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Connections in Networks: Hardness of Feasibility vs. Optimality

Connections in Networks: Hardness of Feasibility vs. Optimality. Jon Conrad, Carla P. Gomes, Willem-Jan van Hoeve, Ashish Sabharwal , Jordan Suter Cornell University CP-AI-OR Conference, May 2007 Brussels, Belgium. Feasibility Testing & Optimization.

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Connections in Networks: Hardness of Feasibility vs. Optimality

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  1. Connections in Networks:Hardness of Feasibility vs. Optimality Jon Conrad, Carla P. Gomes,Willem-Jan van Hoeve, Ashish Sabharwal, Jordan Suter Cornell University CP-AI-OR Conference, May 2007 Brussels, Belgium

  2. Feasibility Testing & Optimization Constraint satisfaction work often focuses on pure feasibility testing: Is there a solution? Find me one! • In principle, can be used for optimization as well • Worst-case complexity classes well understood • Often finer-grained typical-case hardness also known(easy-hard-easy patterns, phase transitions) How does the picture change when problems combine both feasibility and optimization components? • We study this in the context of connection networks • Many positive results; some surprising ones! CP-AI-OR 2007

  3. Outline of the Talk • Worst-case vs. typical-case hardness • Easy-hard-easy patterns; phase transition • The Connection Subgraph Problem • Motivation: economics and social networks • Combining feasibility and optimality components • Theoretical results (NP-hardness of approximation) • Empirical study • Easy-hard-easy patterns for pure optimality • Phase transition • Feasibility testing vs. optimization: a clear winner? CP-AI-OR 2007

  4. Outline of the Talk • Worst-case vs. typical-case hardness • Easy-hard-easy patterns; phase transition • The Connection Subgraph Problem • Motivation: economics and social networks • Combining feasibility and optimality components • Theoretical results (NP-hardness of approximation) • Empirical study • Easy-hard-easy patterns for pure optimality • Phase transition • Feasibility testing vs. optimization: a clear winner? CP-AI-OR 2007

  5. Typical-Case Complexity E.g. consider SAT, the Boolean Satisfiability Problem: Does a given formula have a satisfying truth assignment? • Worst-case complexity: NP-complete • Unless P = NP, cannot solve all instances in poly-time • Of course, need solutions in practice anyway • Typical-case complexity: a more detailed picture • What about a majority of the instances? • How about instances w.r.t. certain interesting parameters?e.g. for SAT: clause-to-variable ratio. • Are some regimes easier than others? • Can such parameters characterize feasibility? CP-AI-OR 2007

  6. Random 3-SAT Random 3-SAT: Easy-Hard-Easy • Key parameter: ratio #constraints / #variables • Easy for very low and very high ratios • Hard in the intermediate region • Complexity peaks at ratio ~ 4.26 Computationalhardness as afunction of a keyproblem parameter [Mitchell, Selman, and Levesque ’92; …] CP-AI-OR 2007

  7. Phase transition Random 3-SAT Coinciding Phase Transition • Before critical ratio: almost all formulas satisfiable • After critical ratio: almost all formulas unsatisfiable • Very sharp transition! From satisfiableto unsatisfiable CP-AI-OR 2007

  8. Typical-Case Complexity Is a similar behavior observed in pure optimization problems? How about problems that combine feasibility and optimization components? Note: very few constraints, e.g., implies easy to solvebut not necessarily easy to optimize! Goal: Obtain further insights into the problem. CP-AI-OR 2007

  9. Typical-Case Complexity • Known: a few results for pure optimization problems • Traveling sales person (TSP) under specialized cost functions like log-normal [Gent,Walsh ’96; Zhang,Korf ’96] • We look at the connection subgraph problem • Motivated by resource environment economics andsocial networks (more on this next) • A generalized variant of the Steiner tree problem • Combines feasibility and optimization components A budget constrainton vertex costs A utility functionto be maximized CP-AI-OR 2007

  10. Outline of the Talk • Worst-case vs. typical-case hardness • Easy-hard-easy patterns; phase transition • The Connection Subgraph Problem • Motivation: economics and social networks • Combining feasibility and optimality components • Theoretical results (NP-hardness of approximation) • Empirical study • Easy-hard-easy patterns for pure optimality • Phase transition • Feasibility testing vs. optimization: a clear winner? CP-AI-OR 2007

  11. Connection Subgraph: Motivation Motivation 1: Resource environment economics • Conservation corridors (a.k.a. movement or wildlife corridors)[Simberloff et al. ’97; Ando et al. ’98; Camm et al. ’02] • Preserve wildlife against land fragmentation • Link zones of biological significance (“reserves”) by purchasing continuous protected land parcels • Limited budget; must maximize environmental benefits/utility Reserve Land parcel CP-AI-OR 2007

  12. Connection Subgraph: Motivation Real problem data: • Goal: preserve grizzly bear population in the U.S.A. by creating movement corridors • 3637 land parcels (6x6 miles) connecting 3 reserves in Wyoming, Montana, and Idaho • Reserves include, e.g., Yellowstone National Park • Budget: ~ $2B CP-AI-OR 2007

  13. Connection Subgraph: Motivation Motivation 2: Social networks • What characterizes the connection between two individuals? The shortest path? Size of the connected component? A “good” connected subgraph?[Faloutsos, McCurley, Tompkins ’04] • If a person is infected with a disease, who else is likely to be? • Which people have unexpected ties to any members of a list of other individuals? • Vertices in graph: people; edges: know each other or not CP-AI-OR 2007

  14. The Connection Subgraph Problem Given • An undirected graph G = (V,E) • Terminal vertices T V • Vertex cost function: c(v); utility function: u(v) • Cost bound / budget C; desired utility U Is there a subgraph H of G such that • H is connected • cost(H)  C; utility(H)  U ? Cost optimization version : given U, minimize cost Utility optimization version : given C, maximize utility CP-AI-OR 2007

  15. Main Results Worst-case complexity of the connection subgraph problem: NP-hard even to approximate Typical-case complexity w.r.t. increasing budget fraction • Without terminals: pure optimization version, always feasible, still a computational easy-hard-easy pattern • With terminals: • Phase transition: Problem turns from mostly infeasible to mostly feasible at budget fraction ~ 0.13 • Computational easy-hard-easy pattern coinciding with the phase transition • Surprisingly, proving optimality can be substantially easier than proving infeasibility in the phase transition region CP-AI-OR 2007

  16. Outline of the Talk • Worst-case vs. typical-case hardness • Easy-hard-easy patterns; phase transition • The Connection Subgraph Problem • Motivation: economics and social networks • Combining feasibility and optimality components • Theoretical results (NP-hardness of approximation) • Empirical study • Easy-hard-easy patterns for pure optimality • Phase transition • Feasibility testing vs. optimization: a clear winner? CP-AI-OR 2007

  17. Theoretical Results: 1 • NP-completeness: reduction from the Steiner Tree problem, preserving the cost function. Idea: • Steiner tree problem already very similar • Simulate edge costs with node costs • Simulate terminal vertices with utility function • NP-complete even without any terminals • Recall: Steiner tree problem poly-time solvable with constant number of terminals • Also holds for planar graphs CP-AI-OR 2007

  18. v1 vn … v2 … v3 Theoretical Results: 2 • NP-hardness of approximating cost optimization (factor 1.36): reduction from the Vertex Cover problem • Reduction motivated by Steiner tree work [Bern, Plassmann ’89] vertex cover of size k iff connection subgraph with cost bound C = k and utility U = m CP-AI-OR 2007

  19. Outline of the Talk • Worst-case vs. typical-case hardness • Easy-hard-easy patterns; phase transition • The Connection Subgraph Problem • Motivation: economics and social networks • Combining feasibility and optimality components • Theoretical results (NP-hardness of approximation) • Empirical study • Easy-hard-easy patterns for pure optimality • Phase transition • Feasibility testing vs. optimization: a clear winner? CP-AI-OR 2007

  20. Experimental Setup • Study parameter: budget fraction(budget as a fraction of the sum of all node costs) How are problem feasibility and hardness affectedas the budget fraction is varied? • Algorithm: CPLEX on a Mixed Integer Programming (MIP) model CP-AI-OR 2007

  21. The MIP Model • Variables: xi {0,1} for each vertex i (included or not) • Cost constraint: i cixi  C • Utility optimization function: maximize i uixi • Connectedness: use a network flow encoding CP-AI-OR 2007

  22. The MIP Model: Connectedness • New source vertex 0, connected to arbitrary terminal t (slightly different construction when no terminals) • Initial flow sent from 0 equals number of vertices • New variables yi,j Z+ for each directed edge (i,j) (flow from i to j) • Flow passes through i iff vi retains 1 unit of flow • Each terminal t retains 1 unit of flow • Conservation of flow constraints CP-AI-OR 2007

  23. Graphs for Evaluation Problem evaluated on semi-structured graphs • m x m lattice / grid graph with k terminals • Inspired by the conservation corridors problem • Place a terminal each on top-left and bottom-right • Maximizes grid use • Place remaining terminals randomly • Assign uniform random costs and utilitiesfrom {0, 1, …, 10} m = 4 k = 4 CP-AI-OR 2007

  24. 10 x 10 8 x 8 Runtime (logscale) 0.01 1 100 10000 6 x 6 0 0.2 0.4 0.6 0.8 Budget fraction Results: without terminals • No terminals  “find the connected component that maximizes the utility within the given budget” • Pure optimization problem; always feasible • Still NP-hard A clear easy-hard-easypattern with uniformrandom costs & utilities Note 1: plot in log-scale for betterviewing of the sharp transitions Note 2: each data point is medianover 100+ random instances CP-AI-OR 2007

  25. Results: with terminals • Easy-hard-easy pattern, peaking at budget fraction ~ 0.13 • Sharp phase transition near 0.13: from infeasible to feasible Note: not in log scale CP-AI-OR 2007

  26. Results: feasibility vs. optimization Split instances into feasible and infeasible; plot median runtime • For feasible ones : computation involves proving optimality • For infeasible ones: computation involves proving infeasibility Infeasible instances take much longer than the feasible ones! CP-AI-OR 2007

  27. With 10 Terminals • The results are even more striking. • Median times: • Hardest instances : 1,200 sec • Hardest feasible instances : 200 sec • Hardest infeasible instances : 30,000 sec (150x) CP-AI-OR 2007

  28. With 20 Terminals • The phenomena still clearly present • Instances a bit easier than for 10 terminals. Median times: • Hardest instances : 340 sec • Hardest feasible instances : 60 sec • Hardest infeasible instances : 7,000 sec (110x) CP-AI-OR 2007

  29. Other Observations • Peak for pure optimality component without terminals (~0.2) is slightly to the right of the peak for feasibility component (~0.13) • Easy-hard-easy pattern also w.r.t. number of terminals • 3 terminals: easy, 10: hard, 20 again easy • Intuitively, more terminals • ----- are harder to connect • +++ leave fewer choices for other vertices to include • Competing constraints  a hard intermediate region CP-AI-OR 2007

  30. Could Other Models / SolversSignificantly Change the Picture? Perhaps, although some other natural options appear unlikely to. • Within Cplex, first check for feasibility then apply optimization • Problem: checking feasibility of the cost constraint equivalent to the metric Steiner tree problem; solvable in O(nk+1), which grows quickly with #terminals. Also, unlikely to be Fixed Parameter Tractable (FPT)[cf. Promel, Steger ’02] • Constraint Prog. (CP) model more promising for feasibility? • Problem: appears promising only as a global constraint, but hard to filter efficiently (unlikely to be FPT); Also, weighted sum not easy to optimize with CP. CP-AI-OR 2007

  31. Summary • Combining feasibility and optimization components can result in intriguing typical-case properties • Connection subgraphs: • NP-hard to approximate • Clear easy-hard-easy patterns and phase transitions • Feasibility testing can be much harder than optimization CP-AI-OR 2007

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