Computational Complexity of Fixed Points and Intersection Points

AbstractWe study the computational complexity of Brouwer′s fixed point theorem and the intersection point theorem in the two-dimensional case. Papadimitriou (1990, in "Proceedings, 31st IEEE Sympos. Found. Comput. Sci.," pp. 794-801) defined a complexity class PDLF to characterize the complexity of the fixed point theorem in the three-dimensional case. We define a subclass PMLF of PDLF and show that the fixed points and the intersection points of polynomial-time computable functions are not polynomial-time computable if PMLF contains a function on unary inputs that is not polynomial-time computable

Utilization-Aware Adaptive Back-Pressure Traffic Signal Control

Back-pressure control of traffic signal, which computes the control phase to apply based on the real-time queue lengths, has been proposed recently. Features of it include (i) provably maximum stability, (ii) low computational complexity, (iii) no requirement of prior knowledge in traffic demand, and (iv) requirement of only local information at each intersection. The latter three points enable it to be completely distributed over intersections. However, one major issue preventing backpressure control from being used in practice is the utilization of the intersection, especially if the control phase period is fixed, as is considered in existing works. In this paper, we propose a utilization-aware adaptive algorithm of back-pressure traffic signal control, which makes the duration of the control phase adaptively dependent on the real-time queue lengths and strives for high utilization of the intersection. While advantages embedded in the back-pressure control are kept, we prove that this algorithm is work-conserving and achieves the maximum utilization. Simulation results on an isolated intersection show that the proposed adaptive algorithm has better control performance than the fixed-period back-pressure control presented in previous works

The Complexity of Drawing Graphs on Few Lines and Few Planes

It is well known that any graph admits a crossing-free straight-line drawing in $\mathbb{R}^3$ and that any planar graph admits the same even in $\mathbb{R}^2$. For a graph $G$ and $d \in \{2,3\}$, let $\rho^1_d(G)$ denote the minimum number of lines in $\mathbb{R}^d$ that together can cover all edges of a drawing of $G$. For $d=2$, $G$ must be planar. We investigate the complexity of computing these parameters and obtain the following hardness and algorithmic results. - For $d\in\{2,3\}$, we prove that deciding whether $\rho^1_d(G)\le k$ for a given graph $G$ and integer $k$ is ${\exists\mathbb{R}}$-complete. - Since $\mathrm{NP}\subseteq{\exists\mathbb{R}}$, deciding $\rho^1_d(G)\le k$ is NP-hard for $d\in\{2,3\}$. On the positive side, we show that the problem is fixed-parameter tractable with respect to $k$. - Since ${\exists\mathbb{R}}\subseteq\mathrm{PSPACE}$, both $\rho^1_2(G)$ and $\rho^1_3(G)$ are computable in polynomial space. On the negative side, we show that drawings that are optimal with respect to $\rho^1_2$ or $\rho^1_3$ sometimes require irrational coordinates. - Let $\rho^2_3(G)$ be the minimum number of planes in $\mathbb{R}^3$ needed to cover a straight-line drawing of a graph $G$. We prove that deciding whether $\rho^2_3(G)\le k$ is NP-hard for any fixed $k \ge 2$. Hence, the problem is not fixed-parameter tractable with respect to $k$ unless $\mathrm{P}=\mathrm{NP}$

Hierarchical structure-and-motion recovery from uncalibrated images

This paper addresses the structure-and-motion problem, that requires to find camera motion and 3D struc- ture from point matches. A new pipeline, dubbed Samantha, is presented, that departs from the prevailing sequential paradigm and embraces instead a hierarchical approach. This method has several advantages, like a provably lower computational complexity, which is necessary to achieve true scalability, and better error containment, leading to more stability and less drift. Moreover, a practical autocalibration procedure allows to process images without ancillary information. Experiments with real data assess the accuracy and the computational efficiency of the method.Comment: Accepted for publication in CVI