25 research outputs found
An Algorithm for the Maximum Weight Independent Set Problem on Outerstring Graphs
Outerstring graphs are the intersection graphs of curves that lie inside a disk such that each curve intersects the boundary of the disk. Outerstring graphs are among the most general classes of intersection graphs studied. To date, no polynomial time algorithm is known for any of the classical graph optimization problems on outerstring graphs; in fact, most are NP-hard. It is known that there is an intersection model for any outerstring graph that consists of polygonal arcs attached to a circle. However, this representation may require an exponential number of segments relative to the size of the graph. Given an outerstring graph and an intersection model consisting of polygonal arcs with a total of N segments, we develop an algorithm that solves the Maximum Weight Independent Set problem in O(N³) time. If the polygonal arcs are restricted to single segments, then outersegment graphs result. For outersegment graphs, we solve the Maximum Weight Independent Set problem in O(n³) time where n is the number of vertices in the graph
Boundary Labeling for Rectangular Diagrams
Given a set of n points (sites) inside a rectangle R and n points (label locations or ports) on its boundary, a boundary labeling problem seeks ways of connecting every site to a distinct port while achieving different labeling aesthetics. We examine the scenario when the connecting lines (leaders) are drawn as axis-aligned polylines with few bends, every leader lies strictly inside R, no two leaders cross, and the sum of the lengths of all the leaders is minimized. In a k-sided boundary labeling problem, where 1 <= k <= 4, the label locations are located on the k consecutive sides of R.
In this paper we develop an O(n^3 log n)-time algorithm for 2-sided boundary labeling, where the leaders are restricted to have one bend. This improves the previously best known O(n^8 log n)-time algorithm of Kindermann et al. (Algorithmica, 76(1):225-258, 2016). We show the problem is polynomial-time solvable in more general settings such as when the ports are located on more than two sides of R, in the presence of obstacles, and even when the objective is to minimize the total number of bends. Our results improve the previous algorithms on boundary labeling with obstacles, as well as provide the first polynomial-time algorithms for minimizing the total leader length and number of bends for 3- and 4-sided boundary labeling. These results settle a number of open questions on the boundary labeling problems (Wolff, Handbook of Graph Drawing, Chapter 23, Table 23.1, 2014)
How to assign volunteers to tasks compatibly ? A graph theoretic and parameterized approach
In this paper we study a resource allocation problem that encodes correlation
between items in terms of \conflict and maximizes the minimum utility of the
agents under a conflict free allocation. Admittedly, the problem is
computationally hard even under stringent restrictions because it encodes a
variant of the {\sc Maximum Weight Independent Set} problem which is one of the
canonical hard problems in both classical and parameterized complexity.
Recently, this subject was explored by Chiarelli et al.~[Algorithmica'22] from
the classical complexity perspective to draw the boundary between {\sf
NP}-hardness and tractability for a constant number of agents. The problem was
shown to be hard even for small constant number of agents and various other
restrictions on the underlying graph. Notwithstanding this computational
barrier, we notice that there are several parameters that are worth studying:
number of agents, number of items, combinatorial structure that defines the
conflict among the items, all of which could well be small under specific
circumstancs. Our search rules out several parameters (even when taken
together) and takes us towards a characterization of families of input
instances that are amenable to polynomial time algorithms when the parameters
are constant. In addition to this we give a superior 2^{m}|I|^{\Co{O}(1)}
algorithm for our problem where denotes the number of items that
significantly beats the exhaustive \Oh(m^{m}) algorithm by cleverly using
ideas from FFT based fast polynomial multiplication; and we identify simple
graph classes relevant to our problem's motivation that admit efficient
algorithms
Intersection Graphs of Rays and Grounded Segments
We consider several classes of intersection graphs of line segments in the
plane and prove new equality and separation results between those classes. In
particular, we show that: (1) intersection graphs of grounded segments and
intersection graphs of downward rays form the same graph class, (2) not every
intersection graph of rays is an intersection graph of downward rays, and (3)
not every intersection graph of rays is an outer segment graph. The first
result answers an open problem posed by Cabello and Jej\v{c}i\v{c}. The third
result confirms a conjecture by Cabello. We thereby completely elucidate the
remaining open questions on the containment relations between these classes of
segment graphs. We further characterize the complexity of the recognition
problems for the classes of outer segment, grounded segment, and ray
intersection graphs. We prove that these recognition problems are complete for
the existential theory of the reals. This holds even if a 1-string realization
is given as additional input.Comment: 16 pages 12 Figure