Exploiting chordal structure in polynomial ideals: a Gr\"obner bases approach

Chordal structure and bounded treewidth allow for efficient computation in numerical linear algebra, graphical models, constraint satisfaction and many other areas. In this paper, we begin the study of how to exploit chordal structure in computational algebraic geometry, and in particular, for solving polynomial systems. The structure of a system of polynomial equations can be described in terms of a graph. By carefully exploiting the properties of this graph (in particular, its chordal completions), more efficient algorithms can be developed. To this end, we develop a new technique, which we refer to as chordal elimination, that relies on elimination theory and Gr\"obner bases. By maintaining graph structure throughout the process, chordal elimination can outperform standard Gr\"obner basis algorithms in many cases. The reason is that all computations are done on "smaller" rings, of size equal to the treewidth of the graph. In particular, for a restricted class of ideals, the computational complexity is linear in the number of variables. Chordal structure arises in many relevant applications. We demonstrate the suitability of our methods in examples from graph colorings, cryptography, sensor localization and differential equations.Comment: 40 pages, 5 figure

Completion and deficiency problems

Given a partial Steiner triple system (STS) of order $n$, what is the order of the smallest complete STS it can be embedded into? The study of this question goes back more than 40 years. In this paper we answer it for relatively sparse STSs, showing that given a partial STS of order $n$ with at most $r \le \varepsilon n^2$ triples, it can always be embedded into a complete STS of order $n+O(\sqrt{r})$, which is asymptotically optimal. We also obtain similar results for completions of Latin squares and other designs. This suggests a new, natural class of questions, called deficiency problems. Given a global spanning property $\mathcal{P}$ and a graph $G$, we define the deficiency of the graph $G$ with respect to the property $\mathcal{P}$ to be the smallest positive integer $t$ such that the join $G\ast K_t$ has property $\mathcal{P}$. To illustrate this concept we consider deficiency versions of some well-studied properties, such as having a $K_k$-decomposition, Hamiltonicity, having a triangle-factor and having a perfect matching in hypergraphs. The main goal of this paper is to propose a systematic study of these problems; thus several future research directions are also given

Many Physical Design Problems are Sparse QCQPs

Physical design refers to mathematical optimization of a desired objective (e.g. strong light--matter interactions, or complete quantum state transfer) subject to the governing dynamical equations, such as Maxwell's or Schrodinger's differential equations. Computing an optimal design is challenging: generically, these problems are highly nonconvex and finding global optima is NP hard. Here we show that for linear-differential-equation dynamics (as in linear electromagnetism, elasticity, quantum mechanics, etc.), the physical-design optimization problem can be transformed to a sparse-matrix, quadratically constrained quadratic program (QCQP). Sparse QCQPs can be tackled with convex optimization techniques (such as semidefinite programming) that have thrived for identifying global bounds and high-performance designs in other areas of science and engineering, but seemed inapplicable to the design problems of wave physics. We apply our formulation to prototypical photonic design problems, showing the possibility to compute fundamental limits for large-area metasurfaces, as well as the identification of designs approaching global optimality. Looking forward, our approach highlights the promise of developing bespoke algorithms tailored to specific physical design problems.Comment: 9 pages, 4 figures, plus references and Supplementary Material