an engquist osher type scheme for conservation laws with discontinuous flux adapted to flux connections we consider scalar conservation laws with the spatially varying flux h x f u 1 h x g u where h x is the heaviside function and f and g are smooth nonlinear functions adimurthi mishra and veerappa gowda j hyperbolic differ equ 2 2005 pp 783 837 pointed out that such a conservation law admits many l 1 contraction semigroups one for each so called connection a b here we define entropy solutions of type a b involving kruzkov type entropy inequalities that can be adapted to any fixed connection a b it is proved that these entropy inequalities imply the l 1 contraction property for l infty solutions in contrast to the piecewise smooth setting of adimurthi mishra and veerappa gowda for a fixed connection these entropy inequalities include a single adapted entropy of the type used by audusse and perthame proc roy soc edinburgh sect a 135 2005 pp 253 265 we prove convergence of a new difference scheme that approximates entropy solutions of type a b for any connection a b if a few parameters are varied the scheme relies on a modification of the standard engquist osher flux is simple as no 2 times2 riemann solver is involved and is designed such that the steady state solution connecting a to b is preserved in contrast to most analyses of similar problems our convergence proof is not based on the singular mapping or compensated compactness methods but on standard spatial variation estimates away from the flux discontinuity some numerical examples are presented

approximate search strategies for weighted trees the problems of classical searching and connected searching of weighted trees are known to be computationally hard in this work we give a polynomial time 3 approximation algorithm that finds a connected search strategy of a given weighted tree this in particular yields constant factor approximation algorithms for the non connected classical searching problems and for the weighted pathwidth problem for this class of graphs

the treewidth of proofs abstract so called ordered variants of the classical notions of pathwidth and treewidth are introduced and proposed as proof theoretically meaningful complexity measures for the directed acyclic graphs underlying proofs ordered pathwidth is roughly the same as proof space and the ordered treewidth of a proof is meant to serve as a measure of how far it is from being treelike length space lower bounds for k dnf refutations are generalized to arbitrary infinity axioms and strengthened in that the space measure is relaxed to ordered treewidth

digraph decompositions and monotonicity in digraph searching graph decompositions such as tree decompositions and associated width measures have been the focus of much attention in structural and algorithmic graph theory in particular it has been found that many otherwise intractable problems become tractable on graph classes of bounded tree width more recently proposals have been made to define a similar notion to tree width for directed graphs several proposals have appeared so far supported by algorithmic applications in this paper we explore the limits of algorithmic applicability of digraph decompositions and show that various natural candidates for problems which potentially could benefit from digraphs having small directed width remain np complete even on almost acyclic graphs closely related to graph and digraph decompositions are graph searching games an important property of graph searching games is monotonicity and a large number of papers addresses the question whether particular variants of these games are monotone however so far for two natural types of graph searching games underlying dag and kelly decompositions the question whether they are monotone was still open we settle this issue by showing that both variants the visible and the inert invisible graph searching games on directed graphs are non monotone

complexity of node coverage games modern software systems may exhibit a nondeterministic behavior due to many unpredictable factors in this work we propose the node coverage game a two player turn based game played on a finite game graph as a formalization of the problem to test such systems each node in the graph represents a functional equivalence class of the software under test sut one player the tester wants to maximize the node coverage measured by the number of nodes visited when exploring the game graphs while his opponent the sut wants to minimize it an optimal test would maximize the cover and it is an interesting problem to find the maximal number of nodes that the tester can guarantee to visit irrespective of the responses of the sut we show that the decision problem of whether the guarantee is less than a given number is np complete we also discuss two extensions of our result and present a testing procedure based on our result when the sut is not so hostile we establish that the decision problem of maximal node coverage guarantee of non deterministic game graphs is np complete we present two variations of node coverage game with np complete decision problem complexity we also provide an adaptive procedure that employs an np procedure to generate test cases with maximal coverage guarantee and reevaluate the mcg and regenerate test cases when the sut is not so hostile

constructing solutions of a single conservation law on construit la solution d une seule loi de conservation u t f u x 0 t 0 x u x o u 0 x x on suppose que f est reguliere et f ne s annulent qu en des points isoles

scalar conservation laws with discontinuous flux function i the viscous profile condition the equation whereh is heaviside s step function appears for example in continuous sedimentation of solid particles in a liquid in two phase flow in traffic flow analysis and in ion etching the discontinuity of the flux function atx 0 causes a discontinuity of a solution which is not uniquely determined by the initial data the equation can be written as a triangular 2 2 non strictly hyperbolic system this augmentation is non unique and a natural definition is given by means of viscous profiles by a viscous profile we mean a stationary solution ofu t f x u xx wheref is a smooth approximation of the discontinuous flux i e h is smoothed in terms of the 2 2 system the discontinuity atx 0 is either a regular lax an under or overcompressive a marginal under or overcompressive or a degenerate shock wave in some cases depending onf andg there is a unique viscous profile e g undercompressive and regular lax waves and in some cases there are infinitely many e g overcompressive waves the main purpose of the paper is to show the equivalence between a previously introduced uniqueness condition for the discontinuity of the solution atx 0 and the viscous profile condition

a difference scheme for conservation laws with a discontinuous flux the nonconvex case in a previous work by the author convergence was established for a simple difference scheme approximating a scalar conservation law where the flux was concave and had a discontinuous spatially varying coefficient j d towers siam j numer anal 38 2000 pp 681 698 the main result of this paper is an extension of that convergence theorem to the situation where the flux may have any finite number of critical points additionally spatially varying source terms are allowed the spatially varying numerical flux is also shown to satisfy maximum and minimum principles and to be total variation decreasing tvd in time

divide and conquer approximation algorithms via spreading metrics we present a novel divide and conquer paradigm for approximating np hard graph optimization problems the paradigm models graph optimization problems that satisfy two properties first a divide and conquer approach is applicable second a fractional spreading metric is computable in polynomial time the spreading metric assigns lengths to either edges or vertices of the input graph such that all subgraphs for which the optimization problem is nontrivial have large diameters in addition the spreading metric provides a lower bound t on the cost of solving the optimization problem we present a polynomial time approximation algorithm for problems modeled by our paradigm whose approximation factor is o min log t log log t log k log log k where k denotes the number of interesting vertices in the problem instance and is at most the number of vertices we present seven problems that can be formulated to fit the paradigm for all these problems our algorithm improves previous results the problems are 1 linear arrangement 2 embedding a graph in a d dimensional mesh 3 interval graph completion 4 minimizing storage time product 5 subset feedback sets in directed graphs and multicuts in circular networks 6 symmetric multicuts in directed networks 7 balanced partitions and p separators for small values of p in directed graphs

semidiscrete central upwind scheme for conservation laws with a discontinuous flux function in space abstract in this paper a modified semidiscrete central upwind scheme is derived for the scalar conservation laws with a discontinuous flux function in space the new scheme is based on dealing with the phase transition at the stationary discontinuity where the unknown variable function is not continuous but the flux function is continuous the main advantages of the new scheme are the same as them of the original semidiscrete central upwind scheme numerical results are displayed to illustrate the efficiency of the methods

fugitive search games on graphs and related parameters abstract the goal of a fugitive search game on a graph is to trap a fugitive that hides on the vertices of the graph by systematically placing searchers on the vertices the fugitive is assumed to have complete knowledge of the graph and of the searchers moves but is restricted to move only along paths whose vertices are not guarded by searchers the search number of the graph is the least number of searchers necessary to trap the fugitive variants of the fugitive search game have been related to important graph parameters like treewidth and pathwidth in this paper we introduce a class of fugitive search games where the searchers do not see the fugitive and the fugitive can only move just before a searcher is placed on the vertex it occupies letting the fugitive s speed i e the maximum number of edges the fugitive can traverse at a single move vary we get different games we show that if the speed of the fugitive is unbounded then the search number minus 1 is equal to the treewidth of the graph while if the speed is 1 then the search number minus 1 is equal to a polynomially computable graph parameter which is called width or alternatively linkage and is studied in the context of the constraint satisfaction and boolean satisfiability problems we also show that in the above two cases the search number remains the same even if we consider only search strategies that at every step further restrict the fugitive s possible resorts this monotonicity property is usually expressed as recontamination does not help moreover we give an equivalent characterization of the search number for any given fugitive speed in terms of an elimination ordering of the vertices of the graph using this characterization we show that for any graph if the length of any chordless cycle is bounded by a constant s s 3 then the treewidth of the graph plus 1 is equal to the search number for fugitive speed s 2

monotonicity of non deterministic graph searching in graph searching a team of searchers are aiming at capturing a fugitive moving in a graph in the initial variant called invisible graph searching the searchers do not know the position of the fugitive until they catch it in another variant the searchers permanently know the position of the fugitive i e the fugitive is visible this latter variant is called visible graph searching a search strategy that catches any fugitive in such a way that the part of the graph reachable by the fugitive never grows is called monotone a priori monotone strategies may require more searchers than general strategies to catch any fugitive this is however not the case for visible and invisible graph searching two important consequences of the monotonicity of visible and invisible graph searching are 1 the decision problem corresponding to the computation of the smallest number of searchers required to clear a graph is in np and 2 computing optimal search strategies is simplified by taking into account that there exist some that never backtrack fomin et al f v fomin p fraigniaud n nisse nondeterministic graph searching from pathwidth to treewidth in proceedings of the 30th international symposium on mathematical foundations of computer science mfcs 05 2005 pp 364 375 introduced an important graph searching variant called non deterministic graph searching that unifies visible and invisible graph searching in this variant the fugitive is invisible and the searchers can query an oracle that permanently knows the current position of the fugitive the question of the monotonicity of non deterministic graph searching was however left open in this paper we prove that non deterministic graph searching is monotone in particular this result is a unified proof of monotonicity for visible and invisible graph searching as a consequence the decision problem corresponding to non deterministic graph searching belongs to np moreover the exact algorithms designed by fomin et al do compute optimal non deterministic search strategies

fast edge searching and fast searching on graphs given a graph g v e in which a fugitive hides on vertices or along edges graph searching problems are usually to find the minimum number of searchers required to capture the fugitive in this paper we consider the problem of finding the minimum number of steps to capture the fugitive we introduce the fast edge searching problem in the edge search model which is the problem of finding the minimum number of steps called the fast edge search time to capture the fugitive we establish relations between the fast edge searching and the fast searching that is the problem of finding the minimum number of searchers to capture the fugitive in the fast search model while the family of graphs whose fast search number is at most k is not minor closed for any positive integer k 2 we show that the family of graphs whose fast edge search time is at most k is minor closed we establish relations between the fast fast edge searching and the node searching these relations allow us to transform the problem of computing node search numbers to the problem of computing fast edge search numbers or fast search numbers using these relations we prove that the problem of deciding given a graph g and an integer k whether the fast edge search number of g is less than or equal to k is np complete and it remains np complete for eulerian graphs we also prove that the problem of determining whether the fast edge search number of g is half of the number of odd vertices in g is np complete and it remains np complete for planar graphs with maximum degree 4 we present a linear time approximation algorithm for the fast edge search time that always delivers solutions of at most 1 v 1 e 1 times the optimal value this algorithm also gives us a tight upper bound on the fast search number of graphs we also show a lower bound on the fast search number using the minimum degree and the number of odd vertices

approximation algorithms for digraph width parameters several problems that are np hard on general graphs are efficiently solvable on graphs with bounded treewidth efforts have been made to generalize treewidth and the related notion of pathwidth to digraphs directed treewidth dag width and kelly width are some such notions which generalize treewidth whereas directed pathwidth generalizes pathwidth each of these digraph width measures have an associated decomposition structure in this paper we present approximation algorithms for all these digraph width parameters in particular we give an o log n approximation algorithm for directed treewidth and an o log 3 2 n approximation algorithm for directed pathwidth dag width and kelly width our algorithms construct the corresponding decompositions whose widths agree with the above mentioned approximation factors

connected graph searching in the graph searching game the opponents are a set of searchers and a fugitive in a graph the searchers try to capture the fugitive by applying some sequence of moves that include placement removal or sliding of a searcher along an edge the fugitive tries to avoid capture by moving along unguarded paths the search number of a graph is the minimum number of searchers required to guarantee the capture of the fugitive in this paper we initiate the study of this game under the natural restriction of connectivity where we demand that in each step of the search the locations of the graph that are clean i e non accessible to the fugitive remain connected we give evidence that many of the standard mathematical tools used so far in classic graph searching fail under the connectivity requirement we also settle the question on the price of connectivity that is how many searchers more are required for searching a graph when the connectivity demand is imposed we make estimations of the price of connectivity on general graphs and we provide tight bounds for the case of trees in particular for an n vertex graph the ratio between the connected searching number and the non connected one is o logn while for trees this ratio is always at most 2 we also conjecture that this constant ratio upper bound for trees holds also for all graphs our combinatorial results imply a complete characterization of connected graph searching on trees it is based on a forbidden graph characterization of the connected search number we prove that the connected search game is monotone for trees i e restricting search strategies to only those where the clean territories increase monotonically does not require more searchers a consequence of our results is that the connected search number can be computed in polynomial time on trees moreover we show how to make this algorithm distributed finally we reveal connections of this parameter to other invariants on trees such as the horton strahler number

scalar conservation laws with discontinuous flux function ii on the stability of the viscous profiles the equation frac partial u partial t frac partial partial x left h x f u left 1 h x right g u right 0 whereh is heaviside s step function appears for example in continuous sedimentation of solid particles in a liquid in two phase flow in traffic flow analysis and in ion etching the discontinuity of the flux function atx 0 causes a discontinuity of a solution which is not uniquely determined by the initial data by a viscous profile of this discontinuity we mean a stationary solution ofu t f x eu xx wheref is a smooth approximation of the discontinuous flux i e h is smoothed we present some results on the stability of the viscous profiles which means that a small disturbance tends to zero uniformly ast this is done by weighted energy methods where the weight depending onf andg plays a crucial role

one sided difference approximations for nonlinear conservation laws we analyze one sided or upwind finite difference approximations to hyperbolic partial differential equations and in particular nonlinear conservation laws second order schemes are designed for which we prove both nonlinear stability and that the entropy condition is satisfied for limit solutions we show that no such stable approximation of order higher than two is possible these one sided schemes have desirable properties for shock calculations we show that the proper switch used to change the direction in the upwind differencing across a shock is of great importance new and simple schemes are developed for which we prove qualitative properties such as sharp monotone shock profiles existence uniqueness and stability of discrete shocks numerical examples are given

exclusive graph searching vs pathwidth in graph searching a team of searchers aims at capturing an invisible fugitive moving arbitrarily fast in a graph equivalently the searchers try to clear a contaminated network the problem is to compute the minimum number of searchers required to accomplish this task several variants of graph searching have been studied mainly because of their close relationship with the pathwidth of a graph in this paper we study the complexity of the exclusive graph searching variant we show that the problem is np hard in planar graphs and it can be solved in linear time in the class of cographs we also show that monotone exclusive graph searching is np complete in split graphs where pathwidth is known to be solvable in polynomial time moreover we prove that monotone exclusive graph searching is in p in a subclass of star like graphs where pathwidth is known to be np hard

on the complexity of the positive semidefinite zero forcing number abstract the positive semidefinite zero forcing number of a graph is a graph parameter that arises from a non traditional type of graph colouring and is related to a more conventional version of zero forcing we establish a relation between the zero forcing and the fast mixed searching which implies some np completeness results for the zero forcing problem relationships between positive semidefinite zero forcing sets and clique coverings are well understood for chordal graphs building upon constructions associated with optimal tree covers and forest covers we present a linear time algorithm for computing the positive semidefinite zero forcing number of chordal graphs we also prove that it is np complete to determine whether a graph has a positive semidefinite zero forcing set with an additional property

new approximation techniques for some linear ordering problems we describe logarithmic approximation algorithms for the np hard graph optimization problems of minimum linear arrangement minimum containing interval graph and minimum storage time product this improves upon the best previous approximation bounds of even naor rao and schieber j acm 47 2000 pp 585 616 for these problems by a factor of omega log log n we use the lower bound provided by the volume w of a spreading metric for each of the ordering problems above as defined by even et al in order to find a solution with cost at most a logarithmic factor times w for these problems we develop a divide and conquer strategy where the cost of a solution to a problem at a recursive level is c plus the cost of a solution to the subproblems at this level and where the spreading metric volume on the subproblems is less than the original volume by omega c log n ensuring that the resulting solution has cost o log n times the original spreading metric volume we note that this is an existentially tight bound on the relationship between the spreading metric volume and the true optimal values for these problems for planar graphs we combine a structural theorem of klein plotkin and rao proceedings of the 25th acm symposium on theory of computing 1993 pp 682 690 with our new recursion technique to show that the spreading metric cost volumes are within an o log log n factor of the cost of an optimal solution for the minimum linear arrangement and the minimum containing interval graph problems

solutions of hamilton jacobi equations and scalar conservation laws with discontinuous space time dependence abstract we establish a unique stable solution to the hamilton jacobi equation u t h k x t u x 0 x t 0 with lipschitz initial condition where k x t is allowed to be discontinuous in the x t plane along a finite number of possibly intersecting curves parameterized by t we assume that for fixed k h k p is convex in p and lim p h k p p the solution is determined by showing that if k is made smooth by convolving k in the x direction with the standard mollifier then the control theory representation of the viscosity solution to the resulting hamilton jacobi equation must converge uniformly as the mollification decreases to a lipschitz continuous solution with an explicit control theory representation this also defines the unique stable solution to the corresponding scalar conservation law u t f k x t u x 0 x t 0 with k discontinuous

connected searching of weighted trees in this paper we consider the problem of connected edge searching of weighted trees barriere et al claim in l barriere p flocchini p fraigniaud n santoro capture of an intruder by mobile agents in spaa 02 proceedings of the fourteenth annual acm symposium on parallel algorithms and architectures acm new york ny usa 2002 pp 200 209 that there exists a polynomial time algorithm for finding an optimal search strategy that is a strategy that minimizes the number of used searchers however due to some flaws in their algorithm the problem turns out to be open it is proven in this paper that the considered problem is strongly np complete even for node weighted trees the weight of each edge is 1 with one vertex of degree greater than 2 it is also shown that there exists a polynomial time algorithm for finding an optimal connected search strategy for a given bounded degree tree with arbitrary weights on the edges and on the vertices this is an fpt algorithm with respect to the maximum degree of a tree

integral closedness of complete intersection ideals the aim of this paper is to give some structure theorem of certain integrally closed ideals in noetherian rings let a be a noetherian ring and i an ideal of a then an element x of a is said to be integral over z if x satisfies an equation x c x c 0 in a with ci e i the integral closure i of z is by definition the set of elements in a which are integral over z and the ideal z is called integrally closed if f i let ua z resp ht i denote the smallest number of elements in systems of generators for z resp the height of i let min a z be the set of minimal prime divisors of i in this paper we shall study the problem when the ideals z of a such that ua z ht z are integrally closed and our result is stated as follows

finite graph automata for linear and boundary graph languages graph grammars can be regarded as a generalization of context free grammars from strings to graphs over the past 30 years a rich theory of graph grammars and their languages has been developed however there are no graph automata there is no duality between generative and recognizing devices as it is known for the chomsky hierarchy of formal languages here we introduce graph automata as devices for the recognition of sets of undirected node labelled graphs a graph automaton consists of a finite state control a finite set of instructions and a collection of heads or guards it reads an input graph in a systematic way and performs a graph search directed by the instructions as our main results we show that finite graph automata recognize exactly the set of graph languages generated by linear nce graph grammars and that alternating finite graph automata recognize exactly the languages of boundary graph grammars finally we generalize some automata theoretic properties from string to graph automata integrate the connectivity of graphs into graph automata and explain why graph automata cannot be generalized to deal with dynamic edge relabellings and ence graph languages

linear rank width and linear clique width of trees we show that for every forest t the linear rank width of t is equal to the path width of t and the linear clique width of t equals the path width of t plus two provided that t contains a path of length three it follows that both linear rank width and linear clique width of forests can be computed in linear time using our characterization of linear rank width of forests we determine the set of minimal excluded acyclic vertex minors for the class of graphs of linear rank width at most k

edge and node searching problems on trees in this paper we show that there is a natural correspondence between a tree s edge search strategy and its node search strategy by doing so we simplify the previous linear time algorithm for determining the edge search number of a tree and improve the running time of a previous algorithm for constructing an edge search strategy or plan for a tree containing n vertices from o n log n to o n time

the vertex separation and search number of a graph abstract we relate two concepts in graph theory and algorithmic complexity namely the search number and the vertex separation of a graph let s g denote the search number and vs g denote the vertex separation of a connected undirected graph g we show that vs g s g vs g 2 and we give a simple transformation from g to g such that vs g s g we characterize those trees having a given vertex separation and describe the smallest such trees we also note that there exist trees for which the difference between search number and vertex separation is indeed 2 we give algorithms that for any tree t compute vs t in linear time and compute an optimal layout with respect to vertex separation in time o n log n vertex separation has previously been related to progressive black white pebble demand and has been shown to be identical to a variant of search number node search number and to path width which has been related to gate matrix layout cost all these properties are known to be computationally intractable for fixed k an o n log 2 n algorithm is known which decides whether a graph has path width at most k

jumping robbers in digraphs abstract in cops and robber games a team of cops tries to capture a robber on a given graph the cops occupy some vertices of the graph and the robber occupies one vertex given concrete rules how the players can move and when the robber is captured the minimum number of cops needed to capture the robber is a natural parameter of the graph richerby and thilikos 25 generalise the game to a game where the cops must capture multiple robbers they show that the number of cops needed to capture r robbers is at most k log r where k is the minimum number of cops who can capture one robber furthermore this bound is tight moreover capturing every new robber may demand some new cops and infinitely many robbers demand as many cops as one invisible robber we generalise the results of richerby and thilikos in two aspects we give a linear bound on the number of additional cops in the game 1 on directed graphs and 2 where the robbers have more power every robber can leave his vertex and jump to another robber we show that our bound is tight and that the reason for a worse lower bound is not the directedness of the graphs but the ability of the robbers to jump we also prove that adding new robbers induces a hierarchy of the cop numbers for a fixed graph which is strict on some graphs and converges to the minimum number of cops needed to capture an invisible robber as in 25

a comparison of two variations of a pebble game on graphs abstract the number of pebbles used in the black black white pebble game corresponds to the storage requirement of the deterministic non deterministic evaluation of a straight line program suppose a distinguished vertex of a directed acyclic graph can be pebbled with k pebbles in the black white pebble game then it can be pebbled with k 1 2k k 1 1 pebbles in the black pebble game

inapproximability results for maximum edge biclique minimum linear arrangement and sparsest cut we consider the minimum linear arrangement problem and the uniform sparsest cut problem so far these two notorious np hard graph problems have resisted all attempts to prove inapproximability results we show that they have no polynomial time approximation scheme unless np complete problems can be solved in randomized subexponential time furthermore we show that the same techniques can be used for the maximum edge biclique problem for which we obtain a hardness factor similar to previous results but under a more standard assumption

conservation law with the flux function discontinuous in the space variable ii we deal with a single conservation law with discontinuous convex concave type fluxes which arise while considering sign changing flux coefficients the main difficulty is that a weak solution may not exist as the rankine hugoniot condition at the interface may not be satisfied for certain choice of the initial data we develop the concept of generalized entropy solutions for such equations by replacing the rankine hugoniot condition by a generalized rankine hugoniot condition the uniqueness of solutions is shown by proving that the generalized entropy solutions form a contractive semi group in l 1 existence follows by showing that a godunov type finite difference scheme converges to the generalized entropy solution the scheme is based on solutions of the associated riemann problem and is neither consistent nor conservative the analysis developed here enables to treat the cases of fluxes having at most one extrema in the domain of definition completely numerical results reporting the performance of the scheme are presented

the searchlight problem for road networks we consider the problem of searching for a mobile intruder hiding in a road network given as the union of two or more lines or two or more line segments in the plane some of the intersections of the road network are occupied by stationary guards equipped with a number of searchlights each of which can emit a single ray of light in any direction along the lines or line segments it is on the goal is to detect the intruder that is to illuminate its location guards may alter the direction in which they aim a searchlight but need to switch it off for some finite time interval to effect the change in contrast the intruder may move with arbitrary speed along the network but cannot pass guards and exploit this time interval to recontaminate previously illuminated sections of the network for various classes of road networks characterized by the number n of lines or line segments comprising it and the number g n 1 of possible locations of guards fixed in advance and guaranteed to give complete coverage we present several upper and lower bounds on the worst case number of searchlights each placed at one of the guard positions required to successfully search a given road network in particular we prove the following results 1 min 2 g 1 n 2 searchlights are sometimes necessary and min 7 3 g n 1 are always sufficient for searching a road network given as the union of n lines 2 g log n g searchlights are sometimes necessary and o g 2 log n searchlights are always sufficient for searching a road network given as the union of n line segments and3 at most one searchlight per guard position and hence a total of at most g searchlights is always sufficient for searching a road network given as the union of axis aligned lines or line segments the proofs of the upper bounds induce algorithms for generating a search schedule for detecting the intruder using the claimed number of searchlights the problem of searching for an intruder hiding in a road network is introduced upper and lower bounds on the worst case number of searchlights are presented the proofs of the upper bounds induce algorithms for generating a search schedule

on minimum cost edge searching we consider the problem of finding edge search strategies of minimum cost the cost of a search strategy is the sum of searchers used in the clearing steps of the search one of the natural questions is whether it is possible to find a search strategy that minimizes both the cost and the number of searchers used to clear a given graph g we call such a strategy ideal we prove by an example that ideal search strategies do not exist in general on the other hand we provide a formula for the cost of clearing complete graphs from our construction it follows that an ideal search strategy of a complete graph does exist and can be calculated efficiently for general graphs g we give a polynomial time o logn approximation algorithm for finding minimum cost search strategies we also prove that recontamination does not help to obtain minimum cost edge search strategies

pathwidth bandwidth and completion problems to proper interval graphs with small cliques we study two related problems motivated by molecular biology given a graph g and a constant k does there exist a supergraph g of g that is a unit interval graph and has clique size at most k given a graph g and a proper k coloring c of g does there exist a supergraph g of g that is properly colored by c and is a unit interval graph we show that those problems are polynomial for fixed k on the other hand we prove that the first problem is equivalent to deciding if the bandwidth of g is at most k 1 hence it is np hard and w t hard for all t we also show that the second problem is w 1 hard this implies that for fixed k both of the problems are unlikely to have an o n alpha algorithm where alpha is a constant independent of k a central tool in our study is a new graph theoretic parameter closely related to pathwidth an unexpected useful consequence is the equivalence of this parameter to the bandwidth of the graph

fast mixed searching and related problems on graphs abstract in this paper we introduce the fast mixed search model which is a combination of the fast search model and the mixed search model given a graph g in which a fugitive hides on vertices or along edges the fast mixed search problem is to find the minimum number of searchers to capture the fugitive in the fast mixed search model we establish a relation between the fast mixed search problem and the induced path cover problem we also consider relations between the fast mixed search problem and other graph search problems we prove that fast mixed searching from leaves is np complete and it remains np complete for planar graphs with maximum degree 3 we also prove that fast mixed searching between leaves is np complete and it remains np complete for graphs with maximum degree 4 we present linear time algorithms for computing the fast mixed search number and optimal search strategies of some classes of graphs including trees cacti and proper interval graphs

a conservation law with point source and discontinuous flux function modelling continuous sedimentation continuous sedimentation of solid particles in a liquid takes place in a clarifier thickener unit which has one feed inlet and two outlets the aim of the paper is to formulate and analyse a dynamic model for this process under idealized physical conditions the conservation of mass yields the scalar conservation law frac partial u x t partial t frac partial partial x f u x t x s t delta x where delta is the dirac measure s is a source and f is a flux function which is discontinuous at three points in the one dimensional space coordinate x under certain regularity assumptions a procedure for constructing a solution locally in time is presented the nonlinear phenomena are complicated and so is the general construction of a solution the problem of nonuniqueness due to the discontinuities of f u cdot is handled by a generalized entropy condition an advantage of this approach is that the a priori boundary conditions at the discontinuities

boundary conditions for nonlinear hyperbolic systems of conservation laws we propose two formulations of the boundary conditions for nonlinear hyperbolic systems of conservation laws a first approach is based on the vanishing viscosity method and a second one is related to the riemann problem the equivalence between these two conditions is studied the latter formulation is extended to treat numerically physically relevant boundary conditions monodimensional experiments are presented

csp duality and trees of bounded pathwidth we study non uniform constraint satisfaction problems definable in monadic datalog stratified by the use of non linearity we show how such problems can be described in terms of homomorphism dualities involving trees of bounded pathwidth and in algebraic terms for this we introduce a new parameter for trees that closely approximates pathwidth and can be characterised via a hypergraph searching game

a geometric optimization approach to detecting and intercepting dynamic targets using a mobile sensor network a methodology is developed to deploy a mobile sensor network for the purpose of detecting and capturing mobile targets in the plane the sensing pursuit problem considered in this paper is analogous to the marco polo game in which a pursuer marco must capture multiple mobile targets that are sensed intermittently and with very limited information the competing objectives exhibited by this problem arise in a number of surveillance and monitoring applications in this paper the mobile sensor network consists of a set of robotic sensors that must track and capture mobile targets based on the information obtained through cooperative detections when these detections form a satisfactory target track a mobile sensor is switched to pursuit mode and deployed to capture the target in minimum time since the sensors are installed on robotic platforms and have limited range the geometry of the platforms and of the sensors fields of view play a key role in obstacle avoidance and target detection a new cell decomposition approach is presented to determine the probability of detection and the cost of operating the sensors from the geometric properties of the network and its workspace the correctness and complexity of the algorithm are analyzed proving that the termination time is a function of the network parameters and of the number of required detections

searching and pebbling abstract we relate the search number of an undirected graph g with the minimum and maximum of the progressive pebble demands of the directed acyclic graphs obtained by orienting g towards this end we introduce node searching a slight variant of searching in which an edge is cleared by placing searchers on both of its endpoints we also show that the minimum number of searchers necessary to node search a graph equals its vertex separator plus one

godunov type methods for conservation laws with a flux function discontinuous in space scalar conservation laws with a flux function discontinuous in space are approximated using a godunov type method for which a convergence theorem is proved the case where the flux functions at the interface intersect is emphasized a very simple formula is given for the interface flux a numerical comparison between the godunov numerical flux and the upstream mobility flux is presented for two phase flow in porous media a consequence of the convergence theorem is an existence theorem for the solution of the scalar conservation laws under consideration furthermore for regular solutions uniqueness has been shown

regularity and large time behaviour of solutions of a conservation law without convexity a l aide de la methode des caracteristiques generalisees on etudie la regularite et le comportement a grand des solutions faibles admissibles d une seule loi de conservation a 1 variable d espace avec un point d inflexion

convergence of a difference scheme for conservation laws with a discontinuous flux convergence is established for a scalar finite difference scheme based on the godunov or engquist osher eo flux for scalar conservation laws having a flux that is spatially dependent through a possibly discontinuous coefficient other works in this direction have established convergence for methods employing the solution of 2 2 riemann problems the algorithm discussed here uses only scalar riemann solvers satisfaction of a set of kruzkov type entropy inequalities is established for the limit solution from which geometric entropy conditions follow assuming a piecewise constant coefficient it is shown that these conditions imply l1 contractiveness for piecewise c1 solutions thus extending a well known theorem

convergence of upwind finite difference schemes for a scalar conservation law with indefinite discontinuities in the flux function we consider the scalar conservation law with flux function discontinuous in the space variable i e begin eqnarray label eq1 u t h x f u 1 h x g u x 0 quad mbox in r times r nonumber u 0 x u 0 x quad mbox in r label 0 1 end eqnarray where h is the heaviside function and f and g are smooth with the assumptions that either f is convex and g is concave or f is concave and g is convex the existence of a weak solution of ref eq1 is proved by showing that upwind finite difference schemes of godunov and enquist osher type converge to a weak solution uniqueness follows from a kruzkhov type entropy condition we also provide explicit solutions to the riemann problem for ref eq1 at the level of numerics we give easy to implement numerical schemes of godunov and enquist osher type the central feature of this paper is the modification of the singular mapping technique the main analytical tool for these types of equations which allows us to show that the numerical schemes converge equations of type ref eq1 with the above hypothesis on the flux may occur when considering the following scalar conservation law with discontinuous flux begin equation label eq2 begin array email protected l u t k x f u x 0 u 0 x u 0 x end array end equation with f convex and k of indefinite sign

a theory of l 1 dissipative solvers for scalar conservation laws with discontinuous flux we propose a general framework for the study of l1 contractive semigroups of solutions to conservation laws with discontinuous flux u t mathfrak f x u x 0 qquad mathfrak f x u left begin array ll f l u x 0 end array right quad quad quad rm cl where the fluxes fl fr are mainly assumed to be continuous developing the ideas of a number of preceding works baiti and jenssen in j differ equ 140 1 161 185 1997 towers in siam j numer anal 38 2 681 698 2000 towers in siam j numer anal 39 4 1197 1218 2001 towers et al in skr k nor vidensk selsk 3 1 49 2003 adimurthi et al in j math kyoto university 43 1 27 70 2003 adimurthi et al in j hyperbolic differ equ 2 4 783 837 2005 audusse and perthame in proc roy soc edinburgh a 135 2 253 265 2005 garavello et al in netw heterog media 2 159 179 2007 burger et al in siam j numer anal 47 1684 1712 2009 we claim that the whole admissibility issue is reduced to the selection of a family of elementary solutions which are piecewise constant weak solutions of the form c x c l11 left x 0 right we refer to such a family as a germ it is well known that cl admits many different l1 contractive semigroups some of which reflect different physical applications we revisit a number of the existing admissibility or entropy conditions and identify the germs that underly these conditions we devote specific attention to the vanishing viscosity germ which is a way of expressing the condition of diehl j hyperbolic differ equ 6 1 127 159 2009 for any given germ we formulate germ based admissibility conditions in the form of a trace condition on the flux discontinuity line x 0 in the spirit of vol pert math ussr sbornik 2 2 225 267 1967 and in the form of a family of global entropy inequalities following kruzhkov math ussr sbornik 10 2 217 243 1970 and carrillo arch ration mech anal 147 4 269 361 1999 we characterize those germs that lead to the l1 contraction property for the associated admissible solutions our approach offers a streamlined and unifying perspective on many of the known entropy conditions making it possible to recover earlier uniqueness results under weaker conditions than before and to provide new results for other less studied problems several strategies for proving the existence of admissible solutions are discussed and existence results are given for fluxes satisfying some additional conditions these are based on convergence results either for the vanishing viscosity method with standard viscosity or with specific viscosities adapted to the choice of a germ or for specific germ adapted finite volume schemes

monotony properties of connected visible graph searching search games are attractive for their correspondence with classical width parameters for instance the invisible search number a k a node search number of a graph is equal to its pathwidth plus 1 and the visible search number of a graph is equal to its treewidth plus 1 the connected variants of these games ask for search strategies that are connected i e at every step of the strategy the searched part of the graph induces a connected subgraph we focus on monotone search strategies i e strategies for which every node is searched exactly once the monotone connected visible search number of an n node graph is at most o logn times its visible search number first we prove that this logarithmic bound is tight precisely we prove that there is an infinite family of graphs for which the ratio monotone connected visible search number over visible search number is w logn second we prove that as opposed to the non connected variant of visible graph searching recontamination helps for connected visible search precisely we prove that for any k 4 there exists a graph with connected visible search number at most k and monotone connected visible search number k

finding hidden independent sets in interval graphs we design efficient competitive algorithms for discovering hidden information using few queries specifically consider a game in a given set of intervals and their implied interval graph g in which our goal is to discover an unknown independent set x by making the fewest queries of the form is point p covered by an interval in x our interest in this problem stems from two applications experimental gene discovery with pcr technology and the game of battleship in a 1 dimensional setting we provide adaptive algorithms for both the verification scenario given an independent set is it x and the discovery scenario find x without any information under some assumptions these algorithms use an asymptotically optimal number of queries in every instance

recontamination does not help to search a graph this paper is concerned with a game on graphs called graph searching the object of this game is to clear all edges of a contaminated graph clearing is achieved by moving searchers a kind of token along the edges of the graph according to clearing rules certain search strategies cause edges that have been cleared to become contaminated again megiddo et al 9 conjectured that every graph can be searched using a minimum number of searchers without this recontamination occurring that is without clearing any edge twice in this paper this conjecture is proved this places the graph searching problem in np completing the proof by megiddo et al that the graph searching problem is np complete furthermore by eliminating the need to consider recontamination this result simplifies the analysis of searcher requirements with respect to other properties of graphs

min cut is np complete for edge weighted trees abstract we show that the min cut linear arrangement problem min cut is np complete for trees with polynomial size edge weights and derive from this the np completeness of min cut for planar graphs with maximum vertex degree 3 this is used to show the np completeness of search number vertex separation progressive black white pebble demand and topological bandwidth for planar graphs with maximum vertex degree 3

fast approximate graph partitioning algorithms we study graph partitioning problems on graphs with edge capacities and vertex weights the problems of b balanced cuts and k balanced partitions are unified into a new problem called minimum capacity rho separators a rho separator is a subset of edges whose removal partitions the vertex set into connected components such that the sum of the vertex weights in each component is at most rho times the weight of the graph we present a new and simple o log n approximation algorithm for minimum capacity rho separators which is based on spreading metrics yielding an o log n approximation algorithm both for b balanced cuts and k balanced partitions in particular this result improves the previous best known approximation factor for k balanced partitions in undirected graphs by a factor of o log k we enhance these results by presenting a version of the algorithm that obtains an o log opt approximation factor the algorithm is based on a technique called spreading metrics that enables us to formulate directly the minimum capacity rho separator problem as an integer program we also introduce a generalization called the simultaneous separator problem where the goal is to find a minimum capacity subset of edges that separates a given collection of subsets simultaneously we extend our results to directed graphs for values of rho geq 1 2 we conclude with an efficient algorithm for computing an optimal spreading metric for rho separators this yields more efficient algorithms for computing b balanced cuts than were previously known

computing the vertex separation of unicyclic graphs we describe an o n log n algorithm for the computation of the vertex separation of unicyclic graphs the algorithm also computes a linear layout with optimal vertex separation in the same time bound pathwidth node search number and vertex separation are different ways of defining the same notion path decompositions and search strategies can be derived from linear layouts the algorithm applies existing linear time techniques for the computation of the vertex separation of trees together with corresponding optimal layouts we reformulate the earlier work on the linear time computation of optimal layouts a polynomial time algorithm for the problem on unicyclic graphs can be inferred from existing more general methods for graphs of fixed treewidth since unicyclic graphs have treewidth two but the time complexity of the resulting method would seem to be an inordinately high order polynomial our algorithm we claim is practical the addition of one edge to a tree does seem to require a considerably more elaborate algorithm

bounds related to projective equivalence classes of ideals let z be a regular ideal in noetherian ring r and let i denote the integral closure of i a situation which is often of interest is when there is an integer b z 2 0 such that i b a e i for all n 2 1 for instance in an analytically unramified local ring such a b z exists for all i now recall that the ideal h is projectively equivalent to i if there are positive integers r and s with h i in section 1 we make the easy observation that the existence of b z as above is an invariant of the projective equivalence class of i that is if such a b z exists and if the ideal h is projectively equivalent to z then there is a b h with h bch n c h for all n 2 1 of course it may happen that b h b z the main purpose of section 1 is to study the stronger condition that there exist a fixed integer b 3 0 such that for all ideals h projectively equivalent to z h c h for all n 1 we make use of a key result namely that given z there exists an integer d such that for any ideal h projectively equivalent to z one can write hd p for some s let s 2 1 be an integer and let the ideal k be a reduction of i this means there is an integer e 20 depending on s and k with k z z in section 2 we discuss a situation in which e can in fact

explicit hopf lax type formulas for hamilton jacobi equations and conservation laws with discontinuous coefficients we deal with a hamilton jacobi equation with a hamiltonian that is discontinuous in the space variable this is closely related to a conservation law with discontinuous flux recently an entropy framework for single conservation laws with discontinuous flux has been developed which is based on the existence of infinitely many stable semigroups of entropy solutions based on an interface connection in this paper we characterize these infinite classes of solutions in terms of explicit hopf lax type formulas which are obtained from the viscosity solutions of the corresponding hamilton jacobi equation with discontinuous hamiltonian this also allows us to extend the framework of infinitely many classes of solutions to the hamilton jacobi equation and obtain an alternative representation of the entropy solutions for the conservation law we have considered the case where both the hamiltonians are convex concave furthermore we also deal with the less explored case of sign changing coefficients in which one of the hamiltonians is convex and the other concave in fact in convex concave case we cannot expect always an existence of a solution satisfying rankine hugoniot condition across the interface therefore the concept of generalised rankine hugoniot condition is introduced and prove existence and uniqueness

on a class of metrics related to graph layout problems we examine the metrics that arise when a finite set of points is embedded in the real line in such a way that the distance between each pair of points is at least 1 these metrics are closely related to some other known metrics in the literature and also to a class of combinatorial optimization problems known as graph layout problems we prove several results about the structure of these metrics in particular it is shown that their convex hull is not closed in general we then show that certain linear inequalities define facets of the closure of the convex hull finally we characterize the unbounded edges of the convex hull and of its closure

mixed searching and proper path width abstract this paper considers a mixed search game which is a natural generalization of edge search and node search games extensively studied so far we establish a relationship between the mixed search number of a graph g and the proper path width of g introduced by the authors in a previous paper we also prove complexity results

computing the chromatic number using graph decompositions via matrix rank abstract computing the smallest number q such that the vertices of a given graph can be properly q colored known as the chromatic number is one of the oldest and most fundamental problems in combinatorial optimization the q coloring problem has been studied intensively using the framework of parameterized algorithmics resulting in a very good understanding of the best possible algorithms for several parameterizations based on the structure of the graph for example algorithms are known to solve the problem on graphs of treewidth tw in time o q tw while a running time of o q e tw is impossible assuming the strong exponential time hypothesis seth while there is an abundance of work for parameterizations based on decompositions of the graph by vertex separators almost nothing is known about parameterizations based on edge separators we fill this gap by studying q coloring parameterized by cutwidth and parameterized by pathwidth in bounded degree graphs our research uncovers interesting new ways to exploit small edge separators we present two algorithms for q coloring parameterized by cutwidth ctw a deterministic one that runs in time o 2 ctw where is the square matrix multiplication exponent and a randomized one with runtime o 2 ctw in sharp contrast to earlier work the running time is independent of q the dependence on cutwidth is optimal we prove that even 3 coloring cannot be solved in o 2 e ctw time assuming seth our algorithms rely on a new rank bound for a matrix that describes compatible colorings combined with a simple communication protocol for evaluating a product of two polynomials this also yields an o d 2 1 pw time randomized algorithm for q coloring on graphs of pathwidth pw and maximum degree d such a runtime was first obtained by bjorklund but only for graphs with few proper colorings we also prove that this result is optimal in the sense that no o d 2 1 e pw time algorithm exists assuming seth

interval completion is fixed parameter tractable we present an algorithm with runtime o k 2k n 3m for the following np complete problem m garey and d johnson computers and intractability a guide to the theory of np completeness w h freeman and co san francisco 1979 problem gt35 given an arbitrary graph g on n vertices and m edges can we obtain an interval graph by adding at most k new edges to g this resolves the long standing open question h kaplan r shamir and r e tarjan siam j comput 28 1999 pp 1906 1922 r g downey and m r fellows parameterized complexity springer verlag new york 1999 m serna and d thilikos bull eur assoc theory comput sci eatcs 86 2005 pp 41 65 g gutin s szeider and a yeo in proceedings iwpec 2006 lecture notes in comput sci 4169 springer verlag berlin 2006 pp 60 71 first posed by kaplan shamir and tarjan of whether this problem was fixed parameter tractable the problem has applications in profile minimization for sparse matrix computations j a george and j w h liu computer solution of large sparse positive definite systems prentice hall englewood cliffs nj 1981 r e tarjan in sparse matrix computations j r bunch and d j rose eds academic press 1976 pp 3 22 and our results show tractability for the case of a small number k of zero elements in the envelope our algorithm performs bounded search among possible ways of adding edges to a graph to obtain an interval graph and combines this with a greedy algorithm when graphs of a certain structure are reached by the search

tradeoffs in process strategy games with application in the wdm reconfiguration problem we consider a variant of the graph searching games that models the routing reconfiguration problem in wdm networks in the digraph processing game a team of agents aims at processing or clearing the vertices of a digraph d we are interested in two different measures 1 the total number of agents used and 2 the total number of vertices occupied by an agent during the processing of d these measures respectively correspond to the maximum number of simultaneous connections interrupted and to the total number of interruptions during a routing reconfiguration in a wdm network previous works have studied the problem of independently minimizing each of these parameters in particular the corresponding minimization problems are apx hard and the first one is known not to be in apx in this paper we give several complexity results and study tradeoffs between these conflicting objectives in particular we show that minimizing one of these parameters while the other is constrained is np complete then we prove that there exist some digraphs for which minimizing one of these objectives arbitrarily impairs the quality of the solution for the other one we show that such bad tradeoffs may happen even for a basic class of digraphs on the other hand we exhibit classes of graphs for which good tradeoffs can be achieved we finally detail the relationship between this game and the routing reconfiguration problem in particular we prove that any instance of the processing game i e any digraph corresponds to an instance of the routing reconfiguration problem

multicommodity max flow min cut theorems and their use in designing approximation algorithms in this paper we establish max flow min cut theorems for several important classes of multicommodity flow problems in particular we show that for any n node multicommodity flow problem with uniform demands the max flow for the problem is within an o log n factor of the upper bound implied by the min cut the result which is existentially optimal establishes an important analogue of the famous 1 commodity max flow min cut theorem for problems with multiple commodities the result also has substantial applications to the field of approximation algorithms for example we use the flow result to design the first polynomial time polylog n times optimal approximation algorithms for well known np hard optimization problems such as graph partitioning min cut linear arrangement crossing number vlsi layout and minimum feedback arc set applications of the flow results to path routing problems network reconfiguration communication in distributed networks scientific computing and rapidly mixing markov chains are also described in the paper categories and subject descriptors f 2 2 analysis of algorithms and problem complexity

the complexity of searching a graph t parsons originally proposed and studied the following pursuit evasion problem on graphs members of a team of searchers traverse the edges of a graph g in pursuit of a fugitive who moves along the edges of the graph with complete knowledge of the locations of the pursuers what is the smallest number s g of searchers that will suffice for guaranteeing capture of the fugitive it is shown that determining whether s g k for a given integer k is np complete for general graphs but can be solved in linear time for trees we also provide a structural characterization of those graphs g with s g k for k 1 2 3

choice of measure source terms in interface coupling for a model problem in gas dynamics this paper is devoted to the mathematical and numerical analysis of a coupling procedure for one dimensional euler systems the two systems have different closure laws and are coupled through a thin fixed interface following the work of 5 we propose to couple these systems by a bounded vector valued dirac measure concentrated at the coupling interface which in the applications may have a physical meaning we show that the proposed framework allows to control the coupling conditions and we propose an approximate riemann solver based on a relaxation approach preserving equilibrium solutions of the coupled problem numerical experiments in constrained optimization problems are then presented to assess the performances of the present method 1 introduction the study of large scale and complex problems exhibiting a wide range of physical space and time scales see for instance 62 35 14 usually requires separate solvers adapted to the resolution of specific scales this is the case of many industrial flows let us quote for example the numerical simulation of two phase flows applied to the burning liquid oxygen hydrogen gas in rocket engines 58 this kind of flow contains both separated and dispersed two phase flows due to atomization and evaporation phenomena this requires appropriate models and solvers for separated and dispersed phases that have to be appropriately coupled another example concerns turbomachine flows which can be modeled by the euler equations of gas dynamics with different closure laws between the stages of the turbine where the conditions of temperature and pressure are strongly heterogeneous the coupling of these different systems is thus necessary to give a complete description of the flow inside the whole turbine the method of interface coupling allows to represent the evolution of such flows where different models are separated by fixed interfaces first coupling conditions are specified at the interface to exchange information between the systems the definition of transmission conditions generally results from physical consideration e g the conservation or the continuity of given variables then the transmission conditions are represented at the discrete level the study of interface coupling for nonlinear hyperbolic systems has received attention for several years in 43 the authors study the scalar case from both mathematical and numerical points of view

central schemes and systems of conservation laws with discontinuous coefficients modeling gravity separation of polydisperse suspensions in this paper two existing one dimensional mathematical models one for continuous sedimentation of monodisperse suspensions and one for settling of polydisperse suspensions are combined into a model of continuous separation of polydisperse mixtures this model can be written as a first order system of conservation laws for the local concentrations of each particle species with a flux vector that depends discontinuously on the space variable this application motivates the extension of the kurganov tadmor central difference scheme to systems with discontinuous flux the new central schemes are based on discretizing an enlarged system in which the discontinuous coefficients are viewed as additional conservation laws these additional conservation laws can either be discretized and the evolution of the discontinuity parameters is calculated in each time step or solved exactly that is the discontinuity parameters are kept constant with respect to time numerical examples and an l1 error study show that the kurganov tadmor scheme with first order in time discretization produces spurious oscillations whereas its semi discrete version discretized by a second order runge kutta scheme produces good results the scheme with discontinuity parameters kept constant is slightly more accurate than when these are evolved numerical examples illustrate the application to separation of polydisperse suspensions

recognizing hyperelliptic graphs in polynomial time abstract based on analogies between algebraic curves and graphs baker and norine introduced divisorial gonality a graph parameter for multigraphs related to treewidth multigraph algorithms and number theory various equivalent definitions of the gonality of an algebraic curve translate to different notions of gonality for graphs called stable gonality and stable divisorial gonality we consider so called hyperelliptic graphs multigraphs of gonality 2 in any meaning of graph gonality and provide a safe and complete set of reduction rules for such multigraphs this results in an algorithm to recognize hyperelliptic graphs in time o m n log n where n is the number of vertices and m the number of edges of the multigraph a corollary is that we can decide with the same runtime whether a two edge connected graph g admits an involution such that the quotient g is a tree

projectively full ideals in noetherian rings ii abstract let r be a noetherian commutative ring with unit 1 0 and let i be a regular proper ideal of r the set p i of integrally closed ideals projectively equivalent to i is linearly ordered by inclusion and discrete there is naturally associated to p i a numerical semigroup s i we have s i n if and only if every element of p i is the integral closure of a power of the largest element j of p i if this holds the ideal j and the set p i are said to be projectively full if i is invertible and r is integrally closed we prove that p i is projectively full we investigate the behavior of projectively full ideals in various types of ring extensions we prove that a normal ideal i of a local ring r m is projectively full if i m 2 and both the associated graded ring g m and the fiber cone ring f i are reduced we present examples of normal local domains r m of altitude two for which the maximal ideal m is not projectively full

hyperbolic conservation laws with space dependent flux i characteristics theory and riemann problem in the paper a kind of one dimensional scalar hyperbolic conservation laws with flux functions dependent on space variable is discussed and analyzed a better understanding about the behavior of wave propagation of the kind problems is presented especially some sufficient and necessary conditions that ensure the unique physically relevant solution to the riemann problem are proposed because the numerical flux obtained from the riemann s solver is theoretically correct and exact to the problem it must also be of high resolution in its nature for comparison some convincing numerical examples from traffic flow problems are given at the end of the paper

a two person game on graphs where each player tries to encircle his opponent s men we present results on a combinatorial game which was proposed to one of the authors by ingo althofer personal communication let g be an undirected finite graph without loops and multiple edges and let k be a positive integer with k 12 g 1 there are two players called white and black both having k men of their color in turn beginning with white the players position their men one at a time on unoccupied vertices of g when all men are placed the players take turns moving a man of their color along an edge to an unoccupied adjacent vertex again beginning with white a player wins if his opponent cannot carry out his next move since none of his men has an unoccupied neighbor if the game does not stop then the outcome is a draw we always assume that both players play optimal among other questions we deal with the following ones 1 1 is it true that for all g and k white cannot win the game 2 2 does there exist a tree t and a positive integer k for which the outcome is a draw let g denote the covering number of g i e g is the minimum number of vertices covering all edges of g we prove that black wins the game if g k we use this result to show that white never wins the game if gis bipartite thus providing a partial answer to the first question we answer the second question in the affirmative by constructing an infinite series of trees for which the outcome is a draw for some k moreover we present results on extremal problems arising in the context of the game we also completely solve the cases when g is a path or a cycle further we completely settle the case k 2 in the proofs of our results matchings and cycles in graphs play a predominant role

the complexity of zero visibility cops and robber in this work we deal with the computational complexity aspects of the zero visibility cops and robber game we provide an algorithm that computes the zero visibility copnumber of a tree in linear time and show that the corresponding decision problem is np complete even for the class of starlike graphs

stable numerical methods for conservation laws with discontinuous flux function abstract we develop numerical methods for solving nonlinear equations of conservation laws with flux function that depends on discontinuous coefficients using a relaxation approximation the nonlinear equation is transformed to a semilinear diagonalizable problem with linear characteristic variables eulerian and lagrangian methods are used for the advection stage while an implicit explicit scheme solves the relaxation stage the main advantages of this approach are neither riemann problem solvers nor linear iterations are required during the solution process moreover the characteristic based relaxation method is unconditionally stable such that no cfl conditions are imposed on the selection of time steps numerical results are shown for models on traffic flows and two phase flows

stability of conservation laws with discontinuous coefficients we prove l1 contractivity of weak solutions to a conservation law with a flux function that may depend discontinuously on the space variable furthermore we show that the l1 difference between solutions to conservation laws with different flux functions is bounded by the total variation with respect to the space variable of the difference between the flux functions

an engquist osher type finite difference scheme with a discontinuous flux function in space abstract an engquist osher type finite difference scheme is derived for dealing with scalar conservation laws having a flux that is spatially dependent through a possibly discontinuous coefficient the new monotone difference scheme is based on introducing a new interface numerical flux function which is called a generalized engquist osher flux by means of this scheme the existence and uniqueness of weak solutions to the scalar conservation laws are obtained and the convergence theorem is established some numerical examples are presented and the corresponding numerical results are displayed to illustrate the efficiency of the methods

on search games that include ambush we present a stochastic game that models ambush search in a finite region q which has area but no other structure the searcher can search a unit area of q in unit time or adopt an ambush mode for a certain period the searcher captures the hider when the searched region contains the hider s location or if the hider moves while the searcher is in ambush mode the payoff in this zero sum game is the capture time our game is motivated by the still unsolved princess and monster game on a star graph with a large number of leaves

projectively equivalent ideals and rees valuations let r be a noetherian ring two ideals i and j in r are projectively equivalent in case the integral closure of ii is equal to the integral closure of jj for some i j n it is known that if i and j are projectively equivalent then the set reesi of rees valuation rings of i is equal to the set reesj of rees valuation rings of j and the values of i and j with respect to these rees valuation rings are proportional we observe that the converse also holds in particular if the ideal i has only one rees valuation ring v then the ideals j projectively equivalent to i are precisely the ideals j such that reesj v in certain cases such as i dimr 1 or ii r is a two dimensional regular local domain we observe that if i has more than one rees valuation ring then there exist ideals j such that reesi reesj but j is not projectively equivalent to i if i and j are regular ideals of r we prove that reesi reesj reesij with equality holding if dimr 2 but not holding in general if dimr 3 we associate to i and to the set p i of integrally closed ideals projectively equivalent to i a numerical semigroup s i n such that s i n if and only if there exists j p i for which p i jn a n n

a runge kutta discontinuous galerkin scheme for hyperbolic conservation laws with discontinuous fluxes the paper proposes a scheme by combining the runge kutta discontinuous galerkin method with a mapping algorithm for solving hyperbolic conservation laws with discontinuous fluxes this hybrid scheme is particularly applied to nonlinear elasticity in heterogeneous media and multi class traffic flow with inhomogeneous road conditions numerical examples indicate the scheme s efficiency in resolving complex waves of the two systems moreover the discussion implies that the so called mapping algorithm can also be combined with any other classical methods for solving similar problems in general

a partial k arboretum of graphs with bounded treewidth the notion of treewidth has seen to be a powerful vehicle for many graph algorithmic studies this survey paper wants to give an overview of many classes of graphs that can be seen to have a uniform upper bound on the treewidth of graphs in the class also some mutual relations between such classes are discussed

an annotated bibliography on guaranteed graph searching graph searching encompasses a wide variety of combinatorial problems related to the problem of capturing a fugitive residing in a graph using the minimum number of searchers in this annotated bibliography we give an elementary classification of problems and results related to graph searching and provide a source of bibliographical references on this field

monotonicity in digraph search problems in this paper we study the monotonicity and complexity of five digraph search problems directed searching mixed directed searching internal directed searching internal strong searching and internal weak searching in the first three search problems both searchers and intruder must follow the edge directions when they move along edges in the internal strong search problem the intruder must move in the edge directions but searchers need not in the internal weak search problem searchers must move in the edge directions but the intruder need not there are three actions for searchers in the first two search problems placing removing and sliding and there are only two actions for searchers in the last three internal search problems placing and sliding note that the internal strong searching is a strong version of the internal directed searching the internal weak searching is a weak version of the internal directed searching and the internal edge searching is an analogy of the internal directed searching on undirected graphs we prove that the first three problems are monotonic and the last two problems are non monotonic respectively it is interesting that the internal directed searching is monotonic while the internal strong searching the internal weak searching and the internal edge searching are all non monotonic we also show that the first four problems are np complete and the last problem is np hard we solve the open problem on whether a non monotonic searching problem can be np complete

complexity and monotonicity results for domination games abstract in this paper we study domination games a class of games introduced by fomin kratsch and muller in 8 domination games are a variant of the well known graph searching games also called cops and robber games where a number of cops tries to capture a robber hiding on the vertices of a graph variants of these games are often used to provide a game theoretic characterization of important graph parameters such as pathwidth treewidth and hypertreewidth we are primarily interested in questions concerning the complexity and monotonicity of these games we show that dominating games are computationally much harder than standard cops and robber games and establish strong non monotonicity results for various notions of monotonicity that arise naturally in the context of domination games answering a question of 8 we show that there are graphs where the shortest winning strategy for a minimal number of cops must necessarily be of exponential length