A Polynomial Time Algorithm for Solving Systems of Linear Inequalities with Two Variables Per Inequality

Aspvall, Bengt; Shiloach, Yossi

doi:10.1137/0209063

Cited by 67 publications

(71 citation statements)

References 5 publications

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“…The transformation works as follows: Proof. It suffices to show that (r 1 r 2 ) r 3 (r 1 r 3 ) (r 2 r 3 ) because (r 1 r 3 ) (r 2 r 3 ) (r 1 r 2 ) r 3 . Consider any a ∈ (r 1 r 2 ) r 3 .…”

Section: Lemma 4 In a Constraint Graph A Strongly Connected Componementioning

confidence: 99%

“…Nelson [26] considers the same fragment and also gives an exponential time algorithm. Aspvall and Shiloach [3] refine Shostak's "loop residue" method and give a polynomial time algorithm for the fragment with two variables. Because constraints with three variables are NP-hard [22], this may be the most general class one can hope for a polynomial time algorithm.…”

Section: Introductionmentioning

confidence: 99%

“…However, solving integer linear constraints is a difficult problem [22], and only very special cases have efficient algorithms [30,3].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Class of Polynomially Solvable Range Constraints for Interval Analysis without Widenings and Narrowings

Su¹,

Wagner²

2004

Tools and Algorithms for the Construction and Analysis of Systems

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Abstract. In this paper, we study the problem of solving integer range constraints that arise in many static program analysis problems. In particular, we present the first polynomial time algorithm for a general class of integer range constraints. In contrast with abstract interpretation techniques based on widenings and narrowings, our algorithm computes, in polynomial time, the optimal solution of the arising fixpoint equations. Our result implies that "precise" range analysis can be performed in polynomial time without widening and narrowing operations.

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Section: Lemma 4 In a Constraint Graph A Strongly Connected Componementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A Class of Polynomially Solvable Range Constraints for Interval Analysis without Widenings and Narrowings

Su¹,

Wagner²

2004

Tools and Algorithms for the Construction and Analysis of Systems

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show abstract

“…Several algorithms have been developed for the LI(2) feasibility problem, examples are [3,6,12,18]. These algorithms either show that the LI(2) system has no feasible solution or provide a single feasible solution.…”

Section: Introductionmentioning

confidence: 99%

Listing Vertices of Simple Polyhedra Associated with Dual LI(2) Systems

Abdullahi

Dyer

Proll

2003

Discrete Mathematics and Theoretical Computer Science

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Abstract. We present an O(nv) Basis Oriented Pivoting (BOP) algorithm for enumerating vertices of simple polyhedra associated with dual LI(2) systems. The algorithm is based on a characterization of their graphical basis structures, whose set of edges are shown to consist of vertex-disjoint components that are either a tree or a subgraph with only one cycle. The algorithm generates vertices via operations on the basis graph, rather than by simplex transformations.

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“…These results are based upon Shostak's Theory for systems of inequalities, as exposed in [10]. We provide the proof in the Appendix.…”

mentioning

confidence: 99%

Maintaining limited-range connectivity among second-order agents

Notarstefano

Savla

Bullo

et al. 2006

2006 American Control Conference

100

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Abstract. In this paper we consider ad-hoc networks of robotic agents with double integrator dynamics. For such networks, the connectivity maintenance problems are: (i) do there exist control inputs for each agent to maintain network connectivity, and (ii) given desired controls for each agent, can one compute the closest connectivity-maintaining controls in a distributed fashion? The proposed solution is based on three contributions. First, we define and characterize admissible sets for double integrators to remain inside disks. Second, we establish an existence theorem for the connectivity maintenance problem by introducing a novel state-dependent graph, called the double-integrator disk graph. Specifically, we show that one can always maintain connectivity by maintaining a spanning tree of this new graph, but one will not always maintain connectivity of a particular agent pair that happens to be connected at one instant of time. Finally, we design a distributed "flow-control" algorithm for distributed computation of connectivity-maintaining controls.1. Introduction. This work is a contribution to the emerging discipline of motion coordination for ad-hoc networks of mobile autonomous agents. This loose terminology refers to groups of robotic agents with limited mobility and communication capabilities. It is envisioned that such networks will perform a variety of useful tasks including surveillance, exploration and environmental monitoring. The interest in this topics arises from the potential advantages of employing arrays of agents rather than single agents in certain applications. For example, from a control viewpoint, a group of agents inherently provides robustness to failures of single agents or of communication links.The motion coordination problem for groups of autonomous agents is a control problem in the presence of communication constraints. Typically, each agent makes decisions based only on partial information about the state of the entire network that is obtained via communication with its immediate neighbors. One important difficulty is that the topology of the communication network depends on the agents' locations and, therefore, changes with the evolution of the network. In order to ensure a desired emergent behavior for a group of agents, it is necessary that the group does not disintegrate into subgroups that are unable to communicate with each other. In other words, some restrictions must be applied on the movement of the agents to ensure connectivity among the members of the group. In terms of design, it is required to constrain the control input such that the resulting topology maintains connectivity throughout its course of evolution. In [2], a connectivity constraint was developed for a group of agents modeled as first-order discrete time dynamic systems. In [2] and in the related references [3,4], this constraint is used to solve rendezvous problems. Connectivity constraints for line-of-sight communication are proposed in

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A Polynomial Time Algorithm for Solving Systems of Linear Inequalities with Two Variables Per Inequality

Cited by 67 publications

References 5 publications

A Class of Polynomially Solvable Range Constraints for Interval Analysis without Widenings and Narrowings

A Class of Polynomially Solvable Range Constraints for Interval Analysis without Widenings and Narrowings

Listing Vertices of Simple Polyhedra Associated with Dual LI(2) Systems

Maintaining limited-range connectivity among second-order agents

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