Download presentation

Presentation is loading. Please wait.

Published byElijah Weedman Modified over 2 years ago

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
May 25, 2007CP-AI-OR 20072 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!

3
May 25, 2007CP-AI-OR 20073 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?

4
May 25, 2007CP-AI-OR 20074 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?

5
May 25, 2007CP-AI-OR 20075 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?

6
May 25, 2007CP-AI-OR 20076 Key parameter: ratio #constraints / #variables Easy for very low and very high ratios Hard in the intermediate region Complexity peaks at ratio ~ 4.26 Random 3-SAT Random 3-SAT: Easy-Hard-Easy Computational hardness as a function of a key problem parameter [Mitchell, Selman, and Levesque ’92; …]

7
May 25, 2007CP-AI-OR 20077 Coinciding Phase Transition Before critical ratio: almost all formulas satisfiable After critical ratio: almost all formulas unsatisfiable Very sharp transition! Random 3-SAT Phase transition From satisfiable to unsatisfiable

8
May 25, 2007CP-AI-OR 20078 Typical-Case Complexity Is a similar behavior observed in pure optimization problems? How about problems that combine feasibility and optimization components? Goal: Obtain further insights into the problem. Note: very few constraints, e.g., implies easy to solve but not necessarily easy to optimize!

9
May 25, 2007CP-AI-OR 20079 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 and social networks (more on this next) A generalized variant of the Steiner tree problem Combines feasibility and optimization components A budget constraint on vertex costs A utility function to be maximized

10
May 25, 2007CP-AI-OR 200710 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?

11
May 25, 2007CP-AI-OR 200711 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

12
May 25, 2007CP-AI-OR 200712 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

13
May 25, 2007CP-AI-OR 200713 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

14
May 25, 2007CP-AI-OR 200714 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

15
May 25, 2007CP-AI-OR 200715 Main Results Worst-case complexity of the connection subgraph problem: NP-hard even to approximate Typical-case complexity w.r.t. increasing budget fraction 1. Without terminals: pure optimization version, always feasible, still a computational easy-hard-easy pattern 2. With terminals: a) Phase transition: Problem turns from mostly infeasible to mostly feasible at budget fraction ~ 0.13 b) Computational easy-hard-easy pattern coinciding with the phase transition c) Surprisingly, proving optimality can be substantially easier than proving infeasibility in the phase transition region

16
May 25, 2007CP-AI-OR 200716 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?

17
May 25, 2007CP-AI-OR 200717 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

18
May 25, 2007CP-AI-OR 200718 v1v1 vnvn v2v2 v3v3 … … 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

19
May 25, 2007CP-AI-OR 200719 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?

20
May 25, 2007CP-AI-OR 200720 Experimental Setup Study parameter: budget fraction (budget as a fraction of the sum of all node costs) How are problem feasibility and hardness affected as the budget fraction is varied? Algorithm: CPLEX on a Mixed Integer Programming (MIP) model

21
May 25, 2007CP-AI-OR 200721 The MIP Model Variables: x i {0,1} for each vertex i (included or not) Cost constraint: i c i x i C Utility optimization function: maximize i u i x i Connectedness: use a network flow encoding

22
May 25, 2007CP-AI-OR 200722 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 y i,j Z + for each directed edge (i,j) (flow from i to j) Flow passes through i iff v i retains 1 unit of flow Each terminal t retains 1 unit of flow Conservation of flow constraints

23
May 25, 2007CP-AI-OR 200723 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 utilities from {0, 1, …, 10} m = 4 k = 4

24
May 25, 2007CP-AI-OR 200724 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 Budget fraction Runtime (logscale) 0 0.2 0.4 0.6 0.8 0.01 1 100 10000 6 x 6 8 x 8 10 x 10 A clear easy-hard-easy pattern with uniform random costs & utilities Note 1: plot in log-scale for better viewing of the sharp transitions Note 2: each data point is median over 100+ random instances

25
May 25, 2007CP-AI-OR 200725 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

26
May 25, 2007CP-AI-OR 200726 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!

27
May 25, 2007CP-AI-OR 200727 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)

28
May 25, 2007CP-AI-OR 200728 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)

29
May 25, 2007CP-AI-OR 200729 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

30
May 25, 2007CP-AI-OR 200730 Could Other Models / Solvers Significantly 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(n k+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.

31
May 25, 2007CP-AI-OR 200731 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

Similar presentations

OK

Balance and Filtering in Structured Satisfiability Problems Henry Kautz University of Washington joint work with Yongshao Ruan (UW), Dimitris Achlioptas.

Balance and Filtering in Structured Satisfiability Problems Henry Kautz University of Washington joint work with Yongshao Ruan (UW), Dimitris Achlioptas.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on turbo generators manufacturers Ppt on marketing mix product Ppt on mathematics programmed instruction Ppt on aerobics music Ppt on diffusion taking place in our daily life Convert word doc to ppt online form Short ppt on rainwater harvesting Ppt on samuel taylor coleridge Food and nutrition for kids ppt on batteries Ppt on resource allocation in software project management