16 research outputs found
Efficient Generation of Stable Planar Cages for Chemistry
In this paper we describe an algorithm which generates all colored planar
maps with a good minimum sparsity from simple motifs and rules to connect them.
An implementation of this algorithm is available and is used by chemists who
want to quickly generate all sound molecules they can obtain by mixing some
basic components.Comment: 17 pages, 7 figures. Accepted at the 14th International Symposium on
Experimental Algorithms (SEA 2015
Evaluating a 2-approximation algorithm for edge-separators in planar graphs
In this paper we report on results obtained by an implementation of a 2-approximation algorithm for edge separators in planar graphs. For 374 out of the 435 instances the algorithm returned the optimum solution. For the remaining instances the solution returned was never more than 10.6\% away from the lower bound on the optimum separator. We also improve the worst-case running time of the algorithm from to and present techniques which improve the running time significantly in practice
A Polynomial-time Bicriteria Approximation Scheme for Planar Bisection
Given an undirected graph with edge costs and node weights, the minimum
bisection problem asks for a partition of the nodes into two parts of equal
weight such that the sum of edge costs between the parts is minimized. We give
a polynomial time bicriteria approximation scheme for bisection on planar
graphs.
Specifically, let be the total weight of all nodes in a planar graph .
For any constant , our algorithm outputs a bipartition of the
nodes such that each part weighs at most and the total cost
of edges crossing the partition is at most times the total
cost of the optimal bisection. The previously best known approximation for
planar minimum bisection, even with unit node weights, was . Our
algorithm actually solves a more general problem where the input may include a
target weight for the smaller side of the bipartition.Comment: To appear in STOC 201
Balanced Partitions of Trees and Applications
We study the k-BALANCED PARTITIONING problem in which the vertices of a graph are to be partitioned into k sets of size at most ceil(n/k) while minimising the cut size, which is the number of edges connecting vertices in different sets. The problem is well studied for general graphs, for which it cannot be approximated within any factor in polynomial time. However, little is known about restricted graph classes. We show that for trees k-BALANCED PARTITIONING remains surprisingly hard. In particular, approximating the cut size is APX-hard even if the maximum degree of the tree is constant. If instead the diameter of the tree is bounded by a constant, we show that it is NP-hard to approximate the cut size within n^c, for any constant c<1. In the face of the hardness results, we show that allowing near-balanced solutions, in which there are at most (1+eps)ceil(n/k) vertices in any of the k sets, admits a PTAS for trees. Remarkably, the computed cut size is no larger than that of an optimal balanced solution. In the final section of our paper, we harness results on embedding graph metrics into tree metrics to extend our PTAS for trees to general graphs. In addition to being conceptually simpler and easier to analyse, our scheme improves the best factor known on the cut size of near-balanced solutions from O(log^{1.5}(n)/eps^2) [Andreev and Räcke TCS 2006] to 0(log n), for weighted graphs. This also settles a question posed by Andreev and Räcke of whether an algorithm with approximation guarantees on the cut size independent from eps exists.ISSN:1868-896
On the computational tractability of a geographic clustering problem arising in redistricting
Redistricting is the problem of dividing a state into a number of
regions, called districts. Voters in each district elect a representative. The
primary criteria are: each district is connected, district populations are
equal (or nearly equal), and districts are "compact". There are multiple
competing definitions of compactness, usually minimizing some quantity.
One measure that has been recently promoted by Duchin and others is number of
cut edges. In redistricting, one is given atomic regions out of which each
district must be built. The populations of the atomic regions are given.
Consider the graph with one vertex per atomic region (with weight equal to the
region's population) and an edge between atomic regions that share a boundary.
A districting plan is a partition of vertices into parts, each connnected,
of nearly equal weight. The districts are considered compact to the extent that
the plan minimizes the number of edges crossing between different parts.
Consider two problems: find the most compact districting plan, and sample
districting plans under a compactness constraint uniformly at random. Both
problems are NP-hard so we restrict the input graph to have branchwidth at most
. (A planar graph's branchwidth is bounded by its diameter.) If both and
are bounded by constants, the problems are solvable in polynomial time.
Assume vertices have weight~1. One would like algorithms whose running times
are of the form for some constant independent of and
, in which case the problems are said to be fixed-parameter tractable with
respect to and ). We show that, under a complexity-theoretic assumption,
no such algorithms exist. However, we do give algorithms with running time
. Thus if the diameter of the graph is moderately small and the
number of districts is very small, our algorithm is useable
Towards a better approximation for sparsest cut?
We give a new -approximation for sparsest cut problem on graphs
where small sets expand significantly more than the sparsest cut (sets of size
expand by a factor bigger, for some small ; this
condition holds for many natural graph families). We give two different
algorithms. One involves Guruswami-Sinop rounding on the level- Lasserre
relaxation. The other is combinatorial and involves a new notion called {\em
Small Set Expander Flows} (inspired by the {\em expander flows} of ARV) which
we show exists in the input graph. Both algorithms run in time . We also show similar approximation algorithms in graphs with
genus with an analogous local expansion condition. This is the first
algorithm we know of that achieves -approximation on such general
family of graphs