Approximating Minimum Cost Connectivity Orientation and Augmentation

We investigate problems addressing combined connectivity augmentation and orientations settings. We give a polynomial-time 6-approximation algorithm for finding a minimum cost subgraph of an undirected graph $G$ that admits an orientation covering a nonnegative crossing $G$-supermodular demand function, as defined by Frank. An important example is $(k,\ell)$-edge-connectivity, a common generalization of global and rooted edge-connectivity. Our algorithm is based on a non-standard application of the iterative rounding method. We observe that the standard linear program with cut constraints is not amenable and use an alternative linear program with partition and co-partition constraints instead. The proof requires a new type of uncrossing technique on partitions and co-partitions. We also consider the problem setting when the cost of an edge can be different for the two possible orientations. The problem becomes substantially more difficult already for the simpler requirement of $k$-edge-connectivity. Khanna, Naor, and Shepherd showed that the integrality gap of the natural linear program is at most $4$ when $k=1$ and conjectured that it is constant for all fixed $k$. We disprove this conjecture by showing an $\Omega(|V|)$ integrality gap even when $k=2$

Total Representations

Almost all representations considered in computable analysis are partial. We provide arguments in favor of total representations (by elements of the Baire space). Total representations make the well known analogy between numberings and representations closer, unify some terminology, simplify some technical details, suggest interesting open questions and new invariants of topological spaces relevant to computable analysis.Comment: 30 page

Area-Universal Rectangular Layouts

A rectangular layout is a partition of a rectangle into a finite set of interior-disjoint rectangles. Rectangular layouts appear in various applications: as rectangular cartograms in cartography, as floorplans in building architecture and VLSI design, and as graph drawings. Often areas are associated with the rectangles of a rectangular layout and it might hence be desirable if one rectangular layout can represent several area assignments. A layout is area-universal if any assignment of areas to rectangles can be realized by a combinatorially equivalent rectangular layout. We identify a simple necessary and sufficient condition for a rectangular layout to be area-universal: a rectangular layout is area-universal if and only if it is one-sided. More generally, given any rectangular layout L and any assignment of areas to its regions, we show that there can be at most one layout (up to horizontal and vertical scaling) which is combinatorially equivalent to L and achieves a given area assignment. We also investigate similar questions for perimeter assignments. The adjacency requirements for the rectangles of a rectangular layout can be specified in various ways, most commonly via the dual graph of the layout. We show how to find an area-universal layout for a given set of adjacency requirements whenever such a layout exists.Comment: 19 pages, 16 figure