arXiv subjects

Logan Grout

Publications and source records attributed to Logan Grout.

6 recordsLinked to original sources

arXiv2022

Extensions of the $(p,q)$-Flexible-Graph-Connectivity model

We present approximation algorithms for network design problems in some models related to the $(p,q)$-FGC model. Adjiashvili, Hommelsheim and M\"uhlenthaler introduced the model of Flexible Graph Connectivity that we denote by FGC. Boyd, Cheriyan, Haddadan and Ibrahimpur introduced a generalization of FGC. Let $p\geq 1$ and $q\geq 0$ be integers. In an instance of the $(p,q)$-Flexible Graph Connectivity problem, denoted $(p,q)$-FGC, we have an undirected connected graph $G = (V,E)$, a partition of $E$ into a set of safe edges and a set of unsafe edges, and nonnegative costs $c\in\mathbb{R}_{\geq0}^E$ on the edges. A subset $F \subseteq E$ of edges is feasible for the $(p,q)$-FGC problem if for any set of unsafe edges, $F'$, with $|F'|\leq q$, the subgraph $(V, F \setminus F')$ is $p$-edge connected. The algorithmic goal is to find a feasible edge-set $F$ that minimizes $c(F) = \sum_{e \in F} c_e$.

cs.DS↗

arXiv2022

Improved Approximation Algorithms by Generalizing the Primal-Dual Method Beyond Uncrossable Functions

We address long-standing open questions raised by Williamson, Goemans, Vazirani and Mihail pertaining to the design of approximation algorithms for problems in network design via the primal-dual method (Combinatorica 15(3):435-454, 1995). Williamson et al. prove an approximation guarantee of two for connectivity augmentation problems where the connectivity requirements can be specified by so-called uncrossable functions. They state: ``Extending our algorithm to handle non-uncrossable functions remains a challenging open problem. The key feature of uncrossable functions is that there exists an optimal dual solution which is laminar. This property characterizes uncrossable functions\dots\ A larger open issue is to explore further the power of the primal-dual approach for obtaining approximation algorithms for other combinatorial optimization problems.'' Our main result proves that the primal-dual algorithm of Williamson et al. achieves an approximation ratio of 16 for a class of functions that generalizes the notion of an uncrossable function. There exist instances that can be handled by our methods where none of the optimal dual solutions has a laminar support. We present three applications of our main result. (1) A 16-approximation algorithm for augmenting a family of small cuts of a graph $G$. (2) A $16 \cdot {\lceil k/u_{min} \rceil}$-approximation algorithm for the Cap-$k$-ECSS problem which is as follows: Given an undirected graph $G = (V,E)$ with edge costs $c \in \mathbb{Q}_{\geq 0}^E$ and edge capacities $u \in \mathbb{Z}_{\geq 0}^E$, find a minimum-cost subset of the edges $F\subseteq E$ such that the capacity of any cut in $(V,F)$ is at least $k$; we use $u_{min}$ to denote the minimum capacity of an edge in $E$. (3) An $O(1)$-approximation algorithm for the model of $(p,2)$-Flexible Graph Connectivity.

cs.DS↗

arXiv2022

Algorithms for 2-connected network design and flexible Steiner trees with a constant number of terminals

The $k$-Steiner-2NCS problem is as follows: Given a constant $k$, and an undirected connected graph $G = (V,E)$, non-negative costs $c$ on $E$, and a partition $(T, V-T)$ of $V$ into a set of terminals, $T$, and a set of non-terminals (or, Steiner nodes), where $|T|=k$, find a minimum-cost two-node connected subgraph that contains the terminals. We present a randomized polynomial-time algorithm for the unweighted problem, and a randomized PTAS for the weighted problem. We obtain similar results for the $k$-Steiner-2ECS problem, where the input is the same, and the algorithmic goal is to find a minimum-cost two-edge connected subgraph that contains the terminals. Our methods build on results by Bj\"orklund, Husfeldt, and Taslaman (ACM-SIAM SODA 2012) that give a randomized polynomial-time algorithm for the unweighted $k$-Steiner-cycle problem; this problem has the same inputs as the unweighted $k$-Steiner-2NCS problem, and the algorithmic goal is to find a minimum-size simple cycle $C$ that contains the terminals ($C$ may contain any number of Steiner nodes).

cs.DS↗

arXiv2021

On a Partition LP Relaxation for Min-Cost 2-Node Connected Spanning Subgraphs

Our motivation is to improve on the best approximation guarantee known for the problem of finding a minimum-cost 2-node connected spanning subgraph of a given undirected graph with nonnegative edge costs. We present an LP (Linear Programming) relaxation based on partition constraints. The special case where the input contains a spanning tree of zero cost is called 2NC-TAP. We present a greedy algorithm for 2NC-TAP, and we analyze it via dual-fitting for our partition LP relaxation. Keywords: 2-node connected graphs, approximation algorithms, connectivity augmentation, greedy algorithm, network design, partition relaxation

cs.DS↗

arXiv2020

The Pseudoforest analogue for the Strong Nine Dragon Tree Conjecture is True

We prove that for any positive integers $k$ and $d$, if a graph $G$ has maximum average degree at most $2k + \frac{2d}{d+k+1}$, then $G$ decomposes into $k+1$ pseudoforests $C_{1},\ldots,C_{k+1}$ such that there is an $i$ such that for every connected component $C$ of $C_{i}$, we have that $e(C) \leq d$.

math.CO↗

arXiv2019

On Decomposing Graphs Into Forests and Pseudoforests

We prove that for $k \in \mathbb{N}$ and $d \leq 2k+2$, if a graph has maximum average degree at most $2k + \frac{2d}{d+k+1}$, then $G$ decomposes into $k+1$ pseudoforests, where one of the pseudoforests has all connected components having at most $d$ edges.

math.CO↗