range information can be deduced from the image it is known that relaxing the no cycles condition renders the problem np complete we show that relaxing the no range information condition also produces an np complete problem

the complexity of the residual node connectedness reliability problem this paper considers a probabilistic network in which the edges are perfectly reliable but the nodes fail with some known probabilities the network is in an operational state if the surviving nodes induce a connected graph the residual node connectedness reliability r g of a network g is the probability that the graph induced by the surviving nodes is connected this reliability measure is very different from the widely studied k terminal network reliability measure it is proven that the problem of computing the residual connectedness reliability is np hard by showing that the problem of counting the number of node induced connected subgraphs of a given graph is bf p complete the problem remains bf p complete for split graphs as well as planar and bipartite graphs

a polyhedral description of kernels let g be a digraph and let g be the linear system consisting of nonnegativity stability and domination inequalities we call g kernel ideal if h defines an integral polytope for each induced subgraph h of g and we call g kernel mengerian if h is totally dual integral tdi for each induced subgraph h of g in this paper we show that a digraph is kernel ideal iff it is kernel mengerian iff it contains none of three forbidden structures our characterization yields a polynomial time algorithm for the minimum weighted kernel problem on kernel ideal digraphs we also prove that it is np hard to find a kernel of minimum size even in a planar bipartite digraph with maximum degree at most three

bottleneck steiner trees in the plane a steiner tree with maximum weight edge minimized is called a bottleneck steiner tree bst the authors propose a theta mod rho mod log mod rho mod time algorithm for constructing a bst on a point set rho with points labeled as steiner or demand a lower bound in the linear decision tree model is also established it is shown that if it is desired to minimize further the number of used steiner points then the problem becomes np complete it is shown that when locations of steiner points are not fixed the problem remains np complete however if the topology of the final tree is given then the problem can be solved in theta mod rho mod log mod rho mod time the bst problem can be used for example in vlsi layout communication network design and facility location problems

approximation algorithms for pspace hard hierarchically and periodically specified problems we study the efficient approximability of basic graph and logic problems in the literature when instances are specified hierarchically as in t lengauer j assoc comput mach 36 1989 pp 474 509 or are specified by one dimensional finite narrow periodic specifications as in e wanke paths and cycles in finite periodic graphs in lecture notes in comp sci 711 springer verlag new york 1993 pp 751 760 we show that for most of the problems pi considered when specified using k level restricted hierarchical specifications or k narrow periodic specifications the following hold let rho be any performance guarantee of a polynomial time approximation algorithm for pi when instances are specified using standard specifications then forall epsilon 0 pi has a polynomial time approximation algorithm with performance guarantee 1 epsilon rho pi has a polynomial time approximation scheme when restricted to planar instances these are the first polynomial time approximation schemes for pspace hard hierarchically or periodically specified problems since several of the problems considered are pspace hard our results provide the first examples of natural pspace hard optimization problems that have polynomial time approximation schemes this answers an open question in condon et al chicago j theoret comput sci 1995 article 4

radiocoloring in planar graphs complexity and approximations the frequency assignment problem fap in radio networks is the problem of assigning frequencies to transmitters by exploiting frequency reuse while keeping signal interference to acceptable levels the fap is usually modelled by variations of the graph coloring problem a radiocoloring rc of a graph g v e is an assignment function v n such that u v 2 when u v are neighbors in g and u v 1 when the distance of u v in g is two the number of discrete frequencies and the range of frequencies used are called order and span respectively the optimization versions of the radiocoloring problem rcp are to minimize the span or the order in this paper we prove that the radiocoloring problem for general graphs is hard to approximate unless np zpp within a factor of n1 2 e for any e 0 where n is the number of vertices of the graph however when restricted to some special cases of graphs the problem becomes easier we prove that the min span rcp is np complete for planar graphs next we provide an o n time algorithm v n which obtains a radiocoloring of a planar graph g that approximates the minimum order within a ratio which tends to 2 where the maximum degree of g finally we provide a fully polynomial randomized approximation scheme fpras for the number of valid radiocolorings of a planar graph g with colors in the case where 4 50

on the proper orientation number of bipartite graphs an orientation of a graph g is a digraph d obtained from g by replacing each edge by exactly one of the two possible arcs with the same endvertices for each v v g the indegree of v in d denoted by d d v is the number of arcs with head v in d an orientation d of g is proper if d d u d d v for all u v e g the proper orientation number of a graph g denoted by g is the minimum of the maximum indegree over all its proper orientations in this paper we prove that g g g 2 1 if g is a bipartite graph and g 4 if g is a tree it is well known that g g for every graph g however we prove that deciding whether g g 1 is already an np complete problem on graphs with g k for every k 3 we also show that it is np complete to decide whether g 2 for planar subcubic graphs g moreover we prove that it is np complete to decide whether g 3 for planar bipartite graphs g with maximum degree 5

the complexity of understanding line drawings of origami scenes this paper deals with the interpretation of line drawings of origami scenes kanade 1980 that is scenes obtained by assembling planar panels of negligible thickness and it addresses the computational complexity of the problem of consistently assigning suitable labels to the segments describing 3d properties as convexity concavity and occlusion labeling problem the main results of the paper are the following a the labeling problem for line drawings of origami scenes is np as for the case of trihedral scenes b the problem remains np even if the location of the vanishing points in the image plane is given whereas for trihedral scenes the problem was polynomially solvable c in case the vanishing points are known the labeling problem can be subdivided into two subproblems the paneling problem and the labeling a paneled line drawing problem which are both polynomially solvable there may be however exponentially many legal panelings to check against labelability

cycle transversals in perfect graphs and cographs a cycle transversal or feedback vertex set of a graph g is a subset t v g such that t v c 0 for every cycle c of g this work considers the problem of finding special cycle transversals in perfect graphs and cographs we prove that finding a minimum cycle transversal t in a perfect graph g is np hard even for bipartite graphs with maximum degree four since g t is acyclic this result implies that finding a maximum acyclic induced subgraph of a perfect graph is also np hard other special properties of t are considered a clique stable respectively cycle transversal or simply cct sct respectively is a cycle transversal which is a clique stable set respectively recognizing graphs which admit a cct can be done in polynomial time however no structural characterization of such graphs is known even for perfect graphs we characterize cographs with cct in terms of forbidden induced subgraphs and describe their structure this leads to linear time recognition of cographs with cct we also prove that deciding whether a perfect graph admits an sct is np complete we characterize cographs with sct in terms of forbidden induced subgraphs this characterization also leads to linear time recognition

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

following a curve with the discrete frechet distance finding the similarity between curves is an important problem that comes up in many areas such as 3d modeling gis applications ordering and reachability a related problem is to find one of the curves given a measure of similarity and another curve given a set of points s a polygonal curve p and an e 0 the discrete set chain matching problem is to find another polygonal curve q such that the nodes of q are points in s and d f p q e here d f is the discrete frechet distance between the two polygonal curves for the first time we study the set chain matching problem based on the discrete frechet distance rather than the continuous frechet distance we further extend the problem based on unique or non unique nodes and on limiting the number of points used we prove that three of the variations of the set chain matching problem are np complete for the version of the problem that is polynomial we give an o p s time greedy solution

computing coverage kernels under restricted settings abstract given a set b of d dimensional boxes i e axis aligned hyperrectangles a minimum coverage kernel is a subset of b of minimum size covering the same region as b computing it is np hard but as for many similar np hard problems e g box cover and orthogonal polygon covering the problem becomes solvable in polynomial time under restrictions on b we show that computing minimum coverage kernels remains np hard even when restricting the graph induced by the input to a highly constrained class of graphs alternatively we present two polynomial time approximation algorithms for this problem one deterministic with an approximation ratio within o log n and one randomized with an improved approximation ratio within o lg opt with high probability

on the complexity of labeling perspective projections of polyhedral scenes abstract this paper investigates the computational time complexity of the labeling problem for line drawings of trihedral scenes it is shown that the class of problems having polynomial complexity is larger than the simple case of line drawings of legoland scenes kirousis and papadimitriou 1988 once the location of the vanishing points in the image plane is known the labeling problem can be solved in time o nn where n is the number of segments and n is the number of vanishing points the vanishing points can be given a priori otherwise can in many cases be detected by standard techniques from the line drawing itself the np completeness of the labeling problem for line drawings of trihedral scenes kirousis and papadimitriou 1988 is then due to the lack of knowledge about the vanishing points which is equivalent to the knowledge of the possible directions for the edges these results help draw a more accurate boundary between the problems in the interpretation of line drawings that are polynomially solvable and those that are np complete

model selection criteria for missing data problems using the em algorithm we consider novel methods for the computation of model selection criteria in missing data problems based on the output of the em algorithm the methodology is very general and can be applied to numerous situations involving incomplete data within an em framework from covariates missing at random in arbitrary regression models to nonignorably missing longitudinal responses and or covariates toward this goal we develop a class of information criteria for missing data problems called ic h q which yields the akaike information criterion and the bayesian information criterion as special cases the computation of ic h q requires an analytic approximation to a complicated function called the h function along with output from the em algorithm used in obtaining maximum likelihood estimates the approximation to the h function leads to a large class of information criteria called ic h k q theoretical properties of ic h k q including consistency are investigated in detail to eliminate the analytic approximation to the h function a computationally simpler approximation to ic h q called ic q is proposed the computation of which depends solely on the q function of the em algorithm advantages and disadvantages of ic h k q and ic q are discussed and examined in detail in the context of missing data problems extensive simulations are given to demonstrate the methodology and examine the small sample and large sample performance of ic h k q and ic q in missing data problems an aids data set also is presented to illustrate the proposed methodology

on the complexity of clustering with relaxed size constraints in fixed dimension abstract we study the computational complexity of the problem of computing an optimal clustering a 1 a 2 a k of a set of points assuming that every cluster size a i belongs to a given set m of positive integers we present a polynomial time algorithm for solving the problem in dimension 1 i e when the points are simply rational values for an arbitrary set m of size constraints which extends to the l 1 norm an analogous procedure known for the euclidean norm moreover we prove that in dimension 2 assuming euclidean norm the problem is strongly np hard with size constraints m 2 4 this result is extended also to the size constraints m 2 3 both in the case of euclidean and l 1 norm