Planning and Optimization E1. Landmarks: MHS & Uniform Cost Partitioning Heuristic Gabriele R¨oger and Thomas Keller

E1. Landmarks: MHS & Uniform Cost Partitioning Heuristic

Gabriele R¨oger and Thomas Keller

Universit¨at Basel

November 12, 2018

Landmarks

Content of this Course

Planning

Classical

Tasks Progression/

Regression Complexity Heuristics

Probabilistic

MDPs Uninformed Search

Heuristic Search Monte-Carlo

Methods

Content of this Course: Heuristics

Heuristics

Delete Relaxation Abstraction

Landmarks

Action Landmarks MHS Heuristic

Uniform Cost Partitioning Cut Landmarks LM-Cut Heuristic Network Flows

Potential Heuristics Cost Partitioning

Landmarks

Basic Idea: Something that must happen in every solution For example

some operator must be applied some atom must be true some formula must be true

→Derive heuristic estimate from this kind of information.

We will only consider disjunctive action landmarks.

Landmarks

Basic Idea: Something that must happen in every solution For example

some operator must be applied some atom must be true some formula must be true

→Derive heuristic estimate from this kind of information.

We will only considerdisjunctive action landmarks.

Disjunctive Action Landmarks

Definition (Disjunctive Action Landmark)

Lets be a state of planning task Π =hV,I,O, γi.

Adisjunctive action landmarkfor s is a set of operatorsL⊆O such that every plan fors (= label path froms to a goal state) contains an operator fromL.

Thecostof landmark Lis cost(L) = mino∈Lcost(o).

Example Task

Two trucks, one airplane

Airplane can fly between locations A3 and B1

Trucks can drive arbitrarily between locations A1, A2, and A3 Package to be transported from A1 to B1

Operators

Load(v, `)andUnload(v, `)for vehiclev and location` Drive(t, `, `<sup>0</sup>)for truckt and locations`, `⁰

Fly(`, `⁰)for locations`, `⁰

A1 A2

A3 B1

t₁

t2

f

Example: Disjunctive Action Landmarks

L1 ={Load(Truck1,A1),Load(Truck2,A1)} and L2 ={Fly(B1, A3)} are disjunctive action landmarks.

A1 A2

A3 B1

t1

t₂

f

What other disjunctive action landmarks are there?

Remarks

Not every landmark is informative.

For example: If the initial state is not already a goal state then the set of all operators is a disjunctive action landmark.

Deciding whether a given operator set is a disjunctive action landmark is as hard as the plan existence problem.

Exploiting Disjunctive Action Landmarks

How can we exploit a given setL of disjunctive action landmarks?

Sum of costsP

L∈Lcost(L)?

not admissible!

Maximize costs maxL∈Lcost(L)?

usually very weak heuristic

better: hitting setsor cost partitioning

Minimum Hitting Set Heuristic

Content of this Course: Heuristics

Heuristics

Delete Relaxation Abstraction

Landmarks

Action Landmarks MHS Heuristic

Uniform Cost Partitioning Cut Landmarks LM-Cut Heuristic Network Flows

Potential Heuristics Cost Partitioning

Hitting Sets

Definition (Hitting Set)

LetX be a set,F ={F₁, . . . ,Fn} ⊆2^X be a family of subsets of X andc :X →R⁺₀ be a cost function for X.

Ahitting setis a subsetH⊆X that “hits” all subsets in F, i.e., H∩F 6=∅ for allF ∈ F. The costofH isP

x∈Hc(x).

Aminimum hitting set (MHS)is a hitting set with minimal cost.

MHS is a “classical” NP-complete problem (Karp, 1972)

Example: Hitting Sets

Example

X ={o₁,o2,o3,o4}

F={{o₄},{o₁,o₂},{o₁,o₃},{o₂,o₃}}

c(o₁) = 3, c(o₂) = 4, c(o₃) = 5, c(o₄) = 0

minimum hitting set: {o₁,o2,o4}with cost 3 + 4 + 0 = 7

Example: Hitting Sets

Example

X ={o₁,o2,o3,o4}

F={{o₄},{o₁,o₂},{o₁,o₃},{o₂,o₃}}

c(o₁) = 3, c(o₂) = 4, c(o₃) = 5, c(o₄) = 0

minimum hitting set: {o₁,o2,o4}with cost 3 + 4 + 0 = 7

Hitting Sets for Disjunctive Action Landmarks

Idea: disjunctive action landmarks are interpreted as Idea: instance of minimum hitting set

Definition (Hitting Set Heuristic)

LetL be a set of disjunctive action landmarks. The hitting set heuristich^MHS(L) is defined as the cost of a minimum hitting set forL with c(o) =cost(o).

Proposition (Hitting Set Heuristic is Admissible)

LetL be a set of disjunctive action landmarks for state s.

Then h^MHS(L)is an admissible estimate for s.

Why?

Hitting Set Heuristic: Discussion

The hitting set heuristic is the best possibleheuristic that only uses the given information. . .

. . . but is NP-hard to compute.

Use approximations that can be efficiently computed.

Now: uniform cost partitioning

Later in the course: optimal cost partitioning

Uniform Cost Partitioning

Content of this Course: Heuristics

Heuristics

Delete Relaxation Abstraction

Landmarks

Action Landmarks MHS Heuristic

Uniform Cost Partitioning Cut Landmarks LM-Cut Heuristic Network Flows

Potential Heuristics Cost Partitioning

Uniform Cost Partitioning (1)

Idea: Distribute cost of operators uniformly among the landmarks.

Definition (Uniform Cost Partitioning Heuristic for Landmarks) LetL be a set of disjunctive action landmarks.

Theuniform cost partitioning heuristic h^UCP(L) is defined as h^UCP(L) =X

L∈L

min

o∈Lc⁰(o) with c⁰(o) =cost(o)/|{L∈ L |o ∈L}|.

Landmarks Minimum Hitting Set Heuristic Uniform Cost Partitioning Summary

Uniform Cost Partitioning (2)

Theorem (Uniform Cost Partitioning Heuristic is Admissible) LetL be a set of disjunctive action landmarks for state s ofΠ.

Then h^UCP(L) is an admissibleheuristic estimate for s.

Letπ =ho₁, . . . ,oni be an optimal plan fors. ForL∈ Ldefine a new cost functioncost_L ascost_L(o) =c⁰(o) ifo ∈L and

costL(o) = 0 otherwise. Let ΠL be a modified version of Π, where for all operatorso the cost is replaced withcost_L(o). We make three independent observations:

1 For L∈ L the valuecost⁰(L) := mino∈Lc⁰(o) is an admissible estimate for s in Π_L.

2 π is also a plan fors in Π_L, so h^∗_Π

L(s)≤Pn

i=1cost_L(o_i).

3 P

L∈Lcost_L(o) =cost(o) for each operatoro.

. . .

Uniform Cost Partitioning (2)

Theorem (Uniform Cost Partitioning Heuristic is Admissible) LetL be a set of disjunctive action landmarks for state s ofΠ.

Then h^UCP(L) is an admissibleheuristic estimate for s.

Proof.

Letπ=ho₁, . . . ,oni be an optimal plan fors. ForL∈ Ldefine a new cost functioncost_L ascost_L(o) =c⁰(o) ifo ∈L and

costL(o) = 0 otherwise. Let ΠL be a modified version of Π, where for all operatorso the cost is replaced withcost_L(o). We make three independent observations:

1 For L∈ Lthe value cost⁰(L) := mino∈Lc⁰(o) is an admissible estimate for s in Π_L.

2 π is also a plan fors in Π_L, so h^∗_Π

L(s)≤Pn

i=1cost_L(o_i).

3 P

L∈Lcost_L(o) =cost(o) for each operator o.

. . .

Uniform Cost Partitioning (3)

Proof (continued).

Together, this leads to the following inequality (subscripts indicate for which task the heuristic is computed):

h_Π^UCP(L) =X

L∈L

cost⁰(L)

(1)

≤ X

L∈L

h^∗_Π

L(s)

(2)

≤ X

L∈L n

X

i=1

cost_L(o_i) =

n

X

i=1

X

L∈L

cost_L(o_i)

(3)=

n

X

i=1

cost(o) =h^∗_Π(s)

Relationship

Theorem

LetL be a set of disjunctive action landmarks for state s.

Thenh^UCP(L)≤h^MHS(L)≤h^∗(s).

(Proof omitted.)

Summary

Summary

Disjunctive action landmark: setL of operators such that every plan uses some operator from L

The costof the landmark is the cost of its cheapest operator.

Hitting setsyield the most accurate heuristic for a given set of disjunctive action landmarks, but the computation is NP-hard.

Uniform cost partitioningis a polynomial approach for the computation of informative heuristics from

disjunctive action landmarks.