On the interplay of combinatorics, geometry, topology and computational complexity

Matematicko-fyzikální fakult

Embeddability in the 3-sphere is decidable

We show that the following algorithmic problem is decidable: given a $2$-dimensional simplicial complex, can it be embedded (topologically, or equivalently, piecewise linearly) in $\mathbf{R}^3$? By a known reduction, it suffices to decide the embeddability of a given triangulated 3-manifold $X$ into the 3-sphere $S^3$. The main step, which allows us to simplify $X$ and recurse, is in proving that if $X$ can be embedded in $S^3$, then there is also an embedding in which $X$ has a short meridian, i.e., an essential curve in the boundary of $X$ bounding a disk in $S^3\setminus X$ with length bounded by a computable function of the number of tetrahedra of $X$.Comment: 54 pages, 26 figures; few faulty references to figures in the first version fixe

Geometric, Algebraic, and Topological Combinatorics

The 2019 Oberwolfach meeting "Geometric, Algebraic and Topological Combinatorics" was organized by Gil Kalai (Jerusalem), Isabella Novik (Seattle), Francisco Santos (Santander), and Volkmar Welker (Marburg). It covered a wide variety of aspects of Discrete Geometry, Algebraic Combinatorics with geometric flavor, and Topological Combinatorics. Some of the highlights of the conference included (1) Karim Adiprasito presented his very recent proof of the $g$-conjecture for spheres (as a talk and as a "Q\&A" evening session) (2) Federico Ardila gave an overview on "The geometry of matroids", including his recent extension with Denham and Huh of previous work of Adiprasito, Huh and Katz

Resilient routing in the internet

Although it is widely known that the Internet is not prone to random failures, unplanned failures due to attacks can be very damaging. This prevents many organisations from deploying beneficial operations through the Internet. In general, the data is delivered from a source to a destination via a series of routers (i.e routing path). These routers employ routing protocols to compute best paths based on routing information they possess. However, when a failure occurs, the routers must re-construct their routing tables, which may take several seconds to complete. Evidently, most losses occur during this period. IP Fast Re-Route (IPFRR), Multi-Topology (MT) routing, and overlays are examples of solutions proposed to handle network failures. These techniques alleviate the packet losses to different extents, yet none have provided optimal solutions. This thesis focuses on identifying the fundamental routing problem due to convergence process. It describes the mechanisms of each existing technique as well as its pros and cons. Furthermore, it presents new techniques for fast re-routing as follows. Enhanced Loop-Free Alternates (E-LFAs) increase the repair coverage of the existing techniques, Loop-Free Alternates (LFAs). In addition, two techniques namely, Full Fast Failure Recovery (F3R) and fast re-route using Alternate Next Hop Counters (ANHC), offer full protection against any single link failures. Nevertheless, the former technique requires significantly higher computational overheads and incurs longer backup routes. Both techniques are proved to be complete and correct while ANHC neither requires any major modifications to the traditional routing paradigm nor incurs significant overheads. Furthermore, in the presence of failures, ANHC does not jeopardise other operable parts of the network. As emerging applications require higher reliability, multiple failures scenarios cannot be ignored. Most existing fast re-route techniques are able to handle only single or dual failures cases. This thesis provides an insight on a novel approach known as Packet Re-cycling (PR), which is capable of handling any number of failures in an oriented network. That is, packets can be forwarded successfully as long as a path between a source and a destination is available. Since the Internet-based services and applications continue to advance, improving the network resilience will be a challenging research topic for the decades to come

A Linear Time Planarity Algorithm for 2-Complexes

A linear time algorithm to decide whether a given finite 2- complex is planar is described. Topological results of Gross, Harary and Rosen are the mathematical basis for the algorithm. Optimal running time is achieved by constructing various lists simultaneously and keeping their orderings compatible. If the complex is simplical with p vertices, then the algorithm has O(p) time and space bounds. The algorithm uses depth-first search both in application of the graph planarity algorithm of Hopcroft and Tarjan and elsewhere