branched polymers and dimensional reduction we establish an exact relation between self avoiding branched polymers in d 2 continuum dimensions and the hard core continuum gas at negative activity in d dimensions we review conjectures and results on critical exponents for d 2 2 3 4 and show that they are corollaries of our result we explain the connection first proposed by parisi and sourlas between branched polymers in d 2 dimensions and the yang lee edge singularity in d dimensions

exact algorithms for kayles in the game of kayles two players select alternatingly a vertex from a given graph g but may never choose a vertex that is adjacent or equal to an already chosen vertex the last player that can select a vertex wins the game in this paper we give an exact algorithm to determine which player has a winning strategy in this game to analyze the running time of the algorithm we introduce the notion of a k set a nonempty set of vertices w v is a k set in a graph g v e if g w is connected and there exists an independent set x such that w v n x the running time of the algorithm is bounded by a polynomial factor times the number of k sets in g we prove that the number of k sets in a graph with n vertices is bounded by o 1 6052 n a computer generated case analysis improves this bound to o 1 6031 n k sets and thus we have an upper bound of o 1 6031 n on the running time of the algorithm for kayles we also show that the number of k sets in a tree is bounded by n 3 n 3 and thus kayles can be solved on trees in o 1 4423 n time we show that apart from a polynomial factor the number of k sets in a tree is sharp as corollaries we obtain that determining which player has a winning strategy in the games g avoid pos dnf 2 and g seek posdnf 3 can also be determined in o 1 6031 n time in g avoid posdnf 2 we have a positive formula f on n boolean variables in disjunctive normal form with two variables per clause initially all variables are false and players alternately set a variable from false to true the first player that makes f true loses the game the game g seek posdnf 3 is similar but now there are three variables per clause and the first player that makes f true wins the game

distribution of transverse distances in directed animals we relate x s the average number of sites at a transverse distance x in the directed animals with s sites in d transverse dimensions to the two point correlation function of a lattice gas with nearest neighbour exclusion in d dimensions for large s x s has the scaling form s rds f x rs where rs is the root mean square radius of gyration of animals of s sites we determine the exact scaling function for d 1 to be f r 2 3 erfc r 3 we also show that x 0 s can be determined in terms of the animal number generating function of the directed animals

yang lee edge singularities from extended activity expansions of the dimer density for bipartite lattices of dimensionality 2 d 7 we have extended in most cases through 24th order the series expansions of the dimer density in powers of the activity in the case of bipartite hyper simple cubic and hyper body centered cubic lattices of dimensionalities 2 d 7 a numerical analysis of these data yields estimates of the exponents characterizing the yang lee edge singularities for lattice ferromagnetic spin models as d varies between the lower and the upper critical dimensionalities our results are consistent with but more extensive and sometimes more accurate than those obtained from the existing dimer series or from the estimates of related exponents for lattice animals branched polymers and fluids we mention also that it is possible to obtain estimates of the dimer constants from our series for the various lattices

dimensional reduction for directed branched polymers dimensional reduction occurs when the critical behaviour of one system can be related to that of another system in a lower dimension we show that this occurs for directed branched polymers dbp by giving an exact relationship between dbp models in d 1 dimensions and repulsive gases at negative activity in d dimensions this implies relations between exponents of the two models d 1 d the exponent describing the singularity of the pressure and d 1 d the correlation length exponent of the repulsive gas it also leads to the relation d 1 1 d where d is the yang lee edge exponent we derive exact expressions for the number of dbp of size n in two dimensions

when a local hamiltonian must be frustration free a broad range of quantum optimization problems can be phrased as the question of whether a specific system has a ground state at zero energy i e whether its hamiltonian is frustration free frustration free hamiltonians in turn play a central role for constructing and understanding new phases of matter in quantum many body physics unfortunately determining whether this is the case is known to be a complexity theoretically intractable problem this makes it highly desirable to search for efficient heuristics and algorithms to at least partially answer this question here we prove a general criterion a sufficient condition under which a local hamiltonian is guaranteed to be frustration free by lifting shearer s theorem from classical probability theory to the quantum world remarkably evaluating this condition proceeds via a fully classical analysis of a hardcore lattice gas at negative fugacity on the hamiltonian s interaction graph which as a statistical mechanics problem is of interest in its own right we concretely apply this criterion to local hamiltonians on various regular lattices while bringing to bear the tools of spin glass physics that permit us to obtain new bounds on the satisfiable to unsatisfiable transition in random quantum satisfiability we are then led to natural conjectures for when such bounds will be tight as well as to a novel notion of universality for these computer science problems besides providing concrete algorithms leading to detailed and quantitative insights this work underscores the power of marrying classical statistical mechanics with quantum computation and complexity theory

some optimal inapproximability results we prove optimal up to an arbitrary e 0 inapproximability results for max e k sat for k 3 maximizing the number of satisfied linear equations in an over determined system of linear equations modulo a prime p and set splitting as a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously in particular for max e2 sat max cut max di cut and vertex cover

new exact algorithms for the 2 constraint satisfaction problem many optimization problems can be phrased in terms of constraint satisfaction in particular max 2 sat and max 2 csp are known to generalize many hard combinatorial problems on graphs algorithms solving the problem exactly have been designed but the running time is improved over trivial brute force solutions only for very sparse instances despite many efforts the only known algorithm 28 solving max 2 csp over n variables in less than o 2 n steps uses exponential space several authors have designed algorithms with running time o 2 n f d where f r 0 1 is a slowly growing function and d is the average variable degree of the input formula the current best known algorithm for max 2 csp 25 runs in time o 2 n 1 2 d 1 and polynomial space in this paper we continue this line of research and design new algorithms for the max 2 sat and max 2 csp problems first we present a general technique for obtaining new bounds on the running time of a simple algorithm for max 2 csp analyzed with respect to the number of vertices from algorithms that are analyzed with respect to the number of constraints the best known bound for the problem is improved to o 2 n 1 3 d 1 for d 3 we further improve the bound for max 2 sat in particular for d 6 we achieve o 2 n 1 3 6 7 7 d 1 as a second result we present an algorithm with asymptotically better running time for the case when the input instance is not very sparse building on recent work of feige and kogan we derive an upper bound on the size of a vertex separator for graphs in terms of the average degree of the graph we then design a simple algorithm solving max 2 csp in time o 2 c d n c email protected for some a 1 and d o n

identity of the universal repulsive core singularity with yang lee edge criticality lattice and continuum fluid models with repulsive core interactions typically display a dominant critical type singularity on the real negative activity axis lai and fisher recently suggested mainly on numerical grounds that this repulsive core singularity is universal and in the same class as the yang lee edge singularities which arise above criticality at complex activities with positive real part a general analytic demonstration of this identification is presented here using a field theory approach with separate representations of the repulsive and attractive parts of the pair interactions

an improved exponential time algorithm for k sat we propose and analyze a simple new randomized algorithm called resolvesat for finding satisfying assignments of boolean formulas in conjunctive normal form the algorithm consists of two stages a preprocessing stage in which resolution is applied to enlarge the set of clauses of the formula followed by a search stage that uses a simple randomized greedy procedure to look for a satisfying assignment currently this is the fastest known probabilistic algorithm for k cnf satisfiability for k g 4 with a running time of o 20 5625n for 4 cnf in addition it is the fastest known probabilistic algorithm for k cnf k g 3 that have at most one satisfying assignment unique k sat with a running time o 2 2 ln 2 1 n p o n e o 20 386 n in the case of 3 cnf the analysis of the algorithm also gives an upper bound on the number of the codewords of a code defined by a k cnf this is applied to prove a lower bounds on depth 3 circuits accepting codes with nonconstant distance in particular we prove a lower bound 21 282 i i i i i i

hard squares with negative activity we show that the hard square lattice gas with activity z 1 has a number of remarkable properties we conjecture that all the eigenvalues of the transfer matrix are roots of unity they fall into groups strings evenly spaced around the unit circle which have interesting number theoretic properties for example the partition function on an m n lattice with periodic boundary condition is identically 1 when m and n are coprime we provide evidence for these conjectures from analytical and numerical arguments

force method revisited computational mechanics is an important field in engineering with a strong foundation in continuum mechanics and finite element analysis the standard and integrated force methods are reviewed and a structural analysis technique using the finite element force method based on the complementary strain energy is proposed similar to the standard force and the integrated force methods the force method based on the complementary strain energy uses the same equilibrium equations however the compatibility conditions are satisfied through the complementary strain energy the gauss elimination technique has been employed successfully to generate automatically a basis determinate structure and redundant members extension of the force method to dynamics and benchmark problems with displacement stress and frequency constraints are presented

a t o 2 n 2 s o 2 n 4 algorithm for certain np complete problems in this paper we develop a general purpose algorithm that can solve a number of np complete problems in time t o 2 n 2 and space s o 2 n 4 the algorithm can be generalized to a fam

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

structural optimization with frequency constraints this paper presents a design optimization algorithm for structural weight minimization with multiple frequency constraints an optimality criterion method based on uniform lagrangian density for resizing and a scaling procedure to locate the constraint boundary were used in optimization multiple frequency constraints of equality and inequality types were addressed the effectiveness of the algorithm was demonstrated by designing a number of truss structures with as many as 489 design variables no attempt was made to reduce the number of design variables by such procedures as linking and or invoking symmetry conditions the design examples include a 10 bar truss 200 bar truss a modified acoss ii and cofs control of flexible structures mast truss all the structures contain nonstructural mass besides their own mass the algorithm is extremely stable and in all cases the optimum designs were obtained in less than 20 iterations regardless of the size of the structure and the number of design variables

improved upper bounds for vertex cover this paper presents an o 1 2738 k kn time polynomial space algorithm for vertex cover improving the previous o 1 286 k kn time polynomial space upper bound by chen kanj and jia most of the previous algorithms rely on exhaustive case by case branching rules and an underlying conservative worst case scenario assumption the contribution of the paper lies in the simplicity uniformity and obliviousness of the algorithm presented several new techniques as well as generalizations of previous techniques are introduced including general folding struction tuples and local amortized analysis the algorithm also improves the o 1 2745 kk 4 kn time exponential space upper bound for the problem by chandran and grandoni

partition function zeros of the one dimensional blume capel model in transfer matrix formalism zeros of the partition function of the one dimensional ferromagnetic and antiferromagnetic blume capel models have been studied by using the transfer matrix method in the thermodynamic limit and for finite size chains the equation for the distribution of zeros of the partition function in the thermodynamic limit is derived the distribution of the yang lee and fisher zeros are studied for a variety of values of the parameters of the model densities of the yang lee and fisher zeros are investigated and a singular behavior of the corresponding densities of zeros at the edge points is shown the edge singularity exponents are calculated analytically for the densities of the yang lee and fisher zeros it is found that for both cases edge singularity exponents are universal and equal to sigma 1 2

an exact algorithm for the maximum leaf spanning tree problem given an undirected graph with n vertices the maximum leaf spanning tree problem is to find a spanning tree with as many leaves as possible when parameterized in the number of leaves k this problem can be solved in time o 4kpoly n using a simple branching algorithm introduced by a subset of the authors kneis et al 2008 16 daligault et al 2010 6 improved the branching and obtained a running time of o 3 72kpoly n in this paper we study the problem from an exponential time viewpoint where it is equivalent to the connected dominating set problem here fomin grandoni and kratsch showed how to break the 2n barrier and proposed an o 1 9407n time algorithm fomin et al 2008 11 based on some useful properties of kneis et al 2008 16 and daligault et al 2010 6 we present a branching algorithm whose running time of o 1 8966n has been analyzed using the measure and conquer technique finally we provide a lower bound of 1 4422n for the worst case running time of our algorithm

an o 2 0 304n algorithm for solving maximum independent set problem a faster algorithm for finding a maximum independent set in a graph is presented the algorithm is an improved version of the one by tarjan and trojanowski 7 a technique to further accelerate this algorithm is also described

new methods for 3 sat decision and worst case analysis abstract we prove the worst case upper bound 1 5045 n for the time complexity of 3 sat decision where n is the number of variables in the input formula introducing new methods for the analysis as well as new algorithmic techniques we add new 2 and 3 clauses called blocked clauses generalizing the extension rule of extended resolution our methods for estimating the size of trees lead to a refined measure of formula complexity of 3 clause sets and can be applied also to arbitrary trees

fast exact algorithm for l 2 1 labeling of graphs an l 2 1 labeling of a graph is a mapping from its vertex set into nonnegative integers such that the labels assigned to adjacent vertices differ by at least 2 and labels assigned to vertices of distance 2 are different the span of such a labeling is the maximum label used and the l 2 1 span of a graph is the minimum possible span of its l 2 1 labelings we show how to compute the l 2 1 span of a connected graph in time o 2 6488 n previously published exact exponential time algorithms were gradually improving the base of the exponential function from 4 to the so far best known 3 with 3 itself seemingly having been the holy grail for quite a while as concerns special graph classes we are able to solve the problem in time o 2 5944 n for claw free graphs and in time o 2 n r 2 nr r for graphs having a dominating set of size r

dimensional reduction formulas for branched polymer correlation functions in bi01 we have proven that the generating function for self avoiding branched polymers in d 2 continuum dimensions is proportional to the pressure of the hard core continuum gas at negative activity in d dimensions this result explains why the critical behavior of branched polymers should be the same as that of the i 3 or yang lee edge field theory in two fewer dimensions as proposed by parisi and sourlas in 1981 in this article we review and generalize the results of bi01 we show that the generating functions for several branched polymers are proportional to correlation functions of the hard core gas we derive ward identities for certain branched polymer correlations we give reduction formulae for multi species branched polymers and the corresponding repulsive gases finally we derive the massive scaling limit for the 2 point function of the one dimensional hard core gas and thereby obtain the scaling form of the 2 point function for branched polymers in three dimensions

a deterministic 2 2 k 1 n algorithm for k sat based on local search local search is widely used for solving the propositional satisfiability problem papadimitriou 1991 showed that randomized local search solves 2 sat in polynomial time recently schoning 1999 proved that a close algorithm for k sat takes time 2 2 k n up to a polynomial factor this is the best known worst case upper bound for randomized 3 sat algorithms cf also recent preprint by schuler et al we describe a deterministic local search algorithm for k sat running in time 2 2 k 1 n up to a polynomial factor the key point of our algorithm is the use of covering codes instead of random choice of initial assignments compared to other weakly exponential algorithms our algorithm is technically quite simple we also describe an improved version of local search for 3 sat the improved algorithm runs in time 1 481n up to a polynomial factor our bounds are better than all previous bounds for deterministic k sat algorithms

an improved deterministic local search algorithm for 3 sat we slightly improve the pruning technique presented in dantsin et al theoret comput sci 289 2002 69 to obtain an o 1 473n deterministic algorithm for 3 sat

improved exact algorithms for mildly sparse instances of max sat abstract we present improved exponential time exact algorithms for max sat our algorithms run in time of the form o 2 1 c n for instances with n variables and m c n clauses in this setting there are three incomparable currently best algorithms a deterministic exponential space algorithm with c 1 o c log c due to dantsin and wolpert 2006 9 a randomized polynomial space algorithm with c 1 o c log 3 c and a deterministic polynomial space algorithm with c 1 o c 2 log 2 c due to sakai seto and tamaki 2015 30 our first result is a deterministic polynomial space algorithm with c 1 o c log c that achieves the previous best time complexity without exponential space or randomization furthermore this algorithm can handle instances with exponentially large weights and hard constraints the previous algorithms and our deterministic polynomial space algorithm run super polynomially faster than 2 n only if m o n 2 our second results are deterministic exponential space algorithms for max sat with c 1 o c log c 2 3 and for max 3 sat with c 1 o c 1 2 that run super polynomially faster than 2 n when m o n 5 2 log 5 2 n and m o n 3 log 2 n respectively our third result is a randomized exponential space algorithm for max sat with c 1 o c 1 2 log 3 2 c that run super polynomially faster than 2 n when m o n 3 log 5 n

spanning forests and the q state potts model in the limit q 0 we study the q state potts model with nearest neighbor coupling v e j 1 in the limit q v 0 with the ratio w v q held fixed combinatorially this limit gives rise to the generating polynomial of spanning forests physically it provides information about the potts model phase diagram in the neighborhood of q v 0 0 we have studied this model on the square and triangular lattices using a transfer matrix approach at both real and complex values of w for both lattices we have computed the symbolic transfer matrices for cylindrical strips of widths 2 l 10 as well as the limiting curves b of partition function zeros in the complex w plane for real w we find two distinct phases separated by a transition point w w0 where w0 1 4 resp w0 0 1753 0 0002 for the square resp triangular lattice for w w0 we find a non critical disordered phase that is compatible with the predicted asymptotic freedom as w for w w0 our results are compatible with a massless berker kadanoff phase with central charge c 2 and leading thermal scaling dimension xt 1 2 marginally irrelevant operator at w w0 we find a first order critical point the first derivative of the free energy is discontinuous at w0 while the correlation length diverges as w w0 and is infinite at w w0 the critical ehavior at w w0 seems to be the same for both lattices and it differs from that of the berker kadanoff phase our results suggest that the central charge is c 1 the leading thermal scaling dimension is xt 1 0 and the critical exponents are 1 d 1 2 and 1

optimal structural design with damping constraint limitations the algorithm optimization presented uses the optimality criteria method so as to minimize the total weight of the structure in the presence of a constraint of the damping type the goal of this study is to increase the structural damping by optimizing the viscoelastic and elastic material distribution over the structure the types of damping considered are hysteristic the gradient of the constraint function in relation to the design variables is calculated with the aid of a complex variable sensitivity method several numerical results are presented to validate the algorithm proposed

an exact algorithm for the minimum dominating clique problem a subset of vertices d v of a graph g v e is a dominating clique if d is a dominating set and a clique of g the existence problem given a graph g is there a dominating clique in g is np complete and thus both the minimum and the maximum dominating clique problems are np hard we present an o 1 3387n time and polynomial space algorithm that for an input graph on n vertices either computes a minimum dominating clique or reports that the graph has no dominating clique the algorithm uses the branch reduce paradigm and its time analysis is based on the measure conquer approach we also establish a lower bound of 1 2599n for the worst case running time of the algorithm finally using memorization we obtain an o 1 3234n time and exponential space algorithm for the same problem

exact algorithms for maximum independent set abstract we show that the maximum independent set problem on an n vertex graph can be solved in 1 1996 n n o 1 time and polynomial space which even is faster than robson s 1 2109 n n o 1 time exponential space algorithm published in 1986 we also obtain improved algorithms for mis in graphs with maximum degree 6 and 7 which run in time of 1 1893 n n o 1 and 1 1970 n n o 1 respectively our algorithms are obtained by using fast algorithms for mis in low degree graphs in a hierarchical way and making a careful analysis on the structure of bounded degree graphs

optimum design of structures with multiple constraints automated design of large aerospace structures requires efficient optimization algorithms because of a large number of design variables and design constraints most of the difficulties associated with large structural design are solution convergence and computer resources requirements practical aerospace structures generally involve limitations on gage sizes displacements stresses and frequencies the objective of this study was to determine what techniques are reliable and efficient for optimization of a complex design problem the study examined the relative numerical performance of various optimization methods as candidates for a hybrid algorithm using optimality criteria and mathematical programming methods several optimization programs were used to design trussand wing type structures and the iteration history for each technique is presented the computer programs using the method of feasible directions and a modified newton s method of unconstrained minimization were the most reliable mathematical programming methods

yang lee zeros of the q state potts model on recursive lattices the yang lee zeros of the q state potts model on recursive lattices are studied for noninteger values of q considering one dimensional 1d lattice as a bethe lattice with coordination number equal to 2 the location of yang lee zeros of 1d ferromagnetic and antiferromagnetic potts models is completely analyzed in terms of neutral periodical points three different regimes for yang lee zeros are found for qg1 and 0lql1 an exact analytical formula for the equation of phase transition points is derived for the 1d case it is shown that yang lee zeros of the q state potts model on a bethe lattice are located on arcs of circles with the radius depending on q and temperature for qg1 complex magnetic field metastability regions are studied for the qg1 and 0lql1 cases the yang lee edge singularity exponents are calculated for both 1d and bethe lattice potts models the dynamics of metastability regions for different values of q is studied numerically

exact algorithms for treewidth and minimum fill in we show that the treewidth and the minimum fill in of an n vertex graph can be computed in time mathcal o 1 8899 n our results are based on combinatorial proofs that an n vertex graph has mathcal o 1 7087 n minimal separators and mathcal o 1 8135 n potential maximal cliques we also show that for the class of asteroidal triple free graphs the running time of our algorithms can be reduced to mathcal o 1 4142 n

a measure conquer approach for the analysis of exact algorithms for more than 40 years branch reduce exponential time backtracking algorithms have been among the most common tools used for finding exact solutions of np hard problems despite that the way to analyze such recursive algorithms is still far from producing tight worst case running time bounds motivated by this we use an approach that we call measure conquer as an attempt to step beyond such limitations the approach is based on the careful design of a nonstandard measure of the subproblem size this measure is then used to lower bound the progress made by the algorithm at each branching step the idea is that a smarter measure may capture behaviors of the algorithm that a standard measure might not be able to exploit and hence lead to a significantly better worst case time analysis in order to show the potentialities of measure conquer we consider two well studied np hard problems minimum dominating set and maximum independent set for the first problem we consider the current best algorithm and prove thanks to a better measure a much tighter running time bound for it for the second problem we describe a new simple algorithm and show that its running time is competitive with the current best time bounds achieved with far more complicated algorithms and standard analysis our examples show that a good choice of the measure made in the very first stages of exact algorithms design can have a tremendous impact on the running time bounds achievable

the hard hexagon partition function for complex fugacity we study the analyticity of the partition function of the hard hexagon model in the complex fugacity plane by computing zeros and transfer matrix eigenvalues for large finite size systems we find that the partition function per site computed by baxter in the thermodynamic limit for positive real values of the fugacity is not sufficient to describe the analyticity in the full complex fugacity plane we also obtain a new algebraic equation for the low density partition function per site