skip to main content

Title: Realization Problems on Reachability Sequences
The classical Erdös-Gallai theorem kicked off the study ofgraph realizability by characterizing degree sequences. We extend this line of research by investigating realizability of directed acyclic graphs (DAGs)given both a local constraint via degree sequences and a global constraint via a sequence of reachability values (number of nodes reachable from a given node). We show that, without degree constraints, DAG reachability realization is solvable in linear time, whereas it is strongly NP-complete given upper bounds on in-degree or out-degree. After defining a suitable notion of bicriteria approximation based on consistency, we give two approximation algorithms achieving O(logn)-reachability consistency and O(logn)-degree consistency; the first, randomized, uses LP (Linear Program) rounding, while the second, deterministic, employs ak-setpacking heuristic. We end with two conjectures that we hope motivate further study of realizability with reachability constraints.
; ;
Award ID(s):
Publication Date:
Journal Name:
Sponsoring Org:
National Science Foundation
More Like this
  1. Consider an algorithm performing a computation on a huge random object (for example a random graph or a "long" random walk). Is it necessary to generate the entire object prior to the computation, or is it possible to provide query access to the object and sample it incrementally "on-the-fly" (as requested by the algorithm)? Such an implementation should emulate the random object by answering queries in a manner consistent with an instance of the random object sampled from the true distribution (or close to it). This paradigm is useful when the algorithm is sub-linear and thus, sampling the entire objectmore »up front would ruin its efficiency. Our first set of results focus on undirected graphs with independent edge probabilities, i.e. each edge is chosen as an independent Bernoulli random variable. We provide a general implementation for this model under certain assumptions. Then, we use this to obtain the first efficient local implementations for the Erdös-Rényi G(n,p) model for all values of p, and the Stochastic Block model. As in previous local-access implementations for random graphs, we support Vertex-Pair and Next-Neighbor queries. In addition, we introduce a new Random-Neighbor query. Next, we give the first local-access implementation for All-Neighbors queries in the (sparse and directed) Kleinberg’s Small-World model. Our implementations require no pre-processing time, and answer each query using O(poly(log n)) time, random bits, and additional space. Next, we show how to implement random Catalan objects, specifically focusing on Dyck paths (balanced random walks on the integer line that are always non-negative). Here, we support Height queries to find the location of the walk, and First-Return queries to find the time when the walk returns to a specified location. This in turn can be used to implement Next-Neighbor queries on random rooted ordered trees, and Matching-Bracket queries on random well bracketed expressions (the Dyck language). Finally, we introduce two features to define a new model that: (1) allows multiple independent (and even simultaneous) instantiations of the same implementation, to be consistent with each other without the need for communication, (2) allows us to generate a richer class of random objects that do not have a succinct description. Specifically, we study uniformly random valid q-colorings of an input graph G with maximum degree Δ. This is in contrast to prior work in the area, where the relevant random objects are defined as a distribution with O(1) parameters (for example, n and p in the G(n,p) model). The distribution over valid colorings is instead specified via a "huge" input (the underlying graph G), that is far too large to be read by a sub-linear time algorithm. Instead, our implementation accesses G through local neighborhood probes, and is able to answer queries to the color of any given vertex in sub-linear time for q ≥ 9Δ, in a manner that is consistent with a specific random valid coloring of G. Furthermore, the implementation is memory-less, and can maintain consistency with non-communicating copies of itself.« less
  2. There has been an increasing interest in using neural networks in closed-loop control systems to improve performance and reduce computational costs for on-line implementation. However, providing safety and stability guarantees for these systems is challenging due to the nonlinear and compositional structure of neural networks. In this paper, we propose a novel forward reachability analysis method for the safety verification of linear time-varying systems with neural networks in feedback interconnection. Our technical approach relies on abstracting the nonlinear activation functions by quadratic constraints, which leads to an outer-approximation of forward reachable sets of the closed-loop system. We show that wemore »can compute these approximate reachable sets using semidefinite programming. We illustrate our method in a quadrotor example, in which we first approximate a nonlinear model predictive controller via a deep neural network and then apply our analysis tool to certify finite-time reachability and constraint satisfaction of the closed-loop system.« less
  3. In this work, we propose a trajectory generation method for robotic systems with contact force constraint based on optimal control and reachability analysis. Normally, the dynamics and constraints of the contact-constrained robot are nonlinear and coupled to each other. Instead of linearizing the model and constraints, we directly solve the optimal control problem to obtain the feasible state trajectory and the control input of the system. A tractable optimal control problem is formulated which is addressed by dual approaches, which are sampling-based dynamic programming and rigorous reachability analysis. The sampling-based method and Partially Observable Markov Decision Process (POMDP) are usedmore »to break down the end-to-end trajectory generation problem via sample-wise optimization in terms of given conditions. The result generates sequential pairs of subregions to be passed to reach the final goal. The reachability analysis ensures that we will find at least one trajectory starting from a given initial state and going through a sequence of subregions. The distinctive contributions of our method are to enable handling the intricate contact constraint coupled with system’s dynamics due to the reduction of computational complexity of the algorithm. We validate our method using extensive numerical simulations with a legged robot.« less
  4. Directed Steiner Tree (DST) is a central problem in combinatorial optimization and theoretical computer science: Given a directed graph G = (V, E) with edge costs c ∈ ℝ_{≥ 0}^E, a root r ∈ V and k terminals K ⊆ V, we need to output a minimum-cost arborescence in G that contains an rrightarrow t path for every t ∈ K. Recently, Grandoni, Laekhanukit and Li, and independently Ghuge and Nagarajan, gave quasi-polynomial time O(log²k/log log k)-approximation algorithms for the problem, which are tight under popular complexity assumptions. In this paper, we consider the more general Degree-Bounded Directed Steiner Treemore »(DB-DST) problem, where we are additionally given a degree bound d_v on each vertex v ∈ V, and we require that every vertex v in the output tree has at most d_v children. We give a quasi-polynomial time (O(log n log k), O(log² n))-bicriteria approximation: The algorithm produces a solution with cost at most O(log nlog k) times the cost of the optimum solution that violates the degree constraints by at most a factor of O(log²n). This is the first non-trivial result for the problem. While our cost-guarantee is nearly optimal, the degree violation factor of O(log²n) is an O(log n)-factor away from the approximation lower bound of Ω(log n) from the Set Cover hardness. The hardness result holds even on the special case of the Degree-Bounded Group Steiner Tree problem on trees (DB-GST-T). With the hope of closing the gap, we study the question of whether the degree violation factor can be made tight for this special case. We answer the question in the affirmative by giving an (O(log nlog k), O(log n))-bicriteria approximation algorithm for DB-GST-T.« less
  5. This paper proposes a method to generate feasible trajectories for robotic systems with predefined sequences of switched contacts. The proposed trajectory generation method relies on sampling-based methods, optimal control, and reach-ability analysis. In particular, the proposed method is able to quickly test whether a simplified model-based planner, such as the Time-to-Velocity-Reversal planner, provides a reachable contact location based on reachability analysis of the multi-body robot system. When the contact location is reachable, we generate a feasible trajectory to change the contact mode of the robotic system smoothly. To perform reachability analysis efficiently, we devise a method to compute forward andmore »backward reachable sets based on element-wise optimization over a finite time horizon. Then, we compute robot trajectories by employing optimal control. The main contributions of this study are the following. Firstly, we guarantee whether planned contact locations via simplified models are feasible by the robot system. Secondly, we generate optimal trajectories subject to various constraints given a feasible contact sequence. Lastly, we improve the efficiency of computing reachable sets for a class of constrained nonlinear systems by incorporating bi-directional propagation (forward and backward). To validate our methods we perform numerical simulations applied to a humanoid robot walking.« less