On bar frameworks, stress matrices and semidefinite programming

Abstract

A bar framework G(p) in r-dimensional Euclidean space is a graph G = (V, E) on the vertices 1, 2, . . . , n, where each vertex i is located at point p i in R r . Given a framework G(p) in R r , a problem of great interest is that of determining whether or not there exists another framework G(q), not obtained from G(p) by a rigid motion, such that ||qThis problem is known as either the global rigidity problem or the universal rigidity problem depending on whether such a framework G(q) is restricted to be in the same r-dimensional space or not. The stress matrix S of a bar framework G(p) plays a key role in these and other related problems.In this paper, we show that semidefinite programming (SDP) can be effectively used to address the universal rigidity problem. In particular, we use the notion of non-degeneracy of SDP to obtain a sufficient condition for universal rigidity, and to re-derive the known sufficient condition for generic universal rigidity. We present new results concerning positive semidefinite stress matrices and we use a semidefinite version of Farkas lemma to characterize bar frameworks that admit a nonzero positive semidefinite stress matrix S. *

Universal Rigidity and Edge Sparsification for Sensor Network Localization

“…In particular, we conclude that =˜˜, wherẽ = [˜1 ⋅ ⋅ ⋅˜], is the unique solution to (2). It then follows from [35,Theorem 2] …”

“…In particular, observe that the complexity of the EES heuristic is much lower than that required for solving the SDP relaxation (2). This justifies the use of the EES heuristic as a preprocessing procedure for speeding up the solution time of (2).…”

“…Although the notion of universal rigidity has been introduced before (see [1,2] and the references therein), it does not seem to be well-studied. For instance, it is not entirely clear what the relationship is between unique -localizability and universal rigidity in ℝ .…”

“…Recall that given an instance ( , (d,d), a, ) of the sensor network localization problem, our goal is to speed up the solution time of its associated SDP relaxation (2). A natural way to achieve this is to reduce the number of constraints in (2), which corresponds to removing edges from .…”

Selected Open Problems in Discrete Geometry and Optimization

“…This is motivated by an earlier result of So and Ye [20], which states that each dual variable in the SDP relaxation of Biswas and Ye [5] corresponds to a stress on an edge of the input graph, and the optimality conditions of the SDP correspond to a certain equilibrium condition on the input graph. The work [20] has since motivated or been used to develop other rigiditytheoretic results (see, e.g., [1,13]), and a natural question would be whether these results have counterparts in the Schatten quasi-norm regularization setting.…”