Eyal Ackerman

Combinatorial model and bounds for target set selection

The adoption of everyday decisions in public affairs, fashion, movie-going, and consumer behavior is now thoroughly believed to migrate in a population through an influential network. The same diffusion process when being imitated by intention is called viral marketing. This process can be modeled by a (directed) graph G=(V,E) with a threshold t(v) for every vertex v?V, where v becomes active once at least t(v) of its (in-)neighbors are already active. A Perfect Target Set is a set of vertices whose activation will eventually activate the entire graph, and the Perfect Target Set Selection Problem (PTSS) asks for the minimum such initial set. It is known (Chen (2008) 6) that PTSS is hard to approximate, even for some special cases such as bounded-degree graphs, or majority thresholds.We propose a combinatorial model for this dynamic activation process, and use it to represent PTSS and its variants by linear integer programs. This allows one to use standard integer programming solvers for solving small-size PTSS instances. We also show combinatorial lower and upper bounds on the size of the minimum Perfect Target Set. Our upper bound implies that there are always Perfect Target Sets of size at most |V|/2 and 2|V|/3 under majority and strict majority thresholds, respectively, both in directed and undirected graphs. This improves the bounds of 0.727|V| and 0.7732|V| found recently by Chang and Lyuu (2010) 5 for majority and strict majority thresholds in directed graphs, and matches their bound under majority thresholds in undirected graphs. Furthermore, our proof is much simpler, and we observe that some of these bounds are tight. One interesting and perhaps surprising implication of our lower bound for undirected graphs, is that it is easy to get a constant factor approximation for PTSS for “relatively balanced” graphs (e.g., bounded-degree graphs, nearly regular graphs) with a “more than majority” threshold (that is, t(v)=??deg(v), for every v?V and some constant ?>1/2), whereas no polylogarithmic approximation exists for “more than majority” graphs.

On the maximum number of edges in quasi-planar graphs

A topological graph is quasi-planar, if it does not contain three pairwise crossing edges. Agarwal et al. [P.K. Agarwal, B. Aronov, J. Pach, R. Pollack, M. Sharir, Quasi-planar graphs have a linear number of edges, Combinatorica 17 (1) (1997) 1-9] proved that these graphs have a linear number of edges. We give a simple proof for this fact that yields the better upper bound of 8n edges for n vertices. Our best construction with 7n-O(1) edges comes very close to this bound. Moreover, we show matching upper and lower bounds for several relaxations and restrictions of this problem. In particular, we show that the maximum number of edges of a simple quasi-planar topological graph (i.e., every pair of edges have at most one point in common) is 6.5n-O(1), thereby exhibiting that non-simple quasi-planar graphs may have many more edges than simple ones.

On the maximum number of edges in topological graphs with no four pairwise crossing edges

We show that the maximum number of edges in a topological graph on n vertices and with no four pairwise crossing edges is O(n).

A bijection between permutations and floorplans, and its applications

A floorplan represents the relative relations between modules on an integrated circuit. Floorplans are commonly classified as slicing, mosaic, or general. Separable and Baxter permutations are classes of permutations that can be defined in terms of forbidden subsequences. It is known that the number of slicing floorplans equals the number of separable permutations and that the number of mosaic floorplans equals the number of Baxter permutations [B. Yao, H. Chen, C.K. Cheng, R.L. Graham, Floorplan representations: complexity and connections, ACM Trans. Design Automation Electron. Systems 8(1) (2003) 55-80]. We present a simple and efficient bijection between Baxter permutations and mosaic floorplans with applications to integrated circuits design. Moreover, this bijection has two additional merits: (1) It also maps between separable permutations and slicing floorplans; and (2) it suggests enumerations of mosaic floorplans according to various structural parameters.

On the number of rectangulations of a planar point set

We investigate the number of different ways in which a rectangle containing a set of n noncorectilinear points can be partitioned into smaller rectangles by n (nonintersecting) segments, such that every point lies on a segment. We show that when the relative order of the points forms a separable permutation, the number of rectangulations is exactly the (n + 1)st Baxter number. We also show that no matter what the order of the points is, the number of guillotine rectangulations is always the nth Schroder number, and the total number of rectangulations is O(20n/n4).

On the size of graphs that admit polyline drawings with few bends and crossing angles

We consider graphs that admit polyline drawings where all crossings occur at the same angle α ∈ (0, π/2). We prove that every graph on n vertices that admits such a polyline drawing with at most two bends per edge has O(n) edges. This result remains true when each crossing occurs at an angle from a small set of angles. We also provide several extensions that might be of independent interest.

A note on 1-planar graphs

A graph is 1-planar if it can be drawn in the plane such that each of its edges is crossed at most once. We prove a conjecture of Czap and Hudak (2013) stating that the edge set of every 1-planar graph can be decomposed into a planar graph and a forest. We also provide simple proofs for the following recent results: (i) an n-vertex graph that admits a 1-planar drawing with straight-line edges has at most 4n-9 edges (Didimo, 2013); and (ii) every drawing of a maximally dense right angle crossing graph is 1-planar (Eades and Liotta, 2013).

On grids in topological graphs

A topological graph is a graph drawn in the plane with vertices represented by points and edges as arcs connecting its vertices. A k-grid in a topological graph is a pair of subsets of the edge set, each of size k, such that every edge in one subset crosses every edge in the other subset. It is known that for a fixed constant k, every n-vertex topological graph with no k-grid has O(n) edges. We conjecture that this statement remains true (1) for topological graphs in which only k-grids consisting of 2k vertex-disjoint edges are forbidden, and (2) for graphs drawn by straight-line edges, with no k-element sets of edges such that every edge in the first set crosses every edge in the other set and each pair of edges within the same set is disjoint. These conjectures are shown to be true apart from log* n and log2 n factors, respectively. We also settle the conjectures for some special cases.

The number of guillotine partitions in d dimensions

Guillotine partitions play an important role in many research areas and application domains, e.g., computational geometry, computer graphics, integrated circuit layout, and solid modeling, to mention just a few. In this paper we present an exact summation formula for the number of structurally-different guillotine partitions in d dimensions by n hyperplanes, and then show that it is @Q((2d-1+2d(d-1))^n/n^3^/^2).

Graphs That Admit Polyline Drawings with Few Crossing Angles

We consider graphs that admit polyline drawings where all crossings occur at the same angle