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22
Solving Connected Subgraph Problems in Wildlife Conservation
"... Abstract. We investigate mathematical formulations and solution techniques for a variant of the Connected Subgraph Problem. Given a connected graph with costs and profits associated with the nodes, the goal is to find a connected subgraph that contains a subset of distinguished vertices. In this wor ..."
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Abstract. We investigate mathematical formulations and solution techniques for a variant of the Connected Subgraph Problem. Given a connected graph with costs and profits associated with the nodes, the goal is to find a connected subgraph that contains a subset of distinguished vertices. In this work we focus on the budgetconstrained version, where we maximize the total profit of the nodes in the subgraph subject to a budget constraint on the total cost. We propose several mixedinteger formulations for enforcing the subgraph connectivity requirement, which plays a key role in the combinatorial structure of the problem. We show that a new formulation based on subtour elimination constraints is more effective at capturing the combinatorial structure of the problem, providing significant advantages over the previously considered encoding which was based on a single commodity flow. We test our formulations on synthetic instances as well as on realworld instances of an important problem in environmental conservation concerning the design of wildlife corridors. Our encoding results in a much tighter LP relaxation, and more importantly, it results in finding better integer feasible solutions as well as much better upper bounds on the objective (often proving optimality or within less than 1 % of optimality), both when considering the synthetic instances as well as the realworld wildlife corridor instances. 1
Prizecollecting Survivable Network Design in Nodeweighted Graphs
"... We consider nodeweighted network design problems, in particular the survivable network design problem (SNDP) and its prizecollecting version (PCSNDP). The input consists of a nodeweighted undirected graph G = (V, E) and integral connectivity requirements r(st) for each pair of nodes st. The go ..."
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We consider nodeweighted network design problems, in particular the survivable network design problem (SNDP) and its prizecollecting version (PCSNDP). The input consists of a nodeweighted undirected graph G = (V, E) and integral connectivity requirements r(st) for each pair of nodes st. The goal is to find a minimum nodeweighted subgraph H of G such that, for each pair st, H contains r(st) edgedisjoint paths between s and t. PCSNDP is a generalization in which the input also includes a penalty π(st) for each pair, and the goal is to find a subgraph H to minimize the sum of the weight of H and the sum of the penalties for all pairs whose connectivity requirements are not fully satisfied by H. Let k = maxst r(st) be the maximum requirement. There has been no nontrivial approximation for nodeweighted PCSNDP for k> 1, the main reason being the lack of an LP relaxation based approach for nodeweighted SNDP. In this paper we describe multirouteflow based relaxations for the two problems and obtain approximation algorithms for PCSNDP through them. The approximation ratios we obtain for PCSNDP are similar to those that were previously known for SNDP via combinatorial algorithms. Specifically, we obtain an O(k 2 log n)approximation in general graphs and an O(k 2)approximation in graphs that exclude a fixed minor. The approximation ratios can be improved by a factor of k but the running times of the algorithms depend polynomially on n k.
The Constrained Virtual Steiner Arborescence Problem: Formal Definition, SingleCommodity Integer Programming Formulation and Computational Evaluation
, 2013
"... Abstract. As the Internet becomes more virtualized and softwaredefined, new functionality is introduced in the network core: the distributed resources available in ISP central offices, universal nodes, or datacenter middleboxes can be used to process (e.g., filter, aggregate or duplicate) data. Bas ..."
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Abstract. As the Internet becomes more virtualized and softwaredefined, new functionality is introduced in the network core: the distributed resources available in ISP central offices, universal nodes, or datacenter middleboxes can be used to process (e.g., filter, aggregate or duplicate) data. Based on this new networking paradigm, we formulate the Constrained Virtual Steiner Arborescence Problem (CVSAP) which asks for optimal locations to perform innetwork processing, in order to jointly minimize processing costs and network traffic while respecting link and node capacities. We prove that CVSAP cannot be approximated (unless NP = P), and accordingly, develop the exact algorithm VirtuCast to compute optimal solutions to CVSAP. VirtuCast consists of: (1) a compact singlecommodity flow Integer Programming (IP) formulation; (2) a flow decomposition algorithm to reconstruct individual routes from the IP solution. The compactness of the IP formulation allows for computing lower bounds even on large instances quickly, speeding up the algorithm. We rigorously prove VirtuCast’s correctness. To complement our theoretical findings, we have implemented VirtuCast and present an extensive computational evaluation, showing that VirtuCast can solve realistically sized instances (close to) optimality. We show that VirtuCast significantly improves upon naive multicommodity formulations and also initiate the study of primal heuristics to generate feasible solutions during the branchandbound process. 1
The Steiner Multigraph Problem: Wildlife Corridor Design for Multiple Species
"... The conservation of wildlife corridors between existing habitat preserves is important for combating the effects of habitat loss and fragmentation facing species of concern. We introduce the Steiner Multigraph Problem to model the problem of minimumcost wildlife corridor design for multiple species ..."
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The conservation of wildlife corridors between existing habitat preserves is important for combating the effects of habitat loss and fragmentation facing species of concern. We introduce the Steiner Multigraph Problem to model the problem of minimumcost wildlife corridor design for multiple species with different landscape requirements. This problem can also model other analogous settings in wireless and social networks. As a generalization of Steiner forest, the goal is to find a minimumcost subgraph that connects multiple sets of terminals. In contrast to Steiner forest, each set of terminals can only be connected via a subset of the nodes. Generalizing Steiner forest in this way makes the problem NPhard even when restricted to two pairs of terminals. However, we show that if the node subsets have a nested structure, the problem admits a fixedparameter tractable algorithm in the number of terminals. We successfully test exact and heuristic solution approaches on a wildlife corridor instance for wolverines and lynx in western Montana, showing that though the problem is computationally hard, heuristics perform well, and provably optimal solutions can still be obtained. 1
Nodeweighted Network Design in Planar and Minorclosed Families of Graphs
"... We consider nodeweighted network design in planar and minorclosed families of graphs. In particular we focus on the edgeconnectivity survivable network design problem (ECSNDP). The input consists of a nodeweighted undirected graph G = (V, E) and integral connectivity requirements r(uv) for ea ..."
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We consider nodeweighted network design in planar and minorclosed families of graphs. In particular we focus on the edgeconnectivity survivable network design problem (ECSNDP). The input consists of a nodeweighted undirected graph G = (V, E) and integral connectivity requirements r(uv) for each pair of nodes uv. The goal is to find a minimum nodeweighted subgraph H of G such that, for each pair uv, H contains r(uv) edgedisjoint paths between u and v. Our main result is an O(k)approximation algorithm for ECSNDP where k = maxuv r(uv) is the maximum requirement. This improves the O(k log n)approximation known for nodeweighted ECSNDP in general graphs [15]. Our algorithm and analysis applies to the more general problem of covering a proper function with maximum requirement k. Our result is inspired by, and generalizes, the work of Demaine, Hajiaghayi and Klein [5] who gave constant factor approximation algorithms for nodeweighted Steiner tree and Steiner forest problems (and more generally covering 01 proper functions) in planar and minorclosed families of graphs.
Parameterized complexity of directed steiner tree on sparse graphs
, 2012
"... Abstract. We study the parameterized complexity of the directed variant of the classical Steiner Tree problem on various classes of directed sparse graphs. While the parameterized complexity of Steiner Tree parameterized by the number of terminals is well understood, not much is known about the para ..."
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Abstract. We study the parameterized complexity of the directed variant of the classical Steiner Tree problem on various classes of directed sparse graphs. While the parameterized complexity of Steiner Tree parameterized by the number of terminals is well understood, not much is known about the parameterization by the number of nonterminals in the solution tree. All that is known for this parameterization is that both the directed and the undirected versions are W[2]hard on general graphs, and hence unlikely to be fixed parameter tractable (FPT). The undirected Steiner Tree problem becomes FPT when restricted to sparse classes of graphs such as planar graphs, but the techniques used to show this result break down on directed planar graphs. In this article we precisely chart the tractability border for Directed Steiner Tree (DST) on sparse graphs parameterized by the number of nonterminals in the solution tree. Specifically, we show that the problem is fixed parameter tractable on graphs excluding a topological minor, but becomes W[2]hard on graphs of degeneracy 2. On the other hand we show that if the subgraph induced by the terminals is required to be acyclic then the problem becomes FPT on graphs of bounded degeneracy. We further show that our algorithm achieves the best possible running time dependence on the solution size and degeneracy of the input graph, under standard complexity theoretic assumptions. Using the ideas developed for DST, we also obtain improved algorithms for Dominating Set on sparse undirected graphs. These algorithms are asymptotically optimal. 1
Approximation Algorithms and Hardness Results for Packing ElementDisjoint Steiner Trees
, 2008
"... We study the problem of packing elementdisjoint Steiner trees in graphs. We are given a graph and a designated subset of terminal nodes, and the goal is to find a maximum cardinality set of elementdisjoint trees such that each tree contains every terminal node. An element means a nonterminal node ..."
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We study the problem of packing elementdisjoint Steiner trees in graphs. We are given a graph and a designated subset of terminal nodes, and the goal is to find a maximum cardinality set of elementdisjoint trees such that each tree contains every terminal node. An element means a nonterminal node or an edge. (Thus, each nonterminal node and each edge must be in at most one of the trees.) We show that the problem is APXhard when there are only three terminal nodes, thus answering an open question. Our main focus is on the special case when the graph is planar. We show that the problem of finding two elementdisjoint Steiner trees in a planar graph is NPhard. Similarly, the problem of finding two edgedisjoint Steiner trees in a planar graph is NPhard. We design an algorithm for planar graphs that achieves an approximation guarantee close to 2. In fact, given a planar graph that is k elementconnected on the terminals (k is an upper bound on the number of elementdisjoint Steiner trees), the algorithm returns ⌊ ⌋ k 2 − 1 elementdisjoint Steiner trees. Using this algorithm, we get an approximation algorithm for the edgedisjoint version of the problem on planar graphs that improves on the previous approximation guarantees. We also show that the natural LP relaxation of the planar problem has an integrality ratio approaching 2.
VirtuCast: Multicast and Aggregation with InNetwork Processing An Exact SingleCommodity Algorithm
"... Abstract. As the Internet becomes more virtualized and softwaredefined, new functionality is introduced in the network core: the distributed resources available in ISP central offices, universal nodes, or datacenter middleboxes can be used to process (e.g., filter, aggregate or duplicate) data. Bas ..."
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Abstract. As the Internet becomes more virtualized and softwaredefined, new functionality is introduced in the network core: the distributed resources available in ISP central offices, universal nodes, or datacenter middleboxes can be used to process (e.g., filter, aggregate or duplicate) data. Based on this new networking paradigm, we formulate the Constrained Virtual Steiner Arborescence Problem (CVSAP) which asks for optimal locations to perform innetwork processing, in order to jointly minimize processing costs and network traffic while respecting link and node capacities. We prove that CVSAP cannot be approximated (unless NP ⊆ P), and accordingly, develop the exact algorithm VirtuCast to compute optimal solutions to CVSAP. VirtuCast consists of: (1) a compact singlecommodity flow Integer Programming (IP) formulation; (2) a flow decomposition algorithm to reconstruct individual routes from the IP solution. The compactness of the IP formulation allows for computing lower bounds even on large instances quickly, speeding up the algorithm significantly. We rigorously prove VirtuCast’s correctness and show its applicability to solve realistically sized instances close to optimality.
Tight bounds for Planar Strongly Connected Steiner Subgraph with fixed number of terminals (and extensions
 In SODA
, 2014
"... Given a vertexweighted directed graph G = (V,E) and a set T = {t1, t2,... tk} of k terminals, the objective of the STRONGLY CONNECTED STEINER SUBGRAPH (SCSS) problem is to find a vertex set H ⊆V of minimum weight such that G[H] contains a ti → t j path for each i 6 = j. The problem is NPhard, but ..."
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Given a vertexweighted directed graph G = (V,E) and a set T = {t1, t2,... tk} of k terminals, the objective of the STRONGLY CONNECTED STEINER SUBGRAPH (SCSS) problem is to find a vertex set H ⊆V of minimum weight such that G[H] contains a ti → t j path for each i 6 = j. The problem is NPhard, but Feldman and Ruhl (FOCS ’99; SICOMP ’06) gave a novel nO(k) algorithm for the SCSS problem, where n is the number of vertices in the graph and k is the number of terminals. We explore how much easier the problem becomes on planar directed graphs. • Our main algorithmic result is a 2O(k logk) ·nO( k) algorithm for planar SCSS, which is an improvement of a factor of O( k) in the exponent over the algorithm of Feldman and Ruhl. • Our main hardness result is a matching lower bound for our algorithm: we show that planar SCSS does not have an f (k) · no( k) algorithm for any computable function f, unless the Exponential Time Hypothesis (ETH) fails. The algorithm eventually relies on the excluded grid theorem for planar graphs, but we stress that it is not simply a straightforward application of treewidthbased techniques: we need several layers of abstraction to arrive to a problem formulation where the speedup due to planarity can be exploited. To obtain the lower bound matching the algorithm, we need a delicate construction of gadgets arranged in a gridlike fashion to tightly control the number of terminals in the created instance.
Approximation Algorithms for NETWORK DESIGN AND ORIENTEERING
, 2010
"... This thesis presents approximation algorithms for some N PHard combinatorial optimization problems on graphs and networks; in particular, we study problems related to Network Design. Under the widelybelieved complexitytheoretic assumption that P ̸ = N P, there are no efficient (i.e., polynomialt ..."
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This thesis presents approximation algorithms for some N PHard combinatorial optimization problems on graphs and networks; in particular, we study problems related to Network Design. Under the widelybelieved complexitytheoretic assumption that P ̸ = N P, there are no efficient (i.e., polynomialtime) algorithms that solve these problems exactly. Hence, if one desires efficient algorithms for such problems, it is necessary to consider approximate solutions: An approximation algorithm for an N PHard problem is a polynomial time algorithm which, for any instance of the problem, finds a solution whose value is guaranteed to be within a multiplicative factor ρ of the value of an optimal solution to that instance. We attempt to design algorithms for which this factor ρ, referred to as the approximation ratio of the algorithm, is as small as possible. The field of Network Design comprises a large class of problems that deal with constructing networks of low cost and/or high capacity, routing data through existing networks, and many related issues. In this thesis, we focus chiefly on designing faulttolerant networks. Two vertices u, v in a network are said to be kedgeconnected if deleting any set of k − 1 edges leaves u and v connected; similarly, they are kvertex connected if deleting any set of k − 1 other vertices or edges leaves u and v connected. We focus on building networks that are highly connected, meaning