integer programs for logic constraint satisfaction abstract logic constraint satisfaction problems are in general np hard and a general deterministic polynomial time algorithm is not known since several logic constraint problems can be reduced in polynomial time to the satisfaction of a conjunctive normal form cnf sat this case is very important we present here a technique to transform a cnf sat problem in an integer optimization problem that can be solved by linear programming the size of the obtained integer program has a polynomial growth in comparison with the original problem size

a hybrid polynomial system solving method for mixed trigonometric polynomial systems mixed trigonometric polynomial systems arise in many fields of science and engineering commonly this class of systems is transformed into polynomial systems by variable substituting and adding some quadratic equations and then solved by some polynomial system solving method in this paper by exploiting the special structure of the additional quadratic equations an efficient hybrid method for solving polynomial systems coming from mixed trigonometric polynomial systems is presented it combines the homotopy method in which the homotopy is a combination of coefficient parameter homotopy and the random product homotopy with decomposition variable substitution and reduction techniques numerical tests are given to show its effectiveness and it is applied to solve a practical problem

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

comments on a counterexample to a diameter algorithm for convex polygons in this paper we report the results of experimental studies of zero level one level and two level search rearrangement backtracking we establish upper and lower limits for the size problem for which one level backtracking is preferred over zero level and two level methods thereby showing that the zero level method is best for very small problems the one level method is best for moderate size problems and the two level method is best for extremely large problems together with our theoretical asymptotic formulas these measurements provide a useful guide for selecting the best search rearrangement method for a particular problem

an analysis of backtracking with search rearrangement the search rearrangement backtracking algorithm of bitner and reingold comm acm 18 1975 pp 651 655 introduces at each level of the backtrack tree a variable with a minimal number of remaining values search order may differ on different branches for conjunctive normal form formulas with v variables s literals per term s geqq 3 and v alpha terms s 2 alpha s the average number of nodes in a search rearrangement backtrack tree is exp theta v s alpha 1 s 2 i e for some positive constants a 1 a 2 and v 0 when v geqq v 0 the number of nodes is between exp a 1 v s alpha 1 s 2 and exp a 2 v s alpha 1 s 2 for 1 alpha leqq s 2 the average number of nodes is between exp theta v s alpha 1 s 2 and exp theta ln v s 1 s 2 v s alpha 1 s 2 this compares with exp theta v s alpha s 1 for ordinary backtracking for 1 alpha s s

search rearrangement backtracking and polynomial average time the average time required for simple search rearrangement backtracking is compared with that for ordinary backtracking when each algorithm is used to find all solutions for random conjunctive normal form predicates the sets of random predicates are characterized by v the number of variables t v the number of clauses and p v the probability that a literal appears in clause for large v if vp v 1n 2 there is a difficult region where the average number of solutions per problem is exponentially small but backtracking requires an exponentially large time the difficult region for search rearrangement backtracking is only slightly smaller than the difficult region for ordinary backtracking it is conjectured that search rearrangement backtracking is exponentially faster than ordinary backtracking for nearly all of the difficult region it is proved that there is no major advantage in using search rearrangement backtracking outside of the difficult region

self adjusting grid methods for one dimensional hyperbolic conservation laws it is shown how to automatically adjust the grid to follow the dynamics of the numerical solution of hyperbolic conservation laws the grid motion is determined by averaging the local characteristic velocities of the equations with respect to the amplitudes of the signals the resulting algorithm is a simple extension of many currently popular godunov type methods computer codes using one of these methods can be easily modified to add the moving mesh as an option numerical examples are given that illustrate the improved accuracy of godunov s and roe s methods on a self adjusting mesh

homotopies exploiting newton polytopes for solving sparse polynomial systems this paper is concerned with the problem of finding all isolated solutions of a polynomial system the bkk bound defined as the mixed volume of the newton polytopes of the polynomials in the system is a sharp upper bound for the number of isolated solutions in mathbb c 0 n mathbb c 0 mathbb c backslash 0 of a polynomial system with a sparse monomial structure first an algorithm is described for computing the bkk bound following the lines of bernshtei n s proof the algorithmic construction of the cheater s homotopy or the coefficient homotopy is obtained the mixed homotopy methods can be combined with the random product start systems based on a generalized bezout number applications illustrate the effectiveness of the new approach

proving unsatisfiability for problems with constant cubic sparsity abstract this paper presents a perspective on a previously studied heuristic tree search algorithm for the np complete 3sat satisfiability problem which synthesizes results by brown franco purdom and others it is shown that when an a priori calculation indicates a random nontrivial 3sat formula is very likely unsatisfiable then the heuristic algorithm is expected to prove unsatisfiability in a search tree with size essentially independent of the problem size and bounded by a constant that depends only on the ratio of the number of clauses to the cube of the number of variables

moving finite elements i we present new and general numerical methods for dealing with problems whose solutions develop sharp transition layers or near shocks these methods allow many nodes automatically to concentrate in the critical regions and move with them for clarity of exposition we concentrate on the space of piecewise linear functions with movable nodes with burgers equation as our test equation but the generalization to much more general spaces and equations including even certain previous moving vorticity blobs of the first author and s doss for the navier stokes equations becomes clear in this paper we present the theoretical and computational details of our scheme along with computational trial runs for burgers equation showing that the nodes do concentrate sharply and move with the shocks as desired the conclusiveness of these preliminary numerical trials is marred somewhat by the fact that we never successfully debugged a newton s method for our implicit stiff ode solver and were thus limited to ver

shock computations with adaptive mesh refinement and moving grids abstract highly accurate shock wave problems are presented using a combination of numerical techniques the new feature of this scheme is the treatment of discontinuities shocks or contact surfaces which are replaced by approximate step functions these approximations permit the solution of the euler equations within the discontinuity and permits a convenient way of treating source terms within a discontinuity the strong conservative law form is used throughout an explicit moving grid technique is used in which a moving frame is found in which the conservation law equations appear stationery in a least squares sense in this moving frame it is possible to drastically reduce or eliminate the temporal truncation errors thus the same time integration scheme can be used throughout the computational domain in the discontinuous as well as continuous regions for accurate physics of wave interactions a nonlinear riemann solver is applied in conjunction with the riemann solver as new waves from such interactions are created an adaptive mesh refinement scheme injects new mesh points automatically to define a post collision wave structure results are presented for the shock tube problem the von neumann spherical blast wave noh s spherical shock problem shock interactions with a set up and step down contact discontinuity

the random product homotopy and deficient polynomial systems most systems ofn polynomial equations inn unknowns arising in applications aredeficient in the sense that they have fewer solutions than that predicted by the total degree of the system we introduce the random product homotopy an efficient homotopy continuation method for numerically determining all isolated solutions of deficient systems in many cases the amount of computation required to find all solutions can be made roughly proportional to the number of solutions

polynomial average time satisfiability problems a class of random conjunctive normal form cnf predicates is characterized by v the number of variables from which literals may be formed giving 2v literals p v the probability that a given literal is in a random clause and t v the number of random independently selected clauses in a random predicate determining satisfiability on such a class using backtracking takes polynomial average time under the following conditions 1 limv vp v 0 and t v ln2 v ln v l p v and 2 lim vp v lim p v 0 and t v ln2 ep v exp 2vp v for any fixed 0 additional cases are also covered an analysis by goldberg showed that problems for which p v is constant can be solved in polynomial average time by a simplified davis putnam procedure we give an algorithm for solving problems with t v ln lnv ln3 in polynomial worst case time thus all problem sets for which p v or t v grows extremely rapidly or extremely slowly can be solved in polynomial average time we also show that backtracking requires exponential average time when limvp v 0 and t v ln2 v ln v 1 p v and when limvp v limv vp v 0 and t v ln2 ge ep v exp 2vp v

a polyhedral method for solving sparse polynomial systems a continuation method is presented for computing all isolated roots of a semimixed sparse system of polynomial equations we introduce mixed subdivisions of newton polytopes and we apply them to give a new proof and algorithm for bernstein s theorem on the expected number of roots this results in a numerical homotopy with the optimal number of paths to be followed in this homotopy there is one starting system for each cell of the mixed subdivision and the roots of these starting systems are obtained by an easy combinatorial construction

the use of nonrational functions to represent steep front solutions to partial differential equations it is shown by calculated examples that it is possible to follow successfully the development and evolution of steep fronts almost to but not including actual discontinuities in the solution of time dependent partial differential equations by means of global non rational representations a characteristic steplike shape forms the basis for the representations the hyperbolic tangent and a function involving a square root both having this characteristic shape gave similar results the method of lines specifically as formulated by gelinas doss and miller j comput phys 40 1981 202 249 was used to form a system of ordinary differential equations governing the time variation of the parameters in the representation a fixed shape moving solution of the wave equation and developing steep fronts in solutions of burgers equation and of the buckley leverett equation were successfully computed the results are all displayed in the form of plotted profiles and are compared with the corresponding results in the literature

the bkk root count in c n the root count developed by bernshtein kushnirenko and khovanskii only counts the number of isolated zeros of a polynomial system in the algebraic torus c n in this paper we modify this bound slightly so that it counts the number of isolated zeros in c n our bound is apparently significantly sharper than the recent root counts found by rojas and in many cases easier to compute as a consequence of our result the huber sturmfels homotopy for finding all the isolated zeros of a polynomial system in c n can be slightly modified to obtain all the isolated zeros in c n

a convex geometric approach to counting the roots of a polynomial system consider a polynomial system f f1 fn in n variables with complex coefficients a standard way to bound the number of isolated roots of f in cn is to multiply and add the degrees of the fi in a straightfoward computation described explicitly by some variant of bezout s theorem unfortunately even if one knows a priori which monomial terms will appear this upper bound can be far from exact in practice one frequently knows which monomial terms will appear in a polynomial system before one actually decides to count or approximate its roots so a fast accurate root count which takes this additional information into account is vital we propose a much tighter upper bound on the number of isolated roots of f in cn and give an explicit combinatorial geometric criterion for when it is exact as a corollary we obtain a geometric characterization of the average case computational complexity of root counting our root count is a formula derived from the theory of toric varieties and its computation involves finding the mixed volume of n shadowed polytopes in rn our formula generalizes the bkk bound bernshtein 1975 kushnirenko 1976 khovanskii 1978 canny and rojas 1991 in four ways 1 for any r 0 n our formula can be used to count the roots whose first r coordinates are nonzero the bkk bound only handles the r n case as a corollary we obtain that for fixed n all of these root counts for generic f can be done in time polynomial in the numbers of monomial terms of the fi 2 we refine bernshtein s 1975 algebraic criterion for exactness of the bkk bound into a complete classification of the subsets of the coefficients of f whose genericity guarantees exactness in our generalized bound 3 our results are stated in terms of arbitrary algebraically closed fields of any characteristic the bkk bound was originally stated only over the complex numbers 4 we generalize all of the above to results on the number of n k dimensional components of the zero set of f1 fk when k n

the moving finite element method applications to general partial differential equations with multiple large gradients abstract the moving finite element mfe method has been reduced to practice in the automatic solution program dyla for general systems of transient partial differential equations pdes in 1 d several test examples are presented which illustrate the unique node movement and systematic control features which are intrinsic in the mfe method computational dilemmas of numerical diffusion gibbs overshooting and undershooting zone tangling and grid remap or re connection aliasing which occur frequently in conventional pde methods are essentially eliminated in the mfe method arbitrarily large gradients or shocks can be solved with extremely high resolution and accuracy for non coincident or even counterdirected propagating wavefronts boundary layers of arbitrarily small dimensions are solved with high accuracy simultaneously with the large scale structures in reactive and non reactive fluid calculations the mfe method requires a small fraction of the grid nodes which are used in conventional pde solution methods because the nodes migrate continuously and systematically to those positions where they are most needed in order to yield accurate pde solutions on entire problem domains courant friedrichs lewy time step limits are exceeded by wide margins by factors of two to several thousand finally the extension of the mfe method to 2 d is briefly discussed

computing isolated roots of sparse polynomial systems in affine space we present a symbolic probabilistic algorithm to compute the isolated roots in c n of sparse polynomial equation systems as some already known numerical algorithms solving this task our procedure is based on polyhedral deformations and homotopies but it amounts to solving a smaller number of square systems of equations and in fewer variables the output of the algorithm is a geometric resolution of a finite set of points including the isolated roots of the system the complexity is polynomial in the size of the combinatorial structure of the system supports up to a pre processing yielding the mixed cells in a subdivision of the family of these supports