3 research outputs found
Fixed-Parameter Tractability of Maximum Colored Path and Beyond
We introduce a general method for obtaining fixed-parameter algorithms for
problems about finding paths in undirected graphs, where the length of the path
could be unbounded in the parameter. The first application of our method is as
follows.
We give a randomized algorithm, that given a colored -vertex undirected
graph, vertices and , and an integer , finds an -path
containing at least different colors in time . This is the
first FPT algorithm for this problem, and it generalizes the algorithm of
Bj\"orklund, Husfeldt, and Taslaman [SODA 2012] on finding a path through
specified vertices. It also implies the first time algorithm for
finding an -path of length at least .
Our method yields FPT algorithms for even more general problems. For example,
we consider the problem where the input consists of an -vertex undirected
graph , a matroid whose elements correspond to the vertices of and
which is represented over a finite field of order , a positive integer
weight function on the vertices of , two sets of vertices , and integers , and the task is to find vertex-disjoint paths
from to so that the union of the vertices of these paths contains an
independent set of of cardinality and weight , while minimizing the
sum of the lengths of the paths. We give a
time randomized algorithm for this problem.Comment: 50 pages, 16 figure
A Fast Parallel Algorithm for Minimum-Cost Small Integral Flows
We present a new approach to the minimum-cost integral flow problem for small values of the flow. It reduces the problem to the tests of simple multivariate polynomials over a finite field of characteristic two for non-identity with zero. In effect, we show that a minimum-cost flow of value k in a network with n vertices, a sink and a source, integral edge capacities and positive integral edge costs polynomially bounded in n can be found by a randomized PRAM, with errors of exponentially small probability in n, running in O(klog(kn)+log(2)(kn)) time and using 2 (k) (kn) (O(1)) processors. Thus, in particular, for the minimum-cost flow of value O(logn), we obtain an RNC2 algorithm, improving upon time complexity of earlier NC and RNC algorithms
A Fast Parallel Algorithm for Minimum-Cost Small Integral Flows
We present a new approach to the minimum-cost integral flow problem for small values of the flow. It reduces the problem to the tests of simple multi-variable polynomials over a finite field of characteristic two for non-identity with zero. In effect, we show that a minimum-cost flow of value k in a network with n vertices, a sink and a source, integral edge capacities and positive integral edge costs polynomially bounded in n can be found by a randomized PRAM, with errors of exponentially small probability in n, running in O(klog(kn) + log2 (kn)) time and using 2 k (kn) O(1) processors. Thus, in particular, for the minimum-cost flow of value O(logn), we obtain an RNC 2 algorithm