158 research outputs found
Approximating Minimum-Cost k-Node Connected Subgraphs via Independence-Free Graphs
We present a 6-approximation algorithm for the minimum-cost -node
connected spanning subgraph problem, assuming that the number of nodes is at
least . We apply a combinatorial preprocessing, based on the
Frank-Tardos algorithm for -outconnectivity, to transform any input into an
instance such that the iterative rounding method gives a 2-approximation
guarantee. This is the first constant-factor approximation algorithm even in
the asymptotic setting of the problem, that is, the restriction to instances
where the number of nodes is lower bounded by a function of .Comment: 20 pages, 1 figure, 28 reference
On the maximum size of a minimal k-edge connected augmentation
AbstractWe present a short proof of a generalization of a result of Cheriyan and Thurimella: a simple graph of minimum degree k can be augmented to a k-edge connected simple graph by adding ⩽knk+1 edges, where n is the number of nodes. One application (from the previous paper) is an approximation algorithm with a guarantee of 1+2k+1 for the following NP-hard problem: given a simple undirected graph, find a minimum-size k-edge connected spanning subgraph. For the special cases of k=4,5,6, this is the best approximation guarantee known
Can a maximum flow be computed in o(nm) time?
We show that a maximum flow in a network with n vertices can be computed deterministically in O(n^{3}/logn) time on a uniform-cost RAM. For dense graphs, this improves the previous best bound of O(n^{3}). The bottleneck in our algorithm is a combinatorial problem on (unweighted) graphs. The number of operations executed on flow variables is O(n^{8/3}(log n)^{4/3}), in contrast with Omega(nm) flow operations for all previous algorithms, where m denotes the number of edges in the network. A randomized version of our algorithm executes O(n^{3/2}m^{1/2}(log n)^{3/2}+n^{2}(log n)^{2}) flow operations with high probability. Specializing to the case in which all capacities are integers bounded by U, we show that a maximum flow can be computed using O(n^{3/2}m^{1/2}+n^{2}(log U)^{1/2}) flow operations. Finally, we argue that several of our results yield optimal parallel algorithms
Approximating flexible graph connectivity via the primal-dual method
We consider the Flexible Graph Connectivity model (denoted FGC) introduced by
Adjiashvili, Hommelsheim and M\"uhlenthaler (IPCO 2020, Mathematical
Programming 2021), and its generalization, -FGC, where and are integers, introduced by Boyd et al.\ (FSTTCS 2021). In the
-FGC model, we have an undirected connected graph ,
non-negative costs on the edges, and a partition of into a set of safe edges and a set of unsafe
edges . A subset of edges is called feasible if
for any set with , the subgraph is -edge connected. The goal is to find a feasible edge-set
of minimum cost.
For the special case of -FGC when , we give an
approximation algorithm, thus improving on the logarithmic approximation ratio
of Boyd et al. (FSTTCS 2021). Our algorithm is based on the primal-dual method
for covering an uncrossable family, due to Williamson et al. (Combinatorica
1995). We conclude by studying weakly uncrossable families, which are a
generalization of the well-known notion of an uncrossable family
Approximation Algorithms for Flexible Graph Connectivity
We present approximation algorithms for several network design problems in
the model of Flexible Graph Connectivity (Adjiashvili, Hommelsheim and
M\"uhlenthaler, "Flexible Graph Connectivity", Math. Program. pp. 1-33 (2021),
and IPCO 2020: pp. 13-26).
Let , and be integers. In an instance of the
-Flexible Graph Connectivity problem, denoted -FGC, we have an
undirected connected graph , a partition of into a set of safe
edges and a set of unsafe edges , and nonnegative costs on
the edges. A subset of edges is feasible for the -FGC
problem if for any subset of unsafe edges with , the subgraph
is -edge connected. The algorithmic goal is to find a
feasible solution that minimizes . We present a
simple -approximation algorithm for the -FGC problem via a reduction
to the minimum-cost rooted -arborescence problem. This improves on the
-approximation algorithm of Adjiashvili et al. Our -approximation
algorithm for the -FGC problem extends to a -approximation
algorithm for the -FGC problem. We present a -approximation algorithm
for the -FGC problem, and an -approximation algorithm for
the -FGC problem. Finally, we improve on the result of Adjiashvili et
al. for the unweighted -FGC problem by presenting a
-approximation algorithm.
The -FGC problem is related to the well-known Capacitated
-Connected Subgraph problem (denoted Cap-k-ECSS) that arises in the area of
Capacitated Network Design. We give a -approximation
algorithm for the Cap-k-ECSS problem, where denotes the maximum
capacity of an edge.Comment: 23 pages, 1 figure, preliminary version in the Proceedings of the
41st IARCS Annual Conference on Foundations of Software Technology and
Theoretical Computer Science (FSTTCS 2021), December 15-17, (LIPIcs, Volume
213, Article No. 9, pp. 9:1-9:14), see
https://doi.org/10.4230/LIPIcs.FSTTCS.2021.9. Related manuscript:
arXiv:2102.0330
An Improved Approximation Algorithm for the Matching Augmentation Problem
We present a 5/3-approximation algorithm for the matching augmentation problem (MAP): given a multi-graph with edges of cost either zero or one such that the edges of cost zero form a matching, find a 2-edge connected spanning subgraph (2-ECSS) of minimum cost.
A 7/4-approximation algorithm for the same problem was presented recently, see Cheriyan, et al., "The matching augmentation problem: a 7/4-approximation algorithm," Math. Program., 182(1):315-354, 2020.
Our improvement is based on new algorithmic techniques, and some of these may lead to advances on related problems
Can a maximum flow be computed in o(nm) time?
We show that a maximum flow in a network with n vertices can be computed deterministically in O(n^{3}/logn) time on a uniform-cost RAM. For dense graphs, this improves the previous best bound of O(n^{3}). The bottleneck in our algorithm is a combinatorial problem on (unweighted) graphs. The number of operations executed on flow variables is O(n^{8/3}(log n)^{4/3}), in contrast with Omega(nm) flow operations for all previous algorithms, where m denotes the number of edges in the network. A randomized version of our algorithm executes O(n^{3/2}m^{1/2}(log n)^{3/2}+n^{2}(log n)^{2}) flow operations with high probability. Specializing to the case in which all capacities are integers bounded by U, we show that a maximum flow can be computed using O(n^{3/2}m^{1/2}+n^{2}(log U)^{1/2}) flow operations. Finally, we argue that several of our results yield optimal parallel algorithms
A 4/3-Approximation Algorithm for the Minimum 2-Edge Connected Multisubgraph Problem in the Half-Integral Case
Given a connected undirected graph on vertices, and
non-negative edge costs , the 2ECM problem is that of finding a
-edge~connected spanning multisubgraph of of minimum cost. The
natural linear program (LP) for 2ECM, which coincides with the subtour LP for
the Traveling Salesman Problem on the metric closure of , gives a
lower bound on the optimal cost. For instances where this LP is optimized by a
half-integral solution , Carr and Ravi (1998) showed that the integrality
gap is at most : they show that the vector dominates a
convex combination of incidence vectors of -edge connected spanning
multisubgraphs of .
We present a simpler proof of the result due to Carr and Ravi by applying an
extension of Lov\'{a}sz's splitting-off theorem. Our proof naturally leads to a
-approximation algorithm for half-integral instances. Given a
half-integral solution to the LP for 2ECM, we give an -time
algorithm to obtain a -edge connected spanning multisubgraph of
whose cost is at most
- …