connections in networks: hardness of feasibility vs. optimality
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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. - PowerPoint PPT PresentationTRANSCRIPT
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
May 25, 2007 CP-AI-OR 2007 2
Feasibility Testing & Optimization
Constraint satisfaction work often focuses on purefeasibility 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!
May 25, 2007 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?
May 25, 2007 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?
May 25, 2007 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?
May 25, 2007 CP-AI-OR 2007 6
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
Computationalhardness as a
function of a keyproblem parameter
[Mitchell, Selman, and Levesque ’92; …]
May 25, 2007 CP-AI-OR 2007 7
Coinciding Phase Transition
Before critical ratio: almost all formulas satisfiable After critical ratio: almost all formulas unsatisfiable Very sharp transition!
Random 3-SATPhase transition
From satisfiableto unsatisfiable
May 25, 2007 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?
Goal: Obtain further insights into the problem.
Note: very few constraints, e.g., implies easy to solvebut not necessarily easy to optimize!
May 25, 2007 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 and
social 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
May 25, 2007 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?
May 25, 2007 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
May 25, 2007 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
May 25, 2007 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
May 25, 2007 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
May 25, 2007 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 fraction1. Without terminals: pure optimization version, always feasible,
still a computational easy-hard-easy pattern
2. With terminals:o Phase transition: Problem turns from mostly infeasible to
mostly feasible at budget fraction ~ 0.13o Computational easy-hard-easy pattern coinciding with the
phase transitiono Surprisingly, proving optimality can be substantially easier
than proving infeasibility in the phase transition region
May 25, 2007 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?
May 25, 2007 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
May 25, 2007 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
May 25, 2007 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?
May 25, 2007 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
May 25, 2007 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
May 25, 2007 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
May 25, 2007 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 utilities
from {0, 1, …, 10}
m = 4 k = 4
May 25, 2007 CP-AI-OR 2007 24
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
Run
time
(logs
cale
)
0 0.2 0.4 0.6 0.8
0.01
1
10
0
10
000
6 x 6
8 x 8
10 x 10
A clear easy-hard-easypattern with uniform
random costs & utilities
Note 1: plot in log-scale for betterviewing of the sharp transitions
Note 2: each data point is medianover 100+ random instances
May 25, 2007 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
May 25, 2007 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!
May 25, 2007 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)
May 25, 2007 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)
May 25, 2007 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
May 25, 2007 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.
May 25, 2007 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