Consider a planar straight line graph (PSLG), G, with k connected components, k 2. We show that if no component is a singleton, we can always find a vertex in one component that sees an entire edge in another component. This implies that when the vertices of G are colored, so that adjacent vertices have different colors, then (1) we can augment G with k−1 edges so that we get a color conforming...

Straight line factoring algorithms include a variant Lenstra’s elliptic curve method. This note proves lower bounds on the length of straight line factoring algorithms.

Solving symbolically polynomial equation systems when intermediate and final polynomials are represented in the usual dense encoding turns out to be very inefficient: the sizes of the systems one can deal with do not respond to realistic needs. Evaluation representations appeared in this frame a decade ago as a new possibility to treat new families of problems. We present a survey of the most r...

We consider decision problems for deterministic pushdown automata over a unary alphabet (udpda, for short). Udpda are a simple computation model that accept exactly the unary regular languages, but can be exponentially more succinct than finite-state automata. We complete the complexity landscape for udpda by showing that emptiness (and thus universality) is P-hard, equivalence and compressed m...

We present a new algorithm for the computation of resultants associated with multihomogeneous (and, in particular, homogeneous) polynomial equation systems using straight-line programs. Its complexity is polynomial in the number of coefficients of the input system and the degree of the resultant computed.

The paper presents a design for the Denali-2 super-optimizer, which will generate minimum-instruction-length machine code for realistic machine architectures using automatic theorem-proving technology: specifically, using E-graph matching (a technique for pattern matching in the presence of equality information) and boolean satisfiability solving. The paper presents a precise definition of the ...

In this paper, we investigate planar upward straight-line grid drawing problems for bounded-degree rooted trees so that a drawing takes up as little area as possible. A planar upward straight-line grid tree drawing satisses the following four constraints: (1) all vertices are placed at distinct grid points (grid), (2) all edges are drawn as straight lines (straight-line), (3) no two edges in th...

In this paper we investigate the computational complexity of the following problem. Given a planar graph in which some vertices have already been placed in the plane (i.e., a partial embedding), place the remaining vertices to form a planar straight-line drawing of the graph. We show that the above extensibility problem, mentioned in the 2003 paper “Selected Open Problems in Graph Drawing” [1],...

We present a new method for solving symbolically zero–dimensional polynomial equation systems in the affine and toric case. The main feature of our method is the use of problem adapted data structures : arithmetic networks and straight–line programs. For sequential time complexity measured by network size we obtain the following result : it is possible to solve any affine or toric zero–dimensio...

The classical Fáry’s theorem from the 1930s states that every planar graph can be drawn as a straight-line drawing. In this paper, we extend Fáry’s theorem to non-planar graphs. More specifically, we study the problem of drawing 1-planar graphs with straight-line edges. A 1-planar graph is a sparse non-planar graph with at most one crossing per edge. We give a characterisation of those 1planar ...