Impact of Structure on Complexity Carla Gomes gomescs

Impact of Structure on Complexity Carla Gomes gomes@cs. cornell. edu Bart Selman selman@cs. cornell. edu Cornell University Intelligent Information Systems Institute Kickoff Meeting AFOSR MURI May 2001

Outline • I - Overview of our approach • II - Structure vs. complexity – results on a abstract domain • III - Examples of Application Domains • IV - Conclusions

Overview of Approach • Overall theme --- exploit impact of structure on computational complexity – Identification of domain structural features • • • tractable vs. intractable subclasses phase transition phenomena backbone balancedness … – Goal: • Use findings in both the design and operation of distributed platform • Principled controlled hardness aware systems

Part I Structure vs. Complexity

Quasigroup Completion Problem (QCP) Given a matrix with a partial assignment of colors (32%colors in this case), can it be completed so that each color occurs exactly once in each row / column (latin square or quasigroup)? Example: 32% preassignment

Structural features of instances provide insights into their hardness namely: – Phase transition phenomena – Backbone – Inherent structure and balance

Are all the Quasigroup Instances (of same size) Equally Difficult? Time performance: 150 1820 165 What is the fundamental difference between instances?

Are all the Quasigroup Instances Equally Difficult? Time performance: 150 Fraction of preassignment: 35% 1820 165 40% 50%

Median Runtime (log scale) Complexity of Quasigroup Completion Critically constrained area Underconstrained area 20% Overconstrained area 42% 50% Fraction of pre-assignment

Complexity Graph Phase Transition Fraction of unsolvable cases Phase transition from almost all solvable to almost all unsolvable Almost all solvable area Almost all unsolvable area Fraction of pre-assignment

Quasigroup Patterns and Problems Hardness is also controlled by structure of constraints, not just percentage of holes Rectangular Pattern Aligned Pattern Tractable Balanced Pattern Very hard

Bandwidth: permute rows and columns of QCP to minimize the width of the diagonal band that covers all the holes. Fact: can solve QCP in time exponential in bandwidth swap

Random vs Balanced Random Balanced

After Permuting Random bandwidth = 2 Balanced bandwidth = 4

Structure vs. Computational Cost Computational cost Balanced QCP Aligned/ Rectangular QCP % of holes Balancing makes the instances very hard - it increases bandwith!

Backbone is the shared structure of all the solutions to a given instance. This instance has 4 solutions: Backbone Total number of backbone variables: 2

Phase Transition in the Backbone (only satisfiable instances) • We have observed a transition in the backbone from a phase where the size of the backbone is around 0% to a phase with backbone of size close to 100%. • The phase transition in the backbone is sudden and it coincides with the hardest problem instances. (Achlioptas, Gomes, Kautz, Selman 00, Monasson et al. 99)

New Phase Transition in Backbone % of Backbone % Backbone Sudden phase transition in Backbone Computational cost Fraction of preassigned cells

Why correlation between backbone and problem hardness? • Small backbone is associated with lots of solutions, widely distributed in the search space, therefore it is easy for the algorithm to find a solution; • Backbone close to 1 - the solutions are tightly clustered, all the constraints “vote” to push the search into that direction; • Partial Backbone - may be an indication that solutions are in different clusters that are widely distributed, with different clauses pushing the search in different directions.

Structural Features The understanding of the structural properties that characterize problem instances such as phase transitions, backbone, balance, and bandwith provides new insights into the practical complexity of computational tasks.

Examples of Application Domains

Fiber Optic Networks • Wavelength Division Multiplexing (WDM) is the most promising technology for the next generation of wide-area backbone networks. • WDM networks use the large bandwidth available in optical fibers by partitioning it into several channels, each at a different wavelength.

Fiber Optic Networks Nodes connect point to point fiber optic links

Fiber Optic Networks Nodes connect point to point fiber optic links Each fiber optic link supports a large number of wavelengths Nodes are capable of photonic switching --dynamic wavelength routing -which involves the setting of the wavelengths.

Routing in Fiber Optic Networks preassigned channels Input Ports 1 Output Ports 1 2 2 3 3 4 4 Routing Node How can we achieve conflict-free routing in each node of the network? Dynamic wavelength routing is a NP-hard problem.

QCP Example Use: Routers in Fiber Optic Networks Dynamic wavelength routing in Fiber Optic Networks can be directly mapped into the Quasigroup Completion Problem. • each channel cannot be repeated in the same input port (row constraints); • each channel cannot be repeated in the same output port (column constraints); 1 2 3 4 Output ports Output Port 1 2 3 4 Input ports Input Port CONFLICT FREE LATIN ROUTER (Barry and Humblet 93, Cheung et al. 90, Green 92, Kumar et al. 99)

IISI, Cornell University ANTs Challenge Problem • Multiple doppler radar sensors track moving targets • Energy limited sensors • Communication constraints • Distributed environment • Dynamic problem

IISI, Cornell University Domain Models • Start with a simple graph model • Successively refine the model in stages to approximate the real situation: – Static weakly-constrained model – Static constraint satisfaction model with communication constraints – Static distributed constraint satisfaction model – Dynamic distributed constraint satisfaction model • Goal: Identify and isolate the sources of combinatorial complexity

IISI, Cornell University Initial Assumptions • Each sensor can only track one target at a time • 3 sensors are required to track a target

IISI, Cornell University Initial Graph Model • Bipartite graph G = (S U T, E) • S is the set of sensor nodes, T the set of target nodes, E the edges indicating which targets are visible to a given sensor • Decision Problem: Can each target be tracked by three sensors?

IISI, Cornell University Initial Graph Model Target visibility Sensor nodes Graph Representation Target nodes

IISI, Cornell University Initial Graph Model w The initial model presented is a bipartite graph, and this problem can be solved using a maximum flow algorithm in polynomial time Sensor Target nodes

IISI, Cornell University Sensor Communication Constraints initial model + communication edges Possible solution w In the graph model, we now have additional edges between sensor nodes

IISI, Cornell University Constrained Graph Model communication edges sensors targets possible solution

Complexity and Phase Transition Phenomena of Sensor Domain

IISI, Cornell University Complexity • Decision Problem: Can each target be tracked by three sensors which can communicate together ? • We have shown that this constraint satisfaction problem (CSP) is NPcomplete, by reduction from the problem of partitioning a graph into isomorphic subgraphs

Average Case complexity and Phase Transition Phenomena

IISI, Cornell University Phase Transition w. r. t. Communication Level: Probability( all targets tracked ) Experiments with a random configuration of 9 sensors and 3 targets such that there is a communication channel between two sensors with probability p Insights into the design and operation of sensor networks w. r. t. communication level Communication edge probability p

IISI, Cornell University Phase Transition w. r. t. Radar Detection Range Probability( all targets tracked ) Experiments with a random configuration of 9 sensors and 3 targets such that each sensor is able to detect targets within a range R Insights into the design and operation of sensor networks w. r. t. radar detection range Normalized Radar Range R

Distributed Model

IISI, Cornell University Distributed CSP Model • In a distributed CSP (DCSP) variables and constraints are distributed among multiple agents. It consists of: – A set of agents 1, 2, … n – A set of CSPs P 1, P 2, … Pn , one for each agent – There are intra-agent constraints and interagent constraints

IISI, Cornell University DCSP Model • We can represent the sensor tracking problem as DCSP using dual representations: – One with each sensor as a distinct agent – One with a distinct tracker agent for each target

Sensor Agents • Binary variables associated with each target • Intra-agent constraints : – Sensor must track at most 1 visible target • Inter-agent constraints: – 3 communicating sensors should track each target t 1 t 2 t 3 t 4 s 1 x 0 x 1 s 2 x x x 1 s 3 x x x 1 s 4 1 0 x 0

Target Tracker Agents • Binary variables associated with each sensor • Intra-agent constraints : – Each target must be tracked by 3 communicating sensors to which it is visible • Inter-agent constraints: – A sensor can only track one target s 1 s 2 s 3 s 4 s 5 s 6 s 7 s 8 s 9 t 1 1 0 1 x x x 1 t 2 x x x 1 1 1 x x x t 3 x x x 1 1 0

Implicit versus Explicit Constraints • Explicit constraint: (correspond to the explicit domain constraints) – no two targets can be tracked by same sensor (e. g. t 2, t 3 cannot share s 4 and t 1, t 3 cannot share s 9) – three sensors are required to track a target (e. g. s 1, s 3, s 9 for t 1) • Implicit constraint: (due to a chain of explicit constraints: (e. g. implicit constraint between s 4 for t 2 and s 9 for t 1 ) s 1 s 2 s 3 s 4 s 5 s 6 s 7 s 8 s 9 t 1 1 0 1 x x x 1 t 2 x x x 1 1 1 x x x t 3 x x x 1 1 0

Communication Costs for Implicit Constraints • Explicit constraints can be resolved by direct communication between agents • Resolving Implicit constraints may require long communication paths through multiple agents scalability problems s 1 s 2 s 3 s 4 s 5 s 6 s 7 s 8 s 9 t 1 1 0 1 x x x 1 t 2 x x x 1 1 1 x x x t 3 x x x 1 1 0

Conclusions and Research Directions

Future directions • Study structural issues and inpact on complexity, as they occur in the distributed environments e. g. : – effect of balancing; – backbone (insights into critical resources); – refinement of phase transition notions considering additional parameters;

DCSP Model • With the DCSP model, we plan to study both per-node computational costs as well as inter-node communication costs • We are in the process of applying known DCSP algorithms to study issues concerning complexity and scalability

Summary • We have made considerable progress in our understanding of the nature of hard computational problems - structure matters! • We have harnessed a variety of mechanisms with proven impact on time-critical problem solving. • A rich spectrum of applications taking advantage of these new methods are on the horizon in planning, scheduling and many other areas. • Future focus on Dynamic Distributed models

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