8 research outputs found
Fast and Deterministic Approximations for k-Cut
In an undirected graph, a k-cut is a set of edges whose removal breaks the graph into at least k connected components. The minimum weight k-cut can be computed in n^O(k) time, but when k is treated as part of the input, computing the minimum weight k-cut is NP-Hard [Goldschmidt and Hochbaum, 1994]. For poly(m,n,k)-time algorithms, the best possible approximation factor is essentially 2 under the small set expansion hypothesis [Manurangsi, 2017]. Saran and Vazirani [1995] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed via O(k) minimum cuts, which implies a O~(km) randomized running time via the nearly linear time randomized min-cut algorithm of Karger [2000]. Nagamochi and Kamidoi [2007] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed deterministically in O(mn + n^2 log n) time. These results prompt two basic questions. The first concerns the role of randomization. Is there a deterministic algorithm for 2-approximate k-cuts matching the randomized running time of O~(km)? The second question qualitatively compares minimum cut to 2-approximate minimum k-cut. Can 2-approximate k-cuts be computed as fast as the minimum cut - in O~(m) randomized time?
We give a deterministic approximation algorithm that computes (2 + eps)-minimum k-cuts in O(m log^3 n / eps^2) time, via a (1 + eps)-approximation for an LP relaxation of k-cut
LP Relaxation and Tree Packing for Minimum k-cuts
Karger used spanning tree packings [Karger, 2000] to derive a near linear-time randomized algorithm for the global minimum cut problem as well as a bound on the number of approximate minimum cuts. This is a different approach from his well-known random contraction algorithm [Karger, 1995; Karger and Stein, 1996]. Thorup developed a fast deterministic algorithm for the minimum k-cut problem via greedy recursive tree packings [Thorup, 2008].
In this paper we revisit properties of an LP relaxation for k-cut proposed by Naor and Rabani [Naor and Rabani, 2001], and analyzed in [Chekuri et al., 2006]. We show that the dual of the LP yields a tree packing, that when combined with an upper bound on the integrality gap for the LP, easily and transparently extends Karger\u27s analysis for mincut to the k-cut problem. In addition to the simplicity of the algorithm and its analysis, this allows us to improve the running time of Thorup\u27s algorithm by a factor of n. We also improve the bound on the number of alpha-approximate k-cuts. Second, we give a simple proof that the integrality gap of the LP is 2(1-1/n). Third, we show that an optimum solution to the LP relaxation, for all values of k, is fully determined by the principal sequence of partitions of the input graph. This allows us to relate the LP relaxation to the Lagrangean relaxation approach of Barahona [Barahona, 2000] and Ravi and Sinha [Ravi and Sinha, 2008]; it also shows that the idealized recursive tree packing considered by Thorup gives an optimum dual solution to the LP. This work arose from an effort to understand and simplify the results of Thorup [Thorup, 2008]
Approximating submodular -partition via principal partition sequence
In submodular -partition, the input is a non-negative submodular function
defined over a finite ground set (given by an evaluation oracle) along
with a positive integer and the goal is to find a partition of the ground
set into non-empty parts in order to minimize
. Narayanan, Roy, and Patkar (Journal of Algorithms, 1996)
designed an algorithm for submodular -partition based on the principal
partition sequence and showed that the approximation factor of their algorithm
is for the special case of graph cut functions (subsequently rediscovered
by Ravi and Sinha (Journal of Operational Research, 2008)). In this work, we
study the approximation factor of their algorithm for three subfamilies of
submodular functions -- monotone, symmetric, and posimodular, and show the
following results:
1. The approximation factor of their algorithm for monotone submodular
-partition is . This result improves on the -factor achievable via
other algorithms. Moreover, our upper bound of matches the recently shown
lower bound under polynomial number of function evaluation queries (Santiago,
IWOCA 2021). Our upper bound of is also the first improvement beyond
for a certain graph partitioning problem that is a special case of monotone
submodular -partition.
2. The approximation factor of their algorithm for symmetric submodular
-partition is . This result generalizes their approximation factor
analysis beyond graph cut functions.
3. The approximation factor of their algorithm for posimodular submodular
-partition is .
We also construct an example to show that the approximation factor of their
algorithm for arbitrary submodular functions is .Comment: Accepted to APPROX'2
Fast Approximation Algorithms for Bounded Degree and Crossing Spanning Tree Problems
We develop fast approximation algorithms for the minimum-cost version of the Bounded-Degree MST problem (BD-MST) and its generalization the Crossing Spanning Tree problem (Crossing-ST). We solve the underlying LP to within a (1+?) approximation factor in near-linear time via the multiplicative weight update (MWU) technique. This yields, in particular, a near-linear time algorithm that outputs an estimate B such that B ? B^* ? ?(1+?)B?+1 where B^* is the minimum-degree of a spanning tree of a given graph. To round the fractional solution, in our main technical contribution, we describe a fast near-linear time implementation of swap-rounding in the spanning tree polytope of a graph. The fractional solution can also be used to sparsify the input graph that can in turn be used to speed up existing combinatorial algorithms. Together, these ideas lead to significantly faster approximation algorithms than known before for the two problems of interest. In addition, a fast algorithm for swap rounding in the graphic matroid is a generic tool that has other applications, including to TSP and submodular function maximization
Counting and enumerating optimum cut sets for hypergraph -partitioning problems for fixed
We consider the problem of enumerating optimal solutions for two hypergraph
-partitioning problems -- namely, Hypergraph--Cut and
Minmax-Hypergraph--Partition. The input in hypergraph -partitioning
problems is a hypergraph with positive hyperedge costs along with a
fixed positive integer . The goal is to find a partition of into
non-empty parts -- known as a -partition -- so as
to minimize an objective of interest.
1. If the objective of interest is the maximum cut value of the parts, then
the problem is known as Minmax-Hypergraph--Partition. A subset of hyperedges
is a minmax--cut-set if it is the subset of hyperedges crossing an optimum
-partition for Minmax-Hypergraph--Partition.
2. If the objective of interest is the total cost of hyperedges crossing the
-partition, then the problem is known as Hypergraph--Cut. A subset of
hyperedges is a min--cut-set if it is the subset of hyperedges crossing an
optimum -partition for Hypergraph--Cut.
We give the first polynomial bound on the number of minmax--cut-sets and a
polynomial-time algorithm to enumerate all of them in hypergraphs for every
fixed . Our technique is strong enough to also enable an -time
deterministic algorithm to enumerate all min--cut-sets in hypergraphs, thus
improving on the previously known -time deterministic algorithm,
where is the number of vertices and is the size of the hypergraph. The
correctness analysis of our enumeration approach relies on a structural result
that is a strong and unifying generalization of known structural results for
Hypergraph--Cut and Minmax-Hypergraph--Partition. We believe that our
structural result is likely to be of independent interest in the theory of
hypergraphs (and graphs).Comment: Accepted to ICALP'22. arXiv admin note: text overlap with
arXiv:2110.1481