We present approximation algorithms for network design problems in some models related to the (p,... more We present approximation algorithms for network design problems in some models related to the (p, q)-FGC model. Adjiashvili, Hommelsheim and Mühlenthaler [2, 1] introduced the model of Flexible Graph Connectivity that we denote by FGC. Boyd, Cheriyan, Haddadan and Ibrahimpur [4] introduced a generalization of FGC. Let p ≥ 1 and q ≥ 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 S and a set of unsafe edges U, and nonnegative costs c ∈ R E ≥0 on the edges. A subset F ⊆ E of edges is feasible for the (p, q)-FGC problem if for any set F ′ ⊆ U with |F ′ | ≤ q, the subgraph (V, F \ F ′) is p-edge connected. The algorithmic goal is to find a feasible edge-set F that minimizes c(F) = e∈F c e. We introduce a generalization of the (p, q)-FGC model, called ({0, 1,. .. , p}, {0,. .. , q})-FGC, where a required level of edge-connectivity p ij ∈ {0,. .. , p} and a fault-tolerance level q ij ∈ {0,. .. , q} is specified for every pair of nodes {i, j}. The goal is to find a subgraph H = (V, F) of minimum cost such that for any pair of nodes {i, j} and any set of at most q ij unsafe edges F ′ ⊆ F , the graph H − F ′ has p ij edgedisjoint (i, j)-paths. Assuming that p = 1 or q = 1 (i.e., when p ij ∈ {0, 1} or when q ij ∈ {0, 1}), we present max(2(p + 1), 2(q + 1))-approximation algorithms for this model by reductions to the capacitated network design problem (Cap-NDP); we apply Jain's iterative rounding method [10] to solve the Cap-NDP instances. We also consider the Flexible Steiner Tree model, denoted FST. We present a straightforward approximation algorithm for FST that achieves approximation ratio ≈ 2.9 via recent results of Ravi, Zhang & Zlatin [14]. Finally, we introduce the NC-FGC model. In an instance of this problem, we have an undirected connected graph G = (V, E), a partition of V into a set of safe nodes V S and a set of unsafe nodes V U , and non-negative costs c ∈ R E ≥0 on the edges; moreover, for every pair of nodes {s, t}, a required level of connectivity r st ∈ Z ≥0 is specified. The goal is to find a subgraph H = (V, F) of minimum cost such that for any pair of nodes {s, t}, and any set of unsafe nodesÛ ⊆ V U − {s, t}, the graph H −Û has min(0, r st − |Û |) edge-disjoint (s, t)-paths. For the (uniform connectivity) p-NC-FGC model, assuming that there is at least one safe node, we show that there is a 2-approximation algorithm via a result of Frank [9, Theorem 4.4].
We address long-standing open questions raised by Williamson, Goemans, Vazirani and Mihail pertai... more 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.. . 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 an O(1)-approximation guarantee via the primal-dual method for a class of functions that generalizes the notion of an uncrossable function. We mention that the support of every optimal dual solution could be non-laminar for instances that can be handled by our methods. We present two applications of our main result: 1. An O(1)-approximation algorithm for augmenting the family of near-minimum cuts of a graph. 2. An O(1)-approximation algorithm for the model of (p, 2)-Flexible Graph Connectivity.
Our motivation is to improve on the best approximation guarantee known for the problem of finding... more 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.
The k-Steiner-2NCS problem is as follows: Given a constant k, and an undirected connected graph G... more 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 the edges, 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 min-cost two-node connected subgraph that contains the terminals. We present a randomized polynomial-time algorithm for the unweighted problem, and a randomized FPTAS 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 min-cost two-edge connected subgraph that contains the terminals. Our methods build on results by Björklund, Husfeldt, and Taslaman (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 min-cost simple cycle C that contains the terminals (C may contain any number of Steiner nodes).
We prove that for k ∈ N and d ≤ 2k + 2, if a graph has maximum average degree at most 2k + 2d d+k... more We prove that for k ∈ N and d ≤ 2k + 2, if a graph has maximum average degree at most 2k + 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.
First and foremost, I would like to thank Joseph Cheriyan, who has been my supervisor, collaborat... more First and foremost, I would like to thank Joseph Cheriyan, who has been my supervisor, collaborator, and mentor even before I was his Master's student. This thesis would not exist without his guidance, feedback, and numerous discussions. Next, I thank my readers, Jochen Könemann and Chaitanya Swamy, for their comments, insights, and time. I would also like to thank Professors David Adjiashvili from ETH Zurich and Zeev Nutov from The Open University of Israel for giving me permission to use figures from their works in this thesis. Finally, I want to thank all of the people who gave me emotional support and friendship throughout my academic career. In particular, I want to thank Nishad Kothari for encouraging me to do Undergraduate research, and all of friends in the Party Office and the PMC. v
arXiv: Data Structures and Algorithms, Nov 15, 2021
Our motivation is to improve on the best approximation guarantee known for the problem of finding... more 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.
We prove that for $k \in \mathbb{N}$ and $d \leq 2k+2$, if a graph has maximum average degree at ... more 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.
We present a factor 4/3 approximation algorithm for the problem of finding a minimum 2-edge conne... more We present a factor 4/3 approximation algorithm for the problem of finding a minimum 2-edge connected spanning subgraph of a given undirected multigraph. The algorithm is based upon a reduction to a restricted class of graphs. In these graphs, the approximation algorithm constructs a 2-edge connected spanning subgraph by modifying the smallest 2-edge cover.
We prove that for any positive integers k and d, if a graph G has maximum average degree at most ... more We prove that for any positive integers k and d, if a graph G has maximum average degree at most 2k + 2d d+k+1 , then G decomposes into k + 1 pseudoforests C1,. .. , C k+1 such that there is an i such that for every connected component C of Ci, we have that e(C) ≤ d.
We present approximation algorithms for network design problems in some models related to the (p,... more We present approximation algorithms for network design problems in some models related to the (p, q)-FGC model. Adjiashvili, Hommelsheim and Mühlenthaler [2, 1] introduced the model of Flexible Graph Connectivity that we denote by FGC. Boyd, Cheriyan, Haddadan and Ibrahimpur [4] introduced a generalization of FGC. Let p ≥ 1 and q ≥ 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 S and a set of unsafe edges U, and nonnegative costs c ∈ R E ≥0 on the edges. A subset F ⊆ E of edges is feasible for the (p, q)-FGC problem if for any set F ′ ⊆ U with |F ′ | ≤ q, the subgraph (V, F \ F ′) is p-edge connected. The algorithmic goal is to find a feasible edge-set F that minimizes c(F) = e∈F c e. We introduce a generalization of the (p, q)-FGC model, called ({0, 1,. .. , p}, {0,. .. , q})-FGC, where a required level of edge-connectivity p ij ∈ {0,. .. , p} and a fault-tolerance level q ij ∈ {0,. .. , q} is specified for every pair of nodes {i, j}. The goal is to find a subgraph H = (V, F) of minimum cost such that for any pair of nodes {i, j} and any set of at most q ij unsafe edges F ′ ⊆ F , the graph H − F ′ has p ij edgedisjoint (i, j)-paths. Assuming that p = 1 or q = 1 (i.e., when p ij ∈ {0, 1} or when q ij ∈ {0, 1}), we present max(2(p + 1), 2(q + 1))-approximation algorithms for this model by reductions to the capacitated network design problem (Cap-NDP); we apply Jain's iterative rounding method [10] to solve the Cap-NDP instances. We also consider the Flexible Steiner Tree model, denoted FST. We present a straightforward approximation algorithm for FST that achieves approximation ratio ≈ 2.9 via recent results of Ravi, Zhang & Zlatin [14]. Finally, we introduce the NC-FGC model. In an instance of this problem, we have an undirected connected graph G = (V, E), a partition of V into a set of safe nodes V S and a set of unsafe nodes V U , and non-negative costs c ∈ R E ≥0 on the edges; moreover, for every pair of nodes {s, t}, a required level of connectivity r st ∈ Z ≥0 is specified. The goal is to find a subgraph H = (V, F) of minimum cost such that for any pair of nodes {s, t}, and any set of unsafe nodesÛ ⊆ V U − {s, t}, the graph H −Û has min(0, r st − |Û |) edge-disjoint (s, t)-paths. For the (uniform connectivity) p-NC-FGC model, assuming that there is at least one safe node, we show that there is a 2-approximation algorithm via a result of Frank [9, Theorem 4.4].
We address long-standing open questions raised by Williamson, Goemans, Vazirani and Mihail pertai... more 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.. . 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 an O(1)-approximation guarantee via the primal-dual method for a class of functions that generalizes the notion of an uncrossable function. We mention that the support of every optimal dual solution could be non-laminar for instances that can be handled by our methods. We present two applications of our main result: 1. An O(1)-approximation algorithm for augmenting the family of near-minimum cuts of a graph. 2. An O(1)-approximation algorithm for the model of (p, 2)-Flexible Graph Connectivity.
Our motivation is to improve on the best approximation guarantee known for the problem of finding... more 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.
The k-Steiner-2NCS problem is as follows: Given a constant k, and an undirected connected graph G... more 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 the edges, 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 min-cost two-node connected subgraph that contains the terminals. We present a randomized polynomial-time algorithm for the unweighted problem, and a randomized FPTAS 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 min-cost two-edge connected subgraph that contains the terminals. Our methods build on results by Björklund, Husfeldt, and Taslaman (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 min-cost simple cycle C that contains the terminals (C may contain any number of Steiner nodes).
We prove that for k ∈ N and d ≤ 2k + 2, if a graph has maximum average degree at most 2k + 2d d+k... more We prove that for k ∈ N and d ≤ 2k + 2, if a graph has maximum average degree at most 2k + 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.
First and foremost, I would like to thank Joseph Cheriyan, who has been my supervisor, collaborat... more First and foremost, I would like to thank Joseph Cheriyan, who has been my supervisor, collaborator, and mentor even before I was his Master's student. This thesis would not exist without his guidance, feedback, and numerous discussions. Next, I thank my readers, Jochen Könemann and Chaitanya Swamy, for their comments, insights, and time. I would also like to thank Professors David Adjiashvili from ETH Zurich and Zeev Nutov from The Open University of Israel for giving me permission to use figures from their works in this thesis. Finally, I want to thank all of the people who gave me emotional support and friendship throughout my academic career. In particular, I want to thank Nishad Kothari for encouraging me to do Undergraduate research, and all of friends in the Party Office and the PMC. v
arXiv: Data Structures and Algorithms, Nov 15, 2021
Our motivation is to improve on the best approximation guarantee known for the problem of finding... more 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.
We prove that for $k \in \mathbb{N}$ and $d \leq 2k+2$, if a graph has maximum average degree at ... more 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.
We present a factor 4/3 approximation algorithm for the problem of finding a minimum 2-edge conne... more We present a factor 4/3 approximation algorithm for the problem of finding a minimum 2-edge connected spanning subgraph of a given undirected multigraph. The algorithm is based upon a reduction to a restricted class of graphs. In these graphs, the approximation algorithm constructs a 2-edge connected spanning subgraph by modifying the smallest 2-edge cover.
We prove that for any positive integers k and d, if a graph G has maximum average degree at most ... more We prove that for any positive integers k and d, if a graph G has maximum average degree at most 2k + 2d d+k+1 , then G decomposes into k + 1 pseudoforests C1,. .. , C k+1 such that there is an i such that for every connected component C of Ci, we have that e(C) ≤ d.
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