Graph searching with advice

Fraigniaud et al. [L. Blin, P. Fraigniaud, N. Nisse, S. Vial, Distributing chasing of network intruders, in: 13th Colloquium on Structural Information and Communication Complexity, SIROCCO, in: LNCS, vol. 4056, Springer-Verlag, 2006, pp. 70–84] introduced a new measure of difficulty for a distributed task in a network. The smallest number of bits of advice   of a distributed problem is the smallest number of bits of information that has to be available to nodes in order to accomplish the task efficiently. Our paper deals with the number of bits of advice required to perform efficiently the graph searching problem in a distributed setting. In this variant of the problem, all searchers are initially placed at a particular node of the network. The aim of the team of searchers is to clear a contaminated graph in a monotone connected way, i.e., the cleared part of the graph is permanently connected, and never decreases while the search strategy is executed. Moreover, the clearing of the graph must be performed using the optimal number of searchers, i.e. the minimum number of searchers sufficient to clear the graph in a monotone connected way in a centralized setting. We show that the minimum number of bits of advice permitting the monotone connected and optimal clearing of a network in a distributed setting is Θ(nlogn)$\Theta (nlogn)$, where n$n$ is the number of nodes of the network. More precisely, we first provide a labelling of the vertices of any graph G$G$, using a total of O(nlogn)$O(nlogn)$ bits, and a protocol using this labelling that enables the optimal number of searchers to clear G$G$ in a monotone connected distributed way. Then, we show that this number of bits of advice is optimal: any distributed protocol requires Ω(nlogn)$\Omega (nlogn)$ bits of advice to clear a network in a monotone connected way, using an optimal number of searchers.     

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 Ω(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 k4, there exists a graph with connected visible search number at most k, and monotone connected visible search number >k     

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.     

On the complexity of distributed stable matching with small messages

We consider the distributed complexity of the stable matching problem (a.k.a. “stable marriage”). In this problem, the communication graph is undirected and bipartite, and each node ranks its neighbors. Given a matching of the nodes, a pair of unmatched nodes is called blocking if they prefer each other to their assigned match. A matching is called stable if it does not induce any blocking pair. In the distributed model, nodes exchange messages in each round over the communication links, until they find a stable matching. We show that if messages may contain at most B bits each, then any distributed algorithm that solves the stable matching problem requires ${\Omega(\sqrt{n/B\log n})}We consider the distributed complexity of the stable matching problem (a.k.a. “stable marriage”). In this problem, the communication graph is undirected and bipartite, and each node ranks its neighbors. Given a matching of the nodes, a pair of unmatched nodes is called blocking if they prefer each other to their assigned match. A matching is called stable if it does not induce any blocking pair. In the distributed model, nodes exchange messages in each round over the communication links, until they find a stable matching. We show that if messages may contain at most B bits each, then any distributed algorithm that solves the stable matching problem requires W(?{n/Blogn}){\Omega(\sqrt{n/B\log n})} communication rounds in the worst case, even for graphs of diameter O(log n), where n is the number of nodes in the graph. Furthermore, the lower bound holds even if we allow the output to contain O(?n){O(\sqrt n)} blocking pairs, and if a pair is considered blocking only if they like each other much more then their assigned match.     

Decomposition of graphs and monotone formula size of homogeneous functions

Summary We show that every graph on n nodes can be partitioned by a number of complete bipartite graphs with O(n ²/log n) nodes with no edge belonging to two of them. Since each partition corresponds directly to a monotone formula for the associated quadratic function we obtain an upper bound for the monotone formula size of quadratic functions. Our method extends to uniform hypergraphs with fixed range and thus to homogeneous functions with fixed length of prime implicants. Finally we give an example of a quadratic function where each monotone formula built from arbitrary partitions of the graph (double edges allowed) is not optimal. That means we disprove the single-level-conjecture for formulae.     

Eventual Leader Election with Weak Assumptions on Initial Knowledge,Communication Reliability,and Synchrony

his paper considers the eventual leader election problem in asynchronous message-passing systems where an arbitrary number t of processes can crash (t < n, where n is the total number of processes). It considers weak assumptions both on the initial knowledge of the processes and on the network behavior. More precisely, initially, a process knows only its identity and the fact that the process identities are different and totally ordered (it knows neither n nor t). Two eventual leader election protocols and a lower bound are presented. The first protocol assumes that a process also knows a lower bound α on the number of processes that do not crash. This protocol requires the following behavioral properties from the underlying network: the graph made up of the correct processes and fair lossy links is strongly connected, and there is a correct process connected to (n − f) − α other correct processes (where f is the actual number of crashes in the considered run) through eventually timely paths (paths made up of correct processes and eventually timely links). This protocol is not communication-efficient in the sense that each correct process has to send messages forever. The second protocol is communication-efficient: after some time, only the final common leader has to send messages forever. This protocol does not require the processes to know α, but requires stronger properties from the underlying network: each pair of correct processes has to be connected by fair lossy links (one in each direction), and there is a correct process whose n − f − 1 output links to the rest of correct processes have to be eventually timely. A matching lower bound result shows that any eventual leader election protocol must have runs with this number of eventually timely links, even if all processes know all the processes identities. In addition to being communication-efficient, the second protocol has another noteworthy efficiency property, namely, be the run finite or infinite, all the local variables and message fields have a finite domain in the run.     

Passivity‐based state synchronization of homogeneous multiagent systems via static protocol in the presence of input saturation

This paper studies semiglobal and global state synchronization of homogeneous multiagent systems with partial‐state coupling (ie, agents are coupled through part of their states) via a static protocol. We consider 2 classes of agents, ie, G‐passive and G‐passifiable via input feedforward, which are subjected to input saturation. The proposed static protocol is purely decentralized, ie, without an additional channel for the exchange of controller states. For semiglobal synchronization, a static protocol is designed for an a priori given set of network graphs with a directed spanning tree. In other words, the static protocol only needs rough information on the network graph, ie, a lower bound for the real part and an upper bound for the modulus, of the nonzero eigenvalues of the corresponding Laplacian matrix. Whereas for global synchronization, only strongly connected and detailed balanced network graphs are considered. In this case, for G‐passive agents, the static protocol does not need any network information, whereas for G‐passifiable agents via input feedforward, the static protocol only needs an upper bound for the modulus of the eigenvalues of the corresponding Laplacian matrix.     

Fixed-Parameter Complexity of Minimum Profile Problems

The profile of a graph is an integer-valued parameter defined via vertex orderings; it is known that the profile of a graph equals the smallest number of edges of an interval supergraph. Since computing the profile of a graph is an NP-hard problem, we consider parameterized versions of the problem. Namely, we study the problem of deciding whether the profile of a connected graph of order n is at most n−1+k, considering k as the parameter; this is a parameterization above guaranteed value, since n−1 is a tight lower bound for the profile. We present two fixed-parameter algorithms for this problem. The first algorithm is based on a forbidden subgraph characterization of interval graphs. The second algorithm is based on two simple kernelization rules which allow us to produce a kernel with linear number of vertices and edges. For showing the correctness of the second algorithm we need to establish structural properties of graphs with small profile which are of independent interest. A preliminary version of the paper is published in Proc. IWPEC 2006, LNCS vol. 4169, 60–71.     

Efficient reliable communication over partially authenticated networks

Reliable communication between processors in a network is a basic requirement for executing any protocol. Dolev [7] and Dolev et al. [8] showed that reliable communication is possible if and only if the communication network is sufficiently connected. Beimel and Franklin [1] showed that the connectivity requirement can be relaxed if some pairs of processors share authentication keys. That is, costly communication channels can be replaced by authentication keys. In this work, we continue this line of research. We consider the scenario where there is a specific sender and a specific receiver in a synchronous network. In this case, the protocol of [1] has n^O(n) rounds even if there is a single Byzantine processor. We present a more efficient protocol with round complexity of (n/t)^O(t), where n is the number of processors in the network and t is an upper bound on the number of Byzantine processors in the network. Specifically, our protocol is polynomial when the number of Byzantine processors is O(1), and for every t its round complexity is bounded by 2^O(n). The same improvements hold for reliable and private communication. The improved protocol is obtained by analyzing the properties of a "communication and authentication graph" that characterizes reliable communication. Received: 13 January 2004, Accepted: 22 September 2004, Published online: 13 January 2005 A preliminary version of this paper appeared in [2].     

Time efficient k-shot broadcasting in known topology radio networks

The paper considers broadcasting protocols in radio networks with known topology that are efficient in both time and energy. The radio network is modelled as an undirected graph G = (V, E) where |V| = n. It is assumed that during execution of the communication task every node in V is allowed to transmit at most once. Under this assumption it is shown that any radio broadcast protocol requires transmission rounds, where D is the diameter of G. This lower bound is complemented with an efficient construction of a deterministic protocol that accomplishes broadcasting in rounds. Moreover, if we allow each node to transmit at most k times, the lower bound on the number of transmission rounds holds. We also provide a randomised protocol that accomplishes broadcasting in rounds. The paper concludes with a number of open problems in the area. The research of L. Gąsieniec, D.R. Kowalski and C. Su supported in part by the Royal Society grant Algorithmic and Combinatorial Aspects of Radio Communication, IJP - 2006/R2. The research of E. Kantor and D. Peleg supported in part by grants from the Minerva Foundation and the Israel Ministry of Science.     

A subexponential bound for linear programming

We present a simple randomized algorithm which solves linear programs withn constraints andd variables in expected $$\min \{ O(d^2 2^d n),e^{2\sqrt {dIn({n \mathord{\left/ {\vphantom {n {\sqrt d }}} \right. \kern-\nulldelimiterspace} {\sqrt d }})} + O(\sqrt d + Inn)} \}$$ time in the unit cost model (where we count the number of arithmetic operations on the numbers in the input); to be precise, the algorithm computes the lexicographically smallest nonnegative point satisfyingn given linear inequalities ind variables. The expectation is over the internal randomizations performed by the algorithm, and holds for any input. In conjunction with Clarkson's linear programming algorithm, this gives an expected bound of $$O(d^2 n + e^{O(\sqrt {dInd} )} ).$$ The algorithm is presented in an abstract framework, which facilitates its application to several other related problems like computing the smallest enclosing ball (smallest volume enclosing ellipsoid) ofn points ind-space, computing the distance of twon-vertex (orn-facet) polytopes ind-space, and others. The subexponential running time can also be established for some of these problems (this relies on some recent results due to Gärtner).     

Smallest Bipartite Bridge-Connectivity Augmentation

This paper addresses two augmentation problems related to bipartite graphs. The first, a fundamental graph-theoretical problem, is how to add a set of edges with the smallest possible cardinality so that the resulting graph is 2-edge-connected, i.e., bridge-connected, and still bipartite. The second problem, which arises naturally from research on the security of statistical data, is how to add edges so that the resulting graph is simple and does not contain any bridges. In both cases, after adding edges, the graph can be either a simple graph or, if necessary, a multi-graph. Our approach then determines whether or not such an augmentation is possible. We propose a number of simple linear-time algorithms to solve both problems. Given the well-known bridge-block data structure for an input graph, the algorithms run in O(log n) parallel time on an EREW PRAM using a linear number of processors, where n is the number of vertices in the input graph. We note that there is already a polynomial time algorithm that solves the first augmentation problem related to graphs with a given general partition constraint in O(n(m+nlog n)log n) time, where m is the number of distinct edges in the input graph. We are unaware of any results for the second problem. H.-W. Wei, W.-C. Lu and T.-s. Hsu research supported in part by NSC of Taiwan Grants 94-2213-E-001-014, 95-2221-E-001-004 and 96-2221-E-001-004.     

An Algorithm for Strongly Connected Component Analysis in n log n Symbolic Steps

We present a symbolic algorithm for strongly connected component decomposition. The algorithm performs Θ(n log n) image and preimage computations in the worst case, where n is the number of nodes in the graph. This is an improvement over the previously known quadratic bound. The algorithm can be used to decide emptiness of Büchi automata with the same complexity bound, improving Emerson and Lei's quadratic bound, and emptiness of Streett automata, with a similar bound in terms of nodes. It also leads to an improved procedure for the generation of nonemptiness witnesses. This work was supported in part by SRC contract 98-DJ-620 and NSF grant CCR-99-71195. This work was done while the author was at the University of Colorado at Boulder.     

Lower bounds for the broadcast problem in mobile radio networks

Summary.  In this paper, we prove a lower bound on the number of rounds required by a deterministic distributed protocol for broadcasting a message in radio networks whose processors do not know the identities of their neighbors. Such an assumption captures the main characteristic of mobile and wireless environments [3], i.e., the instability of the network topology. For any distributed broadcast protocol Π, for any n and for any D≦n/2, we exhibit a network G with n nodes and diameter D such that the number of rounds needed by Π for broadcasting a message in G is Ω(D log n). The result still holds even if the processors in the network use a different program and know n and D. We also consider the version of the broadcast problem in which an arbitrary number of processors issue at the same time an identical message that has to be delivered to the other processors. In such a case we prove that, even assuming that the processors know the network topology, Ω(n) rounds are required for solving the problem on a complete network (D=1) with n processors. Received: August 1994 / Accepted: August 1996     

Connecting face hitting sets in planar graphs

We show that any face hitting set of size n of a connected planar graph with a minimum degree of at least 3 is contained in a connected subgraph of size 5n−6. Furthermore we show that this bound is tight by providing a lower bound in the form of a family of graphs. This improves the previously known upper and lower bound of 11n−18 and 3n respectively by Grigoriev and Sitters. Our proof is valid for simple graphs with loops and generalizes to graphs embedded in surfaces of arbitrary genus.     

On m-restricted edge connectivity of undirected generalized De Bruijn graphs

An m-restricted edge cut is an edge cut of a connected graph whose removal results in components of order at least m, the minimum cardinality over all m-restricted edge cuts of a graph is its m-restricted edge connectivity. It is known that telecommunication networks with topology having larger m-restricted edge connectivity are locally more reliable for all m≤3. This work shows that if n≥7, then undirected generalized binary De Bruijn graph UB_G(2, n) is maximally m-restricted edge connected for all m≤3, where a graph G is maximally m-restricted edge connected if its m-restricted edge connectivity is equal to the minimum number of edges from any connected subgraphs S to G?S.     

How to meet when you forget: log-space rendezvous in arbitrary graphs

Two identical (anonymous) mobile agents start from arbitrary nodes in an a priori unknown graph and move synchronously from node to node with the goal of meeting. This rendezvous problem has been thoroughly studied, both for anonymous and for labeled agents, along with another basic task, that of exploring graphs by mobile agents. The rendezvous problem is known to be not easier than graph exploration. A well-known recent result on exploration, due to Reingold, states that deterministic exploration of arbitrary graphs can be performed in log-space, i.e., using an agent equipped with O(log n) bits of memory, where n is the size of the graph. In this paper we study the size of memory of mobile agents that permits us to solve the rendezvous problem deterministically. Our main result establishes the minimum size of the memory of anonymous agents that guarantees deterministic rendezvous when it is feasible. We show that this minimum size is Θ(log n), where n is the size of the graph, regardless of the delay between the starting times of the agents. More precisely, we construct identical agents equipped with Θ(log n) memory bits that solve the rendezvous problem in all graphs with at most n nodes, if they start with any delay τ, and we prove a matching lower bound Ω(log n) on the number of memory bits needed to accomplish rendezvous, even for simultaneous start. In fact, this lower bound is achieved already on the class of rings. This shows a significant contrast between rendezvous and exploration: e.g., while exploration of rings (without stopping) can be done using constant memory, rendezvous, even with simultaneous start, requires logarithmic memory. As a by-product of our techniques introduced to obtain log-space rendezvous we get the first algorithm to find a quotient graph of a given unlabeled graph in polynomial time, by means of a mobile agent moving around the graph.     

Some remarks on Boolean sums

Summary Neciporuk, Lamagna/Savage and Tarjan determined the monotone network complexity of a set of Boolean sums if any two sums have at most one variable in common. Wegener then solved the case that any two sums have at most k variables in common. We extend his methods and results and consider the case that any set of h +1 distinct sums have at most k variables in common. We use our general results to explicitly construct a set of n Boolean sums over n variables whose monotone complexity is of order n ^5/3. The best previously known bound was of order n ^3/2. Related results were obtained independently by Pippenger.This paper was presented at the MFCS 79 Symposium, Olomouc, Sept. 79     

Exploring Unknown Environments with Obstacles

We study exploration problems where a robot has to construct a complete map of an unknown environment using a path that is as short as possible. In the first problem setting we consider, a robot has to explore n rectangles. We show that no deterministic or randomized online algorithm can be better than Ω(\sqrt n ) -competitive, solving an open problem by Deng et al. [7]. We also generalize this bound to the problem of exploring three-dimensional rectilinear polyhedra without obstacles. In the second problem setting we study, a robot has to explore a grid graph with obstacles in a piecemeal fashion. The piecemeal constraint was defined by Betke et al. [4] and implies that the robot has to return to a start node every so often. Betke et al. gave an efficient algorithm for exploring grids with rectangular obstacles. We present an efficient strategy for piecemeal exploration of grids with arbitrary rectilinear obstacles.