Connections in Networks: Hardness of Feasibility Versus Optimality

Conrad, Jon M.; Gomes, Carla P.; Hoeve, Willem‐Jan van; Sabharwal, Ashish; Suter, Jordan F.

doi:10.1007/978-3-540-72397-4_2

Cited by 20 publications

(29 citation statements)

References 9 publications

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“…We test our formulations on real problem instances concerning the design of a Grizzly Bear Wildlife Corridor connecting three existing reserves [2]. We show that, for critically constrained budgets, the DFJ encoding proposed here can find optimal or close to optimal solutions, dramatically speeding up runtime.…”

Section: Introductionmentioning

confidence: 99%

“…This problem is known to be NP-hard even for the case of no terminals [2]. Removing the connectivity constraint, we have a 0-1 knapsack problem.…”

Section: Introductionmentioning

confidence: 99%

“…The connected subgraph problem in the context of designing wildlife corridors was recently studied in [2,7]. Conrad et al [2] designate one of the terminals as a root node and encode the connectivity constraints as a single commodity flow from the root to the selected nodes in the subgraph.…”

Section: Introductionmentioning

confidence: 99%

“…Conrad et al [2] designate one of the terminals as a root node and encode the connectivity constraints as a single commodity flow from the root to the selected nodes in the subgraph. This encoding is small and easy to enforce.…”

Section: Introductionmentioning

confidence: 99%

“…We provide computational results on the three different encodings of the connectivity constraints: 1) the single-commodity flow (SCF) encoding [2]; 2) a multi-commodity flow (MCF) encoding; 3) a modified directed Dantzig-FulkersonJohnson (DFJ) formulation using node costs. On a benchmark of synthetic instances consisting of grid graphs with random costs and revenues, we show that indeed the multi-commodity encoding provides better LP relaxation bounds than the single commodity flow, and that the directed Dantzig-Fulkerson-Johnson formulation provides the best bounds.…”

Section: Introductionmentioning

confidence: 99%

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Solving Connected Subgraph Problems in Wildlife Conservation

Dilkina

Gomes

2010

Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems

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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 budget-constrained 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 mixed-integer 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 real-world 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 real-world wildlife corridor instances.

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Section: Introductionmentioning

confidence: 99%

“…This problem is known to be NP-hard even for the case of no terminals [2]. Removing the connectivity constraint, we have a 0-1 knapsack problem.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Solving Connected Subgraph Problems in Wildlife Conservation

Dilkina

Gomes

2010

Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems

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Optimal connected subgraphs: Integer programming formulations and polyhedra

2022

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Connectivity is a central concept in combinatorial optimization, graph theory, and operations research. In many applications, one is interested in finding an optimal subset of vertices with the essential requirement that the vertices are connected, but not how they are connected. In other words, it is not relevant which edges are selected to obtain connectivity. This article is concerned with the exact solution of such problems via integer programming. We analyze and compare (mixed) integer programming formulations with respect to the strength of their linear programming relaxations. Along the way, we also provide a tighter (compact) description of the connected subgraph polytope-the convex hull of subsets of vertices that induce a connected subgraph. Furthermore, we give a (compact) complete description of the connected subgraph polytope for graphs with no four independent vertices.

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Vertex covering with capacitated trees

2022

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The covering of a graph with (possibly disjoint) connected subgraphs is a fundamental problem in graph theory. In this paper, we study a version to cover a graph's vertices by connected subgraphs subject to lower and upper weight bounds, and propose a column generation approach to dynamically generate feasible and promising subgraphs. Our focus is on the solution of the pricing problem which turns out to be a variant of the NP-hard Maximum Weight Connected Subgraph Problem. We compare different formulations to handle connectivity, and find that a single-commodity flow formulation performs best. This is notable since the respective literature seems to have widely dismissed this formulation. We improve it to a new coarse-to-fine flow formulation that is theoretically and computationally superior, especially for large instances with many vertices of degree 2 like highway networks, where it provides a speed-up factor of 5 over the non-flow-based formulations. We also propose a preprocessing method that exploits a median property of weight-constrained subgraphs, a primal heuristic, and a local search heuristic. In an extensive computational study we evaluate the presented connectivity formulations on different classes of instances, and demonstrate the effectiveness of the proposed enhancements. Their speed-ups essentially multiply to an overall factor of well over 10. Overall, our approach allows the reliable solution of instances with several hundreds of vertices in a few minutes. These findings are further corroborated in a comparison to existing districting models on a set of test instances from the literature.

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Connections in Networks: Hardness of Feasibility Versus Optimality

Cited by 20 publications

References 9 publications

Solving Connected Subgraph Problems in Wildlife Conservation

Solving Connected Subgraph Problems in Wildlife Conservation

Optimal connected subgraphs: Integer programming formulations and polyhedra

Vertex covering with capacitated trees

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