Maximum Cliques in Graphs with Small Intersection Number and Random Intersection Graphs

In this paper, we relate the problem of finding a maximum clique to the intersection number of the input graph (i.e. the minimum number of cliques needed to edge cover the graph). In particular, we consider the maximum clique problem for graphs with small intersection number and random intersection graphs (a model in which each one of $m$ labels is chosen independently with probability $p$ by each one of $n$ vertices, and there are edges between any vertices with overlaps in the labels chosen). We first present a simple algorithm which, on input $G$ finds a maximum clique in $O(2^{2^m + O(m)} + n^2 \min\{2^m, n\})$ time steps, where $m$ is an upper bound on the intersection number and $n$ is the number of vertices. Consequently, when $m \leq \ln{\ln{n}}$ the running time of this algorithm is polynomial. We then consider random instances of the random intersection graphs model as input graphs. As our main contribution, we prove that, when the number of labels is not too large ($m=n^{\alpha}, 0< \alpha <1$), we can use the label choices of the vertices to find a maximum clique in polynomial time whp. The proof of correctness for this algorithm relies on our Single Label Clique Theorem, which roughly states that whp a "large enough" clique cannot be formed by more than one label. This theorem generalizes and strengthens other related results in the state of the art, but also broadens the range of values considered. As an important consequence of our Single Label Clique Theorem, we prove that the problem of inferring the complete information of label choices for each vertex from the resulting random intersection graph (i.e. the \emph{label representation of the graph}) is \emph{solvable} whp. Finding efficient algorithms for constructing such a label representation is left as an interesting open problem for future research

Hammock-on-ears decomposition: a technique for the efficient parallel solution of shortest paths and other problems

We show how to decompose efficiently in parallel {\em any} graph into a number, $\tilde{\gamma}$, of outerplanar subgraphs (called {\em hammocks}) satisfying certain separator properties. Our work combines and extends the sequential hammock decomposition technique introduced by G.~Frederickson and the parallel ear decomposition technique, thus we call it the {\em hammock-on-ears decomposition}. We mention that hammock-on-ears decomposition also draws from techniques in computational geometry and that an embedding of the graph does not need to be provided with the input. We achieve this decomposition in $O(\log n\log\log n)$ time using $O(n+m)$ CREW PRAM processors, for an $n$-vertex, $m$-edge graph or digraph. The hammock-on-ears decomposition implies a general framework for solving graph problems efficiently. Its value is demonstrated by a variety of applications on a significant class of (di)graphs, namely that of {\em sparse (di)graphs}. This class consists of all (di)graphs which have a $\tilde{\gamma}$ between $1$ and $\Theta(n)$, and includes planar graphs and graphs with genus $o(n)$. We improve previous bounds for certain instances of shortest paths and related problems, in this class of graphs. These problems include all pairs shortest paths, all pairs reachability

The price of optimum in Stackelberg games on arbitrary single commodity networks and latency functions

Let M be a single s-t network of parallel links with load dependent latency functions shared by an infinite number of selfish users. This may yield a Nash equilibrium with unbounded Coordination Ratio [23, 43]. A Leader can decrease the coordination ratio by assigning flow αr on M, and then all Followers assign selfishly the (1 − α)r remaining flow. This is a Stackelberg Scheduling Instance (M, r, α), 0 ≤ α ≤ 1. It was shown [38] that it is weakly NP-hard to compute the optimal Leader’s strategy. For any such network M we efficiently compute the minimum portion βM of flow r&gt; 0 needed by a Leader to induce M ’s optimum cost, as well as her optimal strategy. This shows that the optimal Leader’s strategy on instances (M, r, α ≥ βM) is in P. Unfortunately, Stackelberg routing in more general nets can be arbitrarily hard. Roughgarden pre-sented a modification of Braess’s Paradox graph, such that no strategy controlling αr flow can induce ≤ 1α times the optimum cost. However, we show that our main result also applies to any s-t net G. We take care of the Braess’s graph explicitly, as a convincing example. Finally, we extend this result to k commodities. A conference version of this paper has appeared in [16]. Some preliminary results have also appeare

Efficient Multi-Robot Motion Planning for Unlabeled Discs in Simple Polygons

We consider the following motion-planning problem: we are given $m$ unit discs in a simple polygon with $n$ vertices, each at their own start position, and we want to move the discs to a given set of $m$ target positions. Contrary to the standard (labeled) version of the problem, each disc is allowed to be moved to any target position, as long as in the end every target position is occupied. We show that this unlabeled version of the problem can be solved in $O(n\log n+mn+m^2)$ time, assuming that the start and target positions are at least some minimal distance from each other. This is in sharp contrast to the standard (labeled) and more general multi-robot motion-planning problem for discs moving in a simple polygon, which is known to be strongly NP-hard

Finding a needle in an exponential haystack: Discrete RRT for exploration of implicit roadmaps in multi-robot motion planning

We present a sampling-based framework for multi-robot motion planning which combines an implicit representation of a roadmap with a novel approach for pathfinding in geometrically embedded graphs tailored for our setting. Our pathfinding algorithm, discrete-RRT (dRRT), is an adaptation of the celebrated RRT algorithm for the discrete case of a graph, and it enables a rapid exploration of the high-dimensional configuration space by carefully walking through an implicit representation of a tensor product of roadmaps for the individual robots. We demonstrate our approach experimentally on scenarios of up to 60 degrees of freedom where our algorithm is faster by a factor of at least ten when compared to existing algorithms that we are aware of.Comment: Kiril Solovey and Oren Salzman contributed equally to this pape

Navigation of Distinct Euclidean Particles via Hierarchical Clustering

We present a centralized online (completely reactive) hybrid navigation algorithm for bringing a swarm of n perfectly sensed and actuated point particles in Euclidean d space (for arbitrary n and d) to an arbitrary goal configuration with the guarantee of no collisions along the way. Our construction entails a discrete abstraction of configurations using cluster hierarchies, and relies upon two prior recent constructions: (i) a family of hierarchy-preserving control policies and (ii) an abstract discrete dynamical system for navigating through the space of cluster hierarchies. Here, we relate the (combinatorial) topology of hierarchical clusters to the (continuous) topology of configurations by constructing “portals” — open sets of configurations supporting two adjacent hierarchies. The resulting online sequential composition of hierarchy-invariant swarming followed by discrete selection of a hierarchy “closer” to that of the destination along with its continuous instantiation via an appropriate portal configuration yields a computationally effective construction for the desired navigation policy

On convergence and threshold properties of discrete lotka-volterra population protocols

In this work we focus on a natural class of population protocols whose dynamics are modeled by the discrete version of Lotka-Volterra equations with no linear term. In such protocols, when an agent a of type (species) i interacts with an agent b of type (species) j with a as the initiator, then b’s type becomes i with probability Pij. In such an interaction, we think of a as the predator, b as the prey, and the type of the prey is either converted to that of the predator or stays as is. Such protocols capture the dynamics of some opinion spreading models and generalize the well-known Rock-Paper-Scissors discrete dynamics. We consider the pairwise interactions among agents that are scheduled uniformly at random. We start by considering the convergence time and show that any Lotka-Volterra-type protocol on an n-agent populati

Approximating Fixation Probabilities in the Generalized Moran Process

We consider the Moran process, as generalized by Lieberman, Hauert and Nowak (Nature, 433:312--316, 2005). A population resides on the vertices of a finite, connected, undirected graph and, at each time step, an individual is chosen at random with probability proportional to its assigned 'fitness' value. It reproduces, placing a copy of itself on a neighbouring vertex chosen uniformly at random, replacing the individual that was there. The initial population consists of a single mutant of fitness $r>0$ placed uniformly at random, with every other vertex occupied by an individual of fitness 1. The main quantities of interest are the probabilities that the descendants of the initial mutant come to occupy the whole graph (fixation) and that they die out (extinction); almost surely, these are the only possibilities. In general, exact computation of these quantities by standard Markov chain techniques requires solving a system of linear equations of size exponential in the order of the graph so is not feasible. We show that, with high probability, the number of steps needed to reach fixation or extinction is bounded by a polynomial in the number of vertices in the graph. This bound allows us to construct fully polynomial randomized approximation schemes (FPRAS) for the probability of fixation (when $r\geq 1$) and of extinction (for all $r>0$).Comment: updated to the final version, which appeared in Algorithmic