1. Combinatorial Problems I: Finding Solutions
Ashish Sabharwal
Cornell University
March 3, 2008
2nd Asian-Pacific School on Statistical Physics and Interdisciplinary Applications KITPC/ITP-CAS, Beijing, China

2. Computer Science
Engineering
Mathematics
Cross-fertilization
of ideas for the study
and design of
Intelligent Systems
Operations Research
Economics
Phase transition
Physics
Cognitive Science
Research part of Cornell’s Intelligent Information Systems Institute (IISI)
Director: Carla Gomes

3. Combinatorial Problems
Examples
Routing: Given a partially connected network on N nodes, find the shortest path between X and Y
Traveling Salesperson Problem (TSP): Given a partially connected network on N nodes, find a path that visits every node of the network exactly once [much harder!!]
Scheduling: Given N tasks with earliest start times, completion deadlines, and set of M machines on which they can execute, schedule them so that they all finish by their deadlines

4. Problem Instance, Algorithm
Specific instantiation of the problem
E.g. three instances for the routing problem with N=8 nodes:
Objective: a single, generic algorithm for the problem that can solve any instance of that problem
A sequence of steps, a “recipe”

5. Measuring the Effectiveness of Algorithms
Capture scaling with input size N, rather than runtime on specific instances
The most common notion in Computer Science is worst-case complexity: What is the longest time (or number of steps) the algorithm might take on any input of size N? Perhaps only N steps, 100 N+5 N linear time, O(N) Maybe N2 steps, or N2 + 4 N + 6 quadratic ,O(N2) Maybe N log N cubic, O(N3) … … … Maybe 2N, or 2N + N exponential, O(2N)

6. Polynomial vs. Exponential Complexity
Polynomial time: “tractable”, can hope to solve very large problems with enough computing power
E.g. known routing / shortest path algorithms [O(N3)]
Exponential time: quickly run into scalability issues as N increases
E.g. best known algorithms for TSP

7. Are some problems inherently harder than others
Are some problems inherently harder than others? A large amount of work on answering this question: computational complexity theory

8. Computational Complexity Hierarchy
EXP-complete:
games like Go, …
EXP
Hard
PSPACE-complete:
QBF, adversarial planning, chess (bounded), …
PSPACE
#P-complete/hard:
#SAT, sampling, probabilistic inference, …
P^#P PH
NP-complete:
SAT, scheduling, graph coloring, puzzles, … NP
P-complete:
circuit-value, … P
In P:
sorting, shortest path, …
Easy
Note: widely believed hierarchy; know P≠EXP for sure

9. NP-Completeness
P : class of problems for which a solution can be found in poly time e.g. can find a shortest path in poly time
NP: class of problems for which a solution can be verified in poly time e.g. can’t find a TSP solution in poly time (as far as we know) but, given a candidate solution (a “witness”) can verify the correctness of the witness in poly time “N”: non-deterministic, with the power of “guessing” “P”: polynomial time
NP-complete: the “hardest” problems within NP

10. NP-Completeness One of the biggest discoveries in Computer Science:
All NP-complete problems are equally hard! [worst-case complexity]
An algorithm for any one NP-complete problem can be used to solve any other NP-complete problem with only a polynomial overhead!
There are catalogues of 10,000’s of such problems e.g. “Boolean satisfiability” or SAT, TSP, scheduling, (bounded) planning, chip verification, 0-1 integer programming, graph coloring, logical inference, …
[Similarly for PSPACE-complete, #P-complete, etc.]

11. Can one design a single algorithm that can efficiently solve thousands of different problems of interest?

12. The Quest for Machine Reasoning
A cornerstone of Artificial Intelligence
Objective: Develop foundations and technology to enable effective, practical, large-scale automated reasoning.
Machine Reasoning ( s)
Current reasoning technology
Computational complexity of reasoning
appears to severely limit real-world
applications
Revisiting the challenge:
Significant progress with new ideas / tools for dealing with complexity (scale-up), uncertainty, and multi-agent reasoning

13. General Automated Reasoning
General Inference Engine
Model Generator (Encoder)
Problem instance
Solution
Domain-specific
Generic
e.g. logistics, chess, planning, scheduling, ...
applicable to all domains within range of modeling language
Research objective
Better reasoning and
modeling technology
Impact
Faster solutions
in several domains

14. For N variables: 2N cases drive complexity!
Reasoning Complexity
EXPONENTIAL COMPLEXITY: INHERENT
AN worst case
N= No. of Variables/Objects A= Object states
TIME/SPACE
Granularity   Object states
Current implementations trade
time with soundness
Simple Example:
Variables (binary)
X1 = _ received
X2 = in_ meeting
X3 = urgent
X4 = respond_to_
X5 = near_deadline
X6 = postpone
X7 = air_ticket_info_request
X8 = travel_ request
X9 = info_request
Rules:
X1 & (not X2) & X3  X4
X2  not X4
X5  X3 or X6
4. X7  X8
5. X8  X9
6. X8  X5
7. X6  not X9
Knowledge Base
Question: Given: X1= true; X2 = false; X7=true.
What is X4 = ?
Answer Development: Inference Chain
Step 1: X7  X8 (rule 4)
Step 2: X8  X5 (rule 6)
Step 3: X5  X3 or X6 (rule 3)
Case A: X6 = true
Step 4: X6  not X9
Step 5: X9  not X8
Step 6: Contradiction
Backtrack to M
Case B: X3 = true
X1 & (not X2) & X3  X4
Step 7: X4 = true (Rule 1) M
Search for rules to apply
For N variables: 2N cases drive complexity!
Check Contradictions

15. Exponential Complexity Growth: The Challenge of Complex Domains
Note: rough estimates, for propositional reasoning 1M 5M
War Gaming
10301,020
0.5M 1M
VLSI
Verification
Case complexity
10150,500
100K
450K
Military Logistics
106020
20K
100K
Chess (20 steps deep)
103010
No. of atoms
on the earth
10K
50K
Deep space mission control
Seconds until heat death of sun
1047
100
200
1030
Car repair diagnosis
Protein folding
Calculation (petaflop-year)
Variables
100
10K
20K
100K 1M
Rules (Constraints)
[Credit: Kumar, DARPA; Cited in Computer World magazine]

16. Progress in Last 15 Years Focus: Combinatorial Search Spaces
Specifically, the Boolean satisfiability problem, SAT
Significant progress since the 1990’s.
How much?
Problem size: We went from 100 variables, 200 constraints (early 90’s) to 1,000,000 vars. and 5,000,000 constraints in 15 years. Search space: from 10^15 to 10^300,000. [Aside: “one can encode quite a bit in 1M variables.”]
Tools: 50+ competitive SAT solvers available
Overview of the state of the art: Plenary talk at IJCAI-05 (Selman); Discrete App. Math. article (Kautz-Selman ’06)

17. How Large are the Problems?
A bounded model checking problem:

18. SAT Encoding i.e., ((not x1) or x7) ((not x1) or x6) etc.
(automatically generated from problem specification)
i.e., ((not x1) or x7)
((not x1) or x6)
etc.
x1, x2, x3, etc. are our Boolean variables
(to be set to True or False)
Should x1 be set to False??

19. 10 Pages Later: … i.e., (x177 or x169 or x161 or x153 …
x33 or x25 or x17 or x9 or x1 or (not x185))
clauses / constraints are getting more interesting…
Note x1 …

20. 4,000 Pages Later: …

21. Finally, 15,000 Pages Later:
Search space of truth assignments:
Current SAT solvers solve this instance in
under 30 seconds!

22. SAT Solver Progress Solvers have continually improved over time
Source: Marques-Silva 2002

23. How do SAT Solvers Keep Improving?
From academically interesting to practically relevant.
We now have regular SAT solver competitions.
(Germany ’89, Dimacs ’93, China ’96, SAT-02, SAT-03, …, SAT-07)
E.g. at SAT (Seattle, Aug ’06):
35+ solvers submitted, most of them open source
500+ industrial benchmarks
50,000+ benchmark instances available on the www
This constant improvement in SAT solvers is the key to making, e.g., SAT-based planning very successful.

24. Current Automated Reasoning Tools
Most-successful fully automated methods: based on Boolean Satisfiability (SAT) / Propositional Reasoning
Problems modeled as rules / constraints over Boolean variables
“SAT solver” used as the inference engine
Applications: single-agent search
AI planning
SATPLAN-06, fastest optimal planner; ICAPS-06 competition (Kautz & Selman ’06)
Verification – hardware and software
Major groups at Intel, IBM, Microsoft, and universities such as CMU, Cornell, and Princeton. SAT has become the dominant technology.
Many other domains: Test pattern generation, Scheduling, Optimal Control, Protocol Design, Routers, Multi-agent systems, E-Commerce (E-auctions and electronic trading agents), etc.

25. Recall: General Automated Reasoning
General Inference Engine
Model Generator (Encoder)
Problem instance
Solution
Domain-specific
Generic
e.g. logistics, chess, planning, scheduling, ...
applicable to all domains within range of modeling language
Research objective
Better reasoning and
modeling technology
Impact
Faster solutions
in several domains

26. Automated Reasoning with SAT
A simple but useful modeling language: Boolean formulas
Corresponding inference engine: Satisfiability or SAT algorithm (e.g. complete search, local search, message passing)
Numerous applications: hardware and software verification, planning, scheduling, e-commerce, circuit design, open problems in algebra, …

27. Boolean Logic Defined over Boolean (binary) variables a, b, c, …
Each of these can be True (1, T) or False (0, F)
Variables connected together with logic operators: and, or, not (denoted )
E.g. ((c  d)  f) is True iff either c is True and d is False, or f is True
Fact: All other Boolean logic operators can be expressed with and, or, not
E.g. (a  b) same as (a or b)
Boolean formula, e.g. F = (a or b) and (a and (b or c))
(Truth) Assignment: any setting of the variables to True or False
Satisfying assignment: assignment where the formula evaluates to True
E.g. F has 3 satisfying assignments: (0,1,0), (0,1,1), (1,0,0)

28. Boolean Logic: Example
F = (a or b) and (a and (b or c))
Note: True often written as 1, False as 0
There are 23 = 8 possible truth assignments to a, b, c
(a=0,b=1,c=0) representing (a=False, b=True, c=False)
(a=0,b=0,c=1) …
Truth Table for F a b c F 1
Exactly 3 truth assignments satisfy F
(a=0,b=1,c=0)
(a=0,b=1,c=1)
(a=1,b=0,c=0)

29. Boolean Logic: Expressivity
All discrete single-agent search problems can be cast as a Boolean formula
Variables a, b, c, … often represent “states” of the system, “events”, “actions”, etc.
(more on this later, using Planning as an example)
Very general encoding language. E.g. can handle
Numbers (k-bit binary representation)
Floating-point numbers
Arithmetic operators like +, x, exp(), log() …
SAT encodings (generated automatically from high level languages) routinely used in domains like planning, scheduling, verification, e-commerce, network design, …
Recall Example:
“event”
Variables
X1 = _ received
X2 = in_ meeting
X3 = urgent
X4 = respond_to_
X5 = near_deadline
X6 = postpone
X7 = air_ticket_info_request
X8 = travel_ request
X9 = info_request
“state”
“action”
Rules:
X1 & (not X2) & X3  X4
X2  not X4
X5  X3 or X6
4. X7  X8
5. X8  X9
6. X8  X5
7. X6  not X9
constraint

30. Boolean Logic: Standard Representations
Each problem constraint typically specified as (a set of) clauses:
E.g. (a or b), (c or d or f), (a or c or d), …
Formula in conjunctive normal form, or CNF: a conjunction of clauses
E.g F = (a or b) and (a and (b or c)) changes to
FCNF = (a or b) and (a or b) and (b or c)
Alternative [useful for QBF]: specify each constraint as a term (only “and”, “not”):
E.g. (a and d), (b and a and f), (b and d and e), …
Formula in disjunctive normal form, or DNF: a disjunction of terms
E.g. FDNF = (a and b) or (a and b and c)
clauses (only “or”, “not”)

31. Boolean Satisfiability Testing
The Boolean Satisfiability Problem, or SAT:
Given a Boolean formula F,
find a satisfying assignment for F
or prove that no such assignment exists.
A wide range of applications
Relatively easy to test for small formulas (e.g. with a Truth Table)
However, very quickly becomes hard to solve
Search space grows exponentially with formula size (more on this next)
SAT technology has been very successful in taming this exponential blow up!

32. Fix one variable to True or False
SAT Search Space
All vars free
Fix one variable to True or False
Fix another var
Fix a 3rd var
Fix a 4th var
True
False
SAT Problem: Find a path to a True leaf node.
For N Boolean variables, the raw search space is of size 2N
Grows very quickly with N
Brute-force exhaustive search unrealistic without efficient heuristics, etc.

33. k-CNF, 3-CNF k-CNF: all clauses have k literals 1-CNF SAT: trivial
2-CNF SAT: solvable in O(N2) time [N = num. of variables]
3-CNF SAT: NP-complete
4-CNF SAT: NP-complete …
Note: Any Boolean formula can be converted into CNF.
-- with or without extra variables (without  size increase)

34. Worst-Case Complexity
SAT is an NP-complete problem
Worst-case believed to be exponential (roughly 2N for N variables)
10,000+ problems in CS are NP-complete (e.g. planning, scheduling, protein folding, reasoning)
P vs. NP --- $1M Clay Prize
However, real-world instances are usually not pathological and can often be solved very quickly with the latest technology!
Typical-case complexity provides a more detailed understanding and a more positive picture.
exponential
polynomial

35. Exponential Complexity Growth
Planning (single-agent): find the right sequence of actions
exponential
polynomial
HARD: 10 actions, 10! = 3 x 106 possible plans
Contingency planning (multi-agent):
actions may or may not produce the desired effect! …
1 out of 10
2 out of 9
4 out of 8
REALLY HARD: 10 x 92 x 84 x 78 x … x 2256 = possible contingency plans!

36. Typical-Case Complexity
A key hardness parameter for k-SAT: the ratio of clauses to variables
Add Constraints
Delete Constraints
Problems that are not critically constrained tend to be much easier in practice than the relatively few critically constrained ones
[Mitchell, Selman, and Levesque ’92; Kirkpatrick and Selman – Science ’94]

37. Typical-Case Complexity
Random 3-SAT as of 2004
Random Walk DP
DP’
Walksat SP
Linear time algs.
GSAT
Phase transition
SAT solvers continually getting close to tackling problems in the hardest region!
SP (survey propagation) now handles 1,000,000 variables very near the phase transition region

38. Tractable Sub-Structure Can Dominate and Drastically Reduce Solution Cost!
2+p-SAT model: mix 2-SAT (tractable) and 3-SAT (intractable) clauses
> 40% 3-SAT: exponential scaling
Median runtime
 40% 3-SAT: linear scaling!
Number of variables
(Monasson, Selman et al. – Nature ’99; Achlioptas ’00)

39. How are other NP-complete problems translated into SAT instances
How are other NP-complete problems translated into SAT instances? “SAT encoding”

40. SAT Encoding Example: Planning Domain
Planning Problem  Propositional CNF formula by axiom schemas
Logistics planning: think of a number of trucks and planes that need to transport a bunch of packages from their origin to their destination
Discrete time, modeled by integers
state predicates: indexed by time at which they hold
E.g. at_location(x,,loc,i), free(x,i+1), route(cityA,cityB,i)
action predicates: indexed by time at which action begins
E.g. fly(cityA,cityB,i), pickup(x,loc,i), drive_truck(loc1,loc2,i)
each action takes 1 time step
many actions may occur at the same step

41. Encoding Rules fly(x,y,i)  at(x,i) and route(x,y,i) and at(y,i+1)
Actions imply preconditions and effects
fly(x,y,i)  at(x,i) and route(x,y,i) and at(y,i+1)
Conflicting actions cannot occur at same time (A deletes a precondition of B)
fly(x,y,i) and yz  not fly(x,z,i)
If something changes, an action must have caused it (Explanatory Frame Axioms)
at(x,i) and not at(x,i+1)  y . route(x,y) and fly(x,y,i)
Initial and final states hold
at(NY,0) and and at(LA,9) and ...

42. Using SAT Solvers for Planning
Modeling and Solving a Planning Problem
instantiated
propositional
clauses
instantiate
Problem description in high level language
axiom
schemas
(manual)
length
mapping
interpret
SAT
engine(s)
plan
satisfying
model
(fully automatic)

43. Planning Benchmark Complexity
Logistics domain – a complex, highly-parallel transportation domain
E.g. logistics.d problem:
2,165 possible actions per time slot
optimal solution contains 74 distinct actions over 14 time slots
(out of 5 x 10^46 possible sequential plans of length 14)
Satplan [Selman et al.] approach is currently fastest optimal planning approach. Winner ICAPS-05 & ICAPS-06 international planning competitions.

44. Solution Approaches to SAT

45. Solving SAT: Systematic Search
One possibility: enumerate all truth assignments one-by-one, test whether any satisfies F
Note: testing is easy!
But too many truth assignments (e.g. for N=1000 variables, have  truth assignments) …… 2N

46. Solving SAT: Systematic Search
Smarter approach: the “DPLL” procedure [1960’s]
(Davis, Putnam, Logemann, Loveland)
Assign values to variables one at a time (“partial” assignments)
Simplify F
If contradiction (i.e. some clause becomes False), “backtrack”, flip last unflipped variable’s value, and continue search
Extended with many new techniques ’s of research papers, yearly conference on SAT e.g., extremely efficient data-structures (representation), randomization, restarts, learning “reasons” of failure
Provides proof of unsatisfiability if F is unsat. [“complete method”]
Forms the basis of dozens of very effective SAT solvers! e.g. minisat, zchaff, relsat, rsat, … (open source, available on the www)

47. Solving SAT: Local Search
Search space: all 2N truth assignments for F
Goal: starting from an initial truth assignment A0, compute assignments A1, A2, …, As such that As is a satisfying assignment for F
Ai+1 is computed by a “local transformation” to Ai e.g. A1 = green bit “flips” to red bit A2 = A3 = A4 = … … As = solution found!
No proof of unsatisfiability if F is unsat. [“incomplete method”]
Several SAT solvers based on this approach, e.g. Walksat

48. Solving SAT: Decimation
“Search” space: all 2N truth assignments for F
Goal: attempt to construct a solution in “one-shot” by very carefully setting one variable at a time
Survey Inspired Decimation:
Estimate certain “marginal probabilities” of each variable being True, False, or ‘undecided’ in each solution cluster using Survey Propagation
Fix the variable that is the most biased to its preferred value
Simplify F and repeat
A method rarely used by computer scientists
But has received tremendous success from the physics community on random k-SAT; can easily solve random instances with 1M+ variables!
No searching for solution
No proof of unsatisfiability [“incomplete method”]

49. The Next Two Lectures
Problems beyond SAT / searching for a single solution
#P-complete: count the number of solutions of a SAT instance
#P-hard: sample a solution uniformly at random for a SAT instance
PSPACE-complete: quantified Boolean formula (QBF)

50. Thank you for attending!
Slides:
Ashish Sabharwal :
Bart Selman :