local consistency in parallel constraint satisfaction networks abstract in this paper we present several basic techniques for achieving parallel execution of constraint networks the major result supported by our investigations is that the parallel complexity of constraint networks is critically dependent on subtle properties of the network that do not influence its sequential complexity

exploring unknown environments we consider exploration problems where a robot has to construct a complete map of an unknown environment we assume that the environment is modeled by a directed strongly connected graph the robot s task is to visit all nodes and edges of the graph using the minimum number r of edge traversals deng and papadimitriou proceedings of the 31st symposium on the foundations of computer science 1990 pp 356 361 showed an upper bound for r of do d m and koutsoupias reported by deng and papadimitriou gave a lower bound of omega d 2 m where m is the number of edges in the graph and d is the minimum number of edges that have to be added to make the graph eulerian we give the first subexponential algorithm for this exploration problem which achieves an upper bound of do log d m we also show a matching lower bound of d omega log d m for our algorithm additionally we give lower bounds of 2 omega d m respectively d omega log d m for various other natural exploration algorithms

computing logcfl certificates the complexity class logcfl consists of all languages or decision problems which are logspace reducible to a context free language since logcfl is included in ac1 the problems in logcfl are highly parallelizable by results of ruzzo jcss 21 1980 218 the complexity class logcfl can be characterized as the class of languages accepted by alternating turing machines atms which use logarithmic space and have polynomially sized accepting computation trees we show that for each such atm m recognizing a language a in logcfl it is possible to construct an llogcfl transducer tm such that tm on input w e a outputs an accepting tree for m on w it follows that computing single logcfl certificates is feasible in functional ac1 and is thus highly parallelizable wanke j algorithms 16 1994 470 has recently shown that for any fixed k deciding whether the treewidth of a graph is at most k is in the complexity class logcfl as an application of our general result we show that the task of computing a tree decomposition for a graph of constant treewidth is in functional logcfl and thus in ac1 we also show that the following tasks are all highly parallelizable computing a solution to an acyclic constraint satisfaction problem computing an m coloring for a graph of bounded treewidth computing the chromatic number and minimal colorings for graphs of bounded tree width

tree exploration with advice we study the amount of knowledge about the network that is required in order to efficiently solve a task concerning this network the impact of available information on the efficiency of solving network problems such as communication or exploration has been investigated before but assumptions concerned availability of particular items of information about the network such as the size the diameter or a map of the network in contrast our approach is quantitative we investigate the minimum number of bits of information bits of advice that has to be given to an algorithm in order to perform a task with given efficiency we illustrate this quantitative approach to available knowledge by the task of tree exploration a mobile entity robot has to traverse all edges of an unknown tree using as few edge traversals as possible the quality of an exploration algorithm a is measured by its competitive ratio i e by comparing its cost number of edge traversals to the length of the shortest path containing all edges of the tree depth first search has competitive ratio 2 and in the absence of any information about the tree no algorithm can beat this value we determine the minimum number of bits of advice that has to be given to an exploration algorithm in order to achieve competitive ratio strictly smaller than 2 our main result establishes an exact threshold number of bits of advice that turns out to be roughly loglogd where d is the diameter of the tree more precisely for any constant c we construct an exploration algorithm with competitive ratio smaller than 2 using at most loglogd c bits of advice and we show that every algorithm using loglogd g d bits of advice for any function g unbounded from above has competitive ratio at least 2

two applications of inductive counting for complementation problems following the recent independent proofs of immerman siam j comput 17 1988 pp 935 938 and szelepcsenyi bull european assoc theoret comput sci 33 1987 pp 96 100 that nondeterministic space bounded complexity classes are closed under complementation two further applications of the inductive counting technique are developed first an errorless probabilistic algorithm for the undirected graph s t connectivity problem that runs in o log n space and polynomial expected time is given then it is shown that the class logcfl is closed under complementation the latter is a special case of a general result that shows closure under complementation of classes defined by semi unbounded fan in circuits or equivalently nondeterministic auxiliary pushdown automata or tree size bounded alternating turing machines as one consequence it is shown that small numbers of role switches in two person pebbling can be eliminated

fast collaborative graph exploration we study the following scenario of online graph exploration a team of k agents is initially located at a distinguished vertex r of an undirected graph we ask how many time steps are required to complete exploration i e to make sure that every vertex has been visited by some agent as our main result we provide the first strategy which performs exploration of a graph with n vertices at a distance of at most d from r in time o d using a team of agents of polynomial size k d n 1 n 2 for any 0 our strategy works in the local communication model in which agents can only exchange information when located at a vertex without knowledge of global parameters such as n or d we also obtain almost tight bounds on the asymptotic relation between exploration time and team size for large k in both the local and the global communication model

a general lower bound for collaborative tree exploration abstract we consider collaborative graph exploration with a set of k agents all agents start at a common vertex of an initially unknown graph with n vertices and need to collectively visit all other vertices we assume agents are deterministic moves are simultaneous and we allow agents to communicate globally for this setting we give the first non trivial lower bounds that bridge the gap between small k n and large k n teams of agents remarkably our bounds tightly connect to existing results in both domains first we significantly extend a lower bound of log k log log k by dynia et al on the competitive ratio of a collaborative tree exploration strategy to the range k n log c n for any c n second we provide a tight lower bound on the number of agents needed for any competitive exploration algorithm in particular we show that any collaborative tree exploration algorithm with k d n 1 o 1 agents has a competitive ratio of 1 while dereniowski et al gave an algorithm with k d n 1 e agents and competitive ratio o 1 for any e 0 and with d denoting the diameter of the graph lastly we show that for any exploration algorithm using k n agents there exist trees of arbitrarily large height d that require d 2 rounds and we provide a simple algorithm that matches this bound for all trees

lower and upper competitive bounds for online directed graph exploration abstract we study the problem of exploring all nodes of an unknown directed graph a searcher has to construct a tour that visits all nodes but only has information about the parts of the graph it already visited analogously to the traveling salesman problem the goal is to minimize the cost of such a tour in this article we present upper and lower bounds for the competitive ratio of both the deterministic and the randomized online version of exploring all nodes of directed graphs our bounds are sharp or sharp up to a small constant depending on the specific model as it turns out restricting the diameter the incoming outgoing degree or randomly choosing a starting point does not improve lower bounds beyond a small constant factor even supplying the searcher in a planar euclidean graph with the nodes coordinates does not help essentially exploring a directed graph has a multiplicative overhead linear in the number of nodes furthermore if one wants to search for a specific node in unweighted directed graphs a greedy algorithm with quadratic multiplicative overhead can only be improved by a small constant factor as well

parallel algorithms with optimal speedup for bounded treewidth we describe the first parallel algorithm with optimal speedup for constructing minimum width tree decompositions of graphs of bounded treewidth on n vertex input graphs the algorithm works in o log n 2 time using o n operations on the erew pram we also give faster parallel algorithms with optimal speedup for the problem of deciding whether the treewidth of an input graph is bounded by a given constant and for a variety of problems on graphs of bounded treewidth including all decision problems expressible in monadic second order logic on n vertex input graphs the algorithms use o n operations together with o log n log n time on the erew pram or o log n time on the crcw pram

on the power of structural decompositions of graph based representations of constraint problems the constraint satisfaction problem csp is a central issue of research in artificial intelligence due to its intractability many efforts have been made in order to identify tractable classes of csp instances and in fact deep and useful results have already been achieved in particular this paper focuses on structural decomposition methods which are essentially meant to look for near acyclicity properties of the graphs or hypergraphs that encode the structure of the constraints interactions in general constraint scopes comprise an arbitrary number of variables and thus this structure may be naturally encoded via hypergraphs however in many practical applications decomposition methods are applied over suitable graph representations of the possibly non binary csp instances at hand despite the great interest in such binary approaches a formal analysis of their power in terms of their ability of identifying islands of tractability was missing in the literature the aim of this paper is precisely to fill this gap by studying the relationships among binary structural methods and by providing a clear picture of the tractable fragments of csp that can be specified with respect to each of these decomposition approaches when they are applied to binary representations of non binary csp instances in particular various long standing questions about primal dual and incidence graph encodings are answered the picture is then completed by comparing methods on binary encodings with methods specifically conceived for non binary constraints

exploring an unknown dangerous graph with a constant number of tokens consider a team of asynchronous agents that must explore an unknown graph g in presence of a black hole a node which destroys all incoming agents without leaving any observable trace inter agent communication and coordination is achieved using tokens that agents can pick up carry and drop on the nodes it is known that if the agents have a map of g the problem can be solved with a constant number of tokens the question we study is under what conditions it is possible to explore with o 1 tokens if the agents have no map of g the contribution of this paper is to provide a definite answer to this question we first prove the unexpected negative result that if only a constant number of tokens are available then 1 agents are not sufficient where is the maximum node degree we also prove that regardless of the team size two tokens are not sufficient in other words any solution protocol using a constant number of tokens must use at least 2 agents and at least three tokens we then show that these bounds are tight by presenting a protocol that allows the exploration of an unknown anonymous dangerous graph using only three tokens and 2 agents at the core of our solution is a novel token based asynchronous communication protocol which is of independent interest our algorithm assumes that the number of agents is known in case the number of agents is sufficient but unknown we show that the problem has a simple solution that uses only five tokens

optimal graph exploration without good maps a robot has to visit all nodes and traverse all edges of an unknown undirected connected graph using as few edge traversals as possible the quality of an exploration algorithm a is measured by comparing its cost number of edge traversals to that of the optimal algorithm having full knowledge of the graph the ratio between these costs maximized over all starting nodes in the graph and over all graphs in a given class u is called the overhead of algorithm a for the class u of graphs we consider three scenarios providing the robot with varying amount of information the robot may either know nothing about the explored graph or have an unlabeled isomorphic copy of it an unanchored map or have such a copy with a marked starting node an anchored map for all of the above scenarios we construct natural exploration algorithms that have smallest or in one case close to smallest overhead while for the class of all graphs depth first search turns out to be an optimal algorithm for all scenarios the situation for trees is much different we show that under the scenario without any knowledge dfs is still optimal for trees but this is not the case if a map is available under the scenario with an unanchored map we show that optimal overhead is at least 3 but strictly below 2 and thus dfs is not optimal under the scenario with an anchored map we construct an optimal algorithm for trees and show that its overhead is 3 2 we also consider exploration of the class of lines simple paths in this case depth first search remains optimal for the scenario without any knowledge with overhead 2 under the scenario with an unanchored map we construct an optimal algorithm and show that its overhead is 3 finally under the scenario with an anchored map we construct an optimal algorithm and show that its overhead is 7 5 an important contribution of this paper is establishing lower bounds that prove optimality of these exploration algorithms

non commutative arithmetic circuits depth reduction and size lower bounds abstract we investigate the phenomenon of depth reduction in commutative and non commutative arithmetic circuits we prove that in the commutative setting uniform semi unbounded arithmetic circuits of logarithmic depth are as powerful as uniform arithmetic circuits of polynomial degree and unrestricted depth earlier proofs did not work in the uniform setting this also provides a unified proof of the circuit characterizations of the class logcfl and its counting variant logcfl we show that ac1 has no more power than arithmetic circuits of polynomial size and degree no log log n improving the trivial bound of no log n connections are drawn between tc1 and arithmetic circuits of polynomial size and degree then we consider non commutative computation we show that over the algebra max concat arithmetic circuits of polynomial size and polynomial degree can be reduced to o log2 n depth and even to o log n depth if unbounded fanin gates are allowed this establishes that optlogcfl is in ac1 this is the first depth reduction result for arithmetic circuits over a non commutative semiring and it complements the lower bounds of kosaraju and nisan showing that depth reduction cannot be done in the general non commutative setting we define new notions called short left paths and short right paths and we show that these notions provide a characterization of the classes of arithmetic circuits for which optimal depth reduction is possible this class also can be characterized using the auxpda model finally we characterize the languages generated by efficient circuits over the semiring 2 union concat in terms of simple one way machines and we investigate and extend earlier lower bounds on non commutative circuits

the power of a pebble exploring and mapping an unknown environment is a fundamental problem that is studied in a variety of contexts many results have focused on finding efficient solutions to restricted versions of the problem in this paper we consider a model that makes very limited assumptions about the environment and solve the mapping problem in this general setting we model the environment by an unknown directed graph g and consider the problem of a robot exploring and mapping g the edges emanating from each vertex are numbered from 1 to d but we do not assume that the vertices of g are labeled since the robot has no way of distinguishing between vertices it has no hope of succeeding unless it is given some means of distinguishing between vertices for this reason we provide the robot with a pebble a device that it can place on a vertex and use to identify the vertex later in this paper we show 1 if the robot knows an upper bound on the number of vertices then it can learn the graph efficiently with only one pebble 2 if the robot does not know an upper bound on the number of vertices n then log log n pebbles are both necessary and sufficient in both cases our algorithms are deterministic

the monadic second order logic of graphs i recognizable sets of finite graphs abstract the notion of a recognizable set of finite graphs is introduced every set of finite graphs that is definable in monadic second order logic is recognizable but not vice versa the monadic second order theory of a context free set of graphs is decidable

on the distance spectrum of graphs abstract let g be a connected graph and d g be the distance matrix of g suppose that 1 d 2 d n d are the d eigenvalues of g in this paper we characterize all connected graphs with n d 2 furthermore we characterize all connected graphs of diameter 2 with exactly three d eigenvalues when 1 d is not an integer we also conjecture that the complete k partite graph is determined by its d spectrum

restricted space algorithms for isomorphism on bounded treewidth graphs the graph isomorphism problem restricted to graphs of bounded treewidth or bounded tree distance width are known to be solvable in polynomial time we give restricted space algorithms for these problems proving the following results isomorphism for bounded tree distance width graphs is in l and thus complete for the class we also show that for this kind of graphs a canon can be computed within logspace for bounded treewidth graphs when both input graphs are given together with a tree decomposition the problem of whether there is an isomorphism which respects the decompositions i e when only isomorphisms are considered mapping bags in one decomposition blockwise onto bags in the other decomposition is in l for bounded treewidth graphs when one of the input graphs is given with a tree decomposition the isomorphism problem is in logcfl as a corollary the isomorphism problem for bounded treewidth graphs is in logcfl this improves the known tc 1 upper bound for the problem given by grohe and verbitsky

the complexity of acyclic conjunctive queries this paper deals with the evaluation of acyclic booleanconjunctive queries in relational databases by well known resultsof yannakakis 1981 this problem is solvable in polynomial time its precise complexity however has not been pinpointed so far weshow that the problem of evaluating acyclic boolean conjunctivequeries is complete for logcfl the class of decision problems thatare logspace reducible to a context free language since logcfl iscontained in ac1 and nc2 the evaluation problem of acyclic booleanconjunctive queries is highly parallelizable we present a paralleldatabase algorithm solving this problem with alogarithmic number ofparallel join operations the algorithm is generalized to computingthe output of relevant classes of non boolean queries we also showthat the acyclic versions of the following well known database andai problems are all logcfl complete the query output tuple problemfor conjunctive queries conjunctive query containment clausesubsumption and constraint satisfaction the logcfl completenessresult is extended to the class of queries of bounded tree widthand to other relevant query classes which are more general than theacyclic queries

word problems and membership problems on compressed words we consider a compressed form of the word problem for finitely presented monoids where the input consists of two compressed representations of words over the generators of a monoid m and we ask whether these two words represent the same monoid element of m words are compressed using straight line programs i e context free grammars that generate exactly one word for several classes of finitely presented monoids we obtain completeness results for complexity classes in the range from p to expspace as a by product of our results on compressed word problems we obtain a fixed deterministic context free language with a pspace complete compressed membership problem the existence of such a language was open so far finally we will investigate the complexity of the compressed membership problem for various circuit complexity classes

on the complexity of constrained nash equilibria in graphical games a widely accepted rational behavior for non cooperative players is based on the notion of nash equilibrium although the existence of a nash equilibrium is guaranteed in the mixed framework i e when players select their actions in a randomized manner in many real world applications the existence of any equilibrium is not enough rather it is often desirable to single out equilibria satisfying some additional requirements in order for instance to guarantee a minimum payoff to certain players which we call constrained nash equilibria in this paper a formal framework for specifying these kinds of requirement is introduced and investigated in the context of graphical games where a player p may directly be interested in some of the other players only called the neighbors of p this setting is very useful for modeling large population games where typically each player does not directly depend on all the players and representing her utility function extensively is either inconvenient or infeasible based on this framework the complexity of deciding the existence and of computing constrained equilibria is then investigated in the light of evidencing how the intrinsic difficulty of these tasks is affected by the requirements prescribed at the equilibrium and by the structure of players interactions the analysis is carried out for the setting of mixed strategies as well as for the setting of pure strategies i e when players are forced to deterministically choose the action to perform in particular for this latter case restrictions on players interactions and on constraints are identified that make the computation of nash equilibria an easy problem for which polynomial and highly parallelizable algorithms are presented

how to learn an unknown environment i the rectilinear case we consider the problem faced by a robot that must explore and learn an unknown room with obstacles in it we seek algorithms that achieve a bounded ratio of the worst case distance traversed in order to see all visible points of the environment thus creating a map divided by the optimum distance needed to verify the map if we had it in the beginning the situation is complicated by the fact that the latter off line problem the problem of optimally verifying a map is np hard although we show that there is no such competitive algorithm for general obstacle courses we give a competitive algorithm for the case of a polygonal room with a bounded number of obstacles in it we restrict ourselves to the rectilinear case where each side of the obstacles and the room is parallel to one of the coordinates and the robot must also move either parallel or perpendicular to the sides in a subsequent paper we will discuss the extension to polygons of general shapes we also discuss the off line problem for simple rectilinear polygons and find an optimal solution in the l 1 metric in polynomial time in the case where the entry and the exit are different points

map construction of unknown graphs by multiple agents we consider a distributed version of the graph exploration and mapping problem where a mobile agent has to traverse the edges of an unlabelled i e anonymous graph and return to its starting point building a map of the graph in the process in our case instead of a single agent there are k identical i e mutually indistinguishable agents initially dispersed among the n nodes of the graph the agents can communicate by writing to the small public bulletin boards available at each node the objective is for each agent to build an identically labelled map of the graph we call this the labelled map construction problem this problem is much more difficult than exploration by a single agent because it involves achieving cooperation among multiple agents in fact this problem is deterministically unsolvable in some cases we present deterministic algorithms that successfully and efficiently solve the problem under the condition that the values of n and k are co prime with each other we also show how the problem of labelled map construction is related to other problems like leader election and rendezvous of agents

tight bounds for undirected graph exploration with pebbles and multiple agents we study the problem of deterministically exploring an undirected and initially unknown graph with n vertices either by a single agent equipped with a set of pebbles or by a set of collaborating agents the vertices of the graph are unlabeled and cannot be distinguished by the agents but the edges incident to a vertex have locally distinct labels the graph is explored when all vertices have been visited by at least one agent in this setting it is known that for a single agent without pebbles log n bits of memory are necessary and sufficient to explore any graph with at most n vertices we are interested in how the memory requirement decreases as the agent may mark vertices by dropping and retrieving distinguishable pebbles or when multiple agents jointly explore the graph we give tight results for both questions showing that for a single agent with constant memory log log n pebbles are necessary and sufficient for exploration we further prove that using collaborating agents instead of pebbles does not help as log log n agents with constant memory each are necessary and sufficient for exploration for the upper bounds we devise an algorithm for a single agent with constant memory that explores any n vertex graph using o log log n pebbles even when n is not known a priori the algorithm terminates after polynomial time and returns to the starting vertex we further show that the algorithm can be realized with additional constant memory agents rather than pebbles implying that o log log n agents with constant memory can explore any n vertex graph for the lower bound we show that the number of agents needed for exploring any graph with at most n vertices is already log log n when we allow each agent to have at most o log n 1 e bits of memory for any e 0 our argument also implies that a single agent with sublogarithmic memory needs log log n pebbles to explore any n vertex graph

exploring an unknown dangerous graph using tokens consider a team of one or more mobile agents operating in a graph g unaware of the graph topology and starting from the same node the team must explore the graph this problem known as graph exploration was initially formulated by shannon in 1951 and has been extensively studied since under a variety of conditions most of the existing investigations have assumed that the network is safe for the agents and the vast majority of the solutions presented in the literature succeed in their task only under this assumption recently the exploration problem has been examined also when the network is unsafe the danger examined is the presence in the network of a black hole a node that disposes of any incoming agent without leaving any observable trace of this destruction the goal is for at least one agent to survive and to have all the surviving agents to construct a map of the network indicating the edges leading to the black hole this variant of the problem is also known as a black hole search this problem has been investigated for the most part assuming powerful inter agent communication mechanisms whiteboards at all nodes indeed in this model the black hole search problem can be solved with an optimal team size and performing a polynomial number of moves in this paper we consider the less powerful enhanced token model each agent has available a token that can be carried placed on a node or on a link and can be removed from it all tokens are identical and no other form of marking or communication is available we constructively prove that the black hole search problem can be solved also in this model furthermore this can be done using a team of agents of optimal size and performing a polynomial number of moves our algorithm works even if the agents are asynchronous and if both the agents and the nodes are anonymous

broadcast in the rendezvous model in many large distributed or mobile networks broadcast algorithms are used to update information stored at the nodes in this paper we propose a new model of communication based on rendezvous and analyze a multi hop distributed algorithm to broadcast a message in a synchronous setting in the rendezvous model two neighbors u and v can communicate if and only if u calls v and v calls u simultaneously thus nodes u and v obtain a rendezvous at a meeting point if m is the number of meeting points the network can be modeled by a graph of n vertices and m edges at each round every vertex chooses a random neighbor and there is a rendezvous if an edge has been chosen by its two extremities rendezvous enable an exchange of information between the two entities we get sharp lower and upper bounds on the time complexity in terms of number of rounds to broadcast we show that for any graph the expected number of rounds is between ln n and o n2 for these two bounds we prove that there exist some graphs for which the expected number of rounds is either o ln n or n2 for specific topologies additional bounds are given