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This work is licensed under a Creative Commons Attribution-Share Alike 3.0 Unported License . CS 312: Algorithm Analysis. Lecture #33: Branch and Bound, Job Assignment. Slides by: Eric Ringger, with contributions from Mike Jones, Eric Mercer, and Sean Warnick. Announcements.

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Cs 312 algorithm analysis

This work is licensed under a Creative Commons Attribution-Share Alike 3.0 Unported License.

CS 312: Algorithm Analysis

Lecture #33: Branch and Bound,

Job Assignment

Slides by: Eric Ringger, with contributions from Mike Jones, Eric Mercer, and Sean Warnick


Announcements
Announcements

  • Homework #23 due now

  • Project #7: TSP

    • Budget about 15 hours

    • ASAP: Read Instructions

    • Today: We discuss main ideas

    • Thursday: Help session 10am Tues 1066 TMCB

    • Read the helpful “B&B for TSP Notes” linked from the schedule in the reading column

    • Thurs: Early day

    • Friday: due


Objectives
Objectives

  • Understand the difference between backtracking and branch and bound

  • Understand bounding functions

  • Develop a branch and bound algorithm for the Job Assignment Problem




Bounding function
Bounding Function

Given some state s in the search space, compute a bound B(s) on the cost/goodness of all solutions that descend from that state.

cost of actual solution is somewhere in here

better

worse

bound B(s)

“From this state s, I can do no better than B(s).”


Bssf best solution so far
BSSF = Best Solution So Far

cost of actual solution is somewhere in here

better

worse

Cost of BSSF

bound B(s)


Pruning scenario 1
Pruning Scenario #1

cost of actual solution is somewhere in here

better

worse

Cost of BSSF

bound B(s)

Would you explore this state s with bound B(s)?

BSSF = “Best Solution So Far”


Pruning scenario 2
Pruning Scenario #2

cost of actual solution is somewhere in here

better

worse

bound B(s’)

Cost of BSSF

Would you explore this state s’with bound B(s’)?

BSSF = “Best Solution So Far”


Job assignment problem
Job Assignment Problem

  • Given n tasks and n agents.

    • Each agent has a cost to complete each task

    • Assign each agent one task; each task one agent;

      • A 1:1 mapping

    • Minimize cost


Minimization
Minimization

worse

  • The cost of the BSSF

    • Is an Upper Bound on the optimal solution

  • B(): Bounding function for evaluating any state s

    • Is a Lower Bound on potential solutions reachable from s

    • Usually involves solving a relaxed form of the problem

  • s0: Initial state

  • LB = B(s0)

    • Is a Lower Bound on all potential solutions reachable from the initial state

  • Tight bounds:

    • Why would you want the upper and lower bounds to be tight?

Cost ofBSSF

LB

better



Example1
Example

How to represent a state?


Example2
Example

First, generate a solution (not necessarily optimal)

and call that your best solution so far (BSSF). How?


Hint

First, generate a solution (not necessarily optimal)

and call that your best solution so far (BSSF). How?


BSSF

BSSF: 73

  • First, generate a solution (not necessarily optimal)

  • and call that your best solution so far (BSSF). How?

  • One possible way:Assign jobs along the diagonals – take the lower one.

  • Something quick (at worst linear?).


Lower bound
Lower Bound

BSSF: 73

Next, compute a lower bound on all solutions. How?


Hint

  • Bounding function:

  • Easy to evaluate

  • True bound

BSSF: 73

Next, compute a lower bound on all solutions. How?


Lower bound1
Lower Bound

BSSF: 73

  • Next, compute a lower bound on all solutions. How?

  • Add the smallest entry in each column

  • LB = 58

  • Other alternatives?


Lower bound2
Lower Bound

BSSF: 73

  • Next, compute a lower bound on all solutions. How?

  • Add the smallest entry in each column

  • LB = 58

  • Our situation:


Lower bound3
Lower Bound

BSSF: 73

  • Next, compute a lower bound on all solutions. How?

  • Add the smallest entry in each column

  • LB = 58

  • What if the LB == BSSF?

  • What if LB > BSSF?


State space search
State-Space Search

BSSF: 73

Now, make an assignment to agent A

A:1

(58)


Example3
Example

BSSF: 73

  • Then apply the bounding function:

  • add the smallest values in each column.

  • This is a lower bound on the cost of any solutionwith job 1 assigned to A.

A:1 (60)

(58)


Example4
Example

BSSF: 73

A:1 (60)

A:2 (58)

same for assigning job 2 to A.

(58)


Example5
Example

BSSF: 73

A:1 (60)

A:2 (58)

A:3 (65)

(58)


Example6
Example

BSSF: 73

A:1 (60)

A:2 (58)

A:3 (65)

A:4 (78)

(58)

What can we say about the last option?


Example7
Example

BSSF: 73

  • How to proceed from here? Many options:

  • Breadth-first

  • Depth-first

  • Most promising first

A:1 (60)

A:2 (58)

A:3 (65)

A:4 (78)

(58)


Example8
Example

BSSF: 73

  • How to proceed from here? Many options:

  • Breadth-first

  • Depth-first

  • Most promising first – we’ll try this one today

  • Other possibilities

A:1 (60)

A:2 (58)

A:3 (65)

A:4 (78)

(58)


Example9
Example

BSSF: 73

A:1 (60)

A:2 (58)

A:3 (65)

A:4 (78)

(58)


Example10
Example

BSSF: 73

A:1 (60)

A:2 (58)

A:3 (65)

A:4 (78)

A:2, B:1 (68)

(58)


Example11
Example

BSSF: 73

A:1 (60)

A:2 (58)

A:3 (65)

A:4 (78)

A:2, B:1 (68)

A:2, B:3 (59)

A:2, B:4 (64)

(58)


Example12
Example

BSSF: 73

A:1 (60)

A:2 (58)

A:3 (65)

A:4 (78)

A:2, B:1 (68)

A:2, B:3 (59)

A:2, B:4 (64)

A:2, B:3, C:1, D:4 (64)

A:2, B:3, C:4, D:1 (65)

(58)

When you reach a solution, update the BSSF if better.


Example13
Example

BSSF: 64

A:1 (60)

A:2 (58)

A:3 (65)

A:4 (78)

A:2, B:1 (68)

A:2, B:3 (59)

A:2, B:4 (64)

A:2, B:3, C:1, D:4 (64)

A:2, B:3, C:4, D:1 (65)

(58)

When you reach a solution, update the BSSF if better. Then what?


Example14
Example

BSSF: 64

A:1 (60)

A:2 (58)

A:3 (65)

A:4 (78)

A:2, B:1 (68)

A:2, B:3 (59)

A:2, B:4 (64)

A:2, B:3, C:1, D:4 (64)

A:2, B:3, C:4, D:1 (65)

(58)

Prune!


Example15
Example

BSSF: 64

A:1 (60)

A:2 (58)

A:3 (65)

A:4 (78)

A:2, B:1 (68)

A:2, B:3 (59)

A:2, B:4 (64)

A:2, B:3, C:1, D:4 (64)

A:2, B:3, C:4, D:1 (65)

(58)


Example16
Example

That’s the basic idea. Many details have been left out.

Let’s add a few.


Details
Details

  • What should each state contain?

    • Just as for backtracking search, we need enough information in each state to generate its children in the state space

    • The value of the bound on this state

  • How to store the set of states visited but remaining to be explored (i.e., “frontier of the search”)?

    • A set, sometimes called the “agenda” or “open list”

    • DFS: use a stack

    • BFS: use a queue

    • Most Promising First: use a priority queue

    • Other possibilities (e.g., hybrids)


Recall iterative dfs
Recall: Iterative DFS

function DFS(v)

P  empty-stack

visited(v)  true

P.push(v)

while !P.empty() do

while there existsw adjacent to P.top()

(in ascending order) such that !visited(w) do

visited(w)  true

P.push(w) // w is the new P.top()

P.pop()


Recall breadth first search
Recall: Breadth First Search

function BFS(v)

Q  empty-queue

visited(v)  true

Q.enqueue(v)

while !Q.empty() do

u  Q.first()

Q.dequeue()

for each w adjacent to u

(in ascending order) do

if !visited(w) then

visited(w)  true

Q.enqueue(w)


B b using a priority queue work in progress
B&B using a Priority Queue:Work in Progress

functionBandB-draft(v)

Q  empty-priority-queue

visited(v)  true

Q.enqueue(v)

while !Q.empty() do

u  Q.first()

Q.dequeue()

for each w adjacent to u (in ascending order)do

if !visited(w) then

visited(w)  true

Q.enqueue(w)

Remember: a priority queue is not the only possible representation of the agenda.

What’s missing?


Use bound as priority
Use Bound as Priority

function BandB-draft(v)

Q  empty-priority-queue

visited(v)  true

v.b  bound(v) // LB

Q.enqueue(v, v.b)

while !Q.empty() do

u  Q.first()

Q.dequeue()

for each w adjacent to u (in ascending order)do

if !visited(w) then

visited(w)  true

w.b  bound(w)

Q.enqueue(w, w.b)

What’s missing?


Adapt for on the fly state space search
Adapt for On-the-flyState-Space Search

function BandB-draft(v)

Q  empty-priority-queue

// visited(v)  true

v.b  bound(v)

Q.enqueue(v, v.b)

while !Q.empty() do

u  Q.first()

Q.dequeue()

children = generate_children_ascending(u)

for each w in childrendo

// if !visited(w) then

// visited(w)  true

w.b  bound(w)

Q.enqueue(w, w.b)

What’s missing?


Keep check bssf
Keep & Check BSSF

function BandB-draft(v)

BSSF  quick-solution(v) // BSSF.cost holds cost

Q  empty-priority-queue

v.b  bound(v)

Q.enqueue(v, v.b)

while !Q.empty() do

u  Q.first()

Q.dequeue()

  • children = generate_children_ascending(u)

    for each w in childrendo

    w.b  bound(w)

    if (w.b < BSSF.cost) then

    if criterion(w)then

    BSSF  w

    else ifpartial_criterion(w) then

    Q.enqueue(w, w.b)

    return BSSF

What’s missing?


V1 prune unpromising nodes lazily
V1: Prune Unpromising Nodes Lazily

functionBandB(v)

BSSF  quick-solution(v) // BSSF.cost holds cost

Q  empty-priority-queue

v.b  bound(v)

Q.enqueue(v, v.b)

while !Q.empty() do

u  Q.first()

Q.dequeue()

ifu.b < BSSF.costthen

  • children = generate_children_ascending(u)

    for each node w in childrendo

    w.b  bound(w)

    if(w.b < BSSF.cost)

    if criterion(w) then

    BSSF  w

    else ifpartial_criterion(w) then

    Q.enqueue(w, w.b)

    return BSSF

Remember: this version only works for best-first; i.e., when a priority queue is the agenda.

What’s missing?


Know when to stop
Know when to Stop!

function BandB(v)

BSSF  quick-solution(v) // BSSF.cost holds cost

Q  empty-priority-queue

v.b  bound(v)

Q.enqueue(v, v.b)

while !Q.empty() do

u  Q.first()

Q.dequeue()

if u.b < BSSF.cost then

  • children = generate_children_ascending(u)

    for each node w in childrendo

    w.b  bound(w)

    if (w.b < BSSF.cost)

    if criterion(w) then

    BSSF  w

    else if partial_criterion(w) then

    Q.enqueue(w, w.b)

    else break

    return BSSF

Remember: this version only works for best-first; i.e., when a priority queue is the agenda.

What’s missing?


V2 prune early when bssf is updated
V2: Prune Earlywhen BSSF is updated

function BandB(v)

BSSF  quick-solution(v) // BSSF.cost holds cost

Q  empty-priority-queue

v.b  bound(v)

Q.enqueue(v, v.b)

while !Q.empty() do

u  Q.first()

Q.dequeue()

  • children = generate_children_ascending(u)

    for each w in childrendo

    w.b  bound(w)

    if (w.b < BSSF.cost) then

    if criterion(w) then

    BSSF  w

    • Q.prune(BSSF.cost)

      else if partial_criterion(w) then

      Q.enqueue(w, w.b)

      return BSSF

What’s missing?


Generalize agenda
Generalize: Agenda

function BandB(v)

BSSF  quick-solution(v) // BSSF.cost holds cost

Agenda.clear()

v.b  bound(v)

Agenda.add(v, v.b)

while !Agenda.empty() do

u  Agenda.first()

Agenda.remove_first()

children = generate_children_ascending(u)

for each w in childrendo

w.b  bound(w)

if (w.b < BSSF.cost) then

if criterion(w) then

BSSF  w

  • Agenda.prune(BSSF.cost)

    else if partial_criterion(w) then

    Agenda.add(w, w.b)

    return BSSF


Critical elements of a b b algorithm
Critical Elements of a B&B Algorithm

  • BSSF

  • Bounding function

  • Representation of the frontier on an agenda

  • Select state from the agenda

    • Generate children

      • By copying parent and making one more decision

    • Calculate the bound

    • Handle children that are solutions

    • Place promising children on the agenda

    • Prune the losers

  • Repeat until

    • Agenda is empty

Required for Project #7


B b for minimization
B&B for Minimization

Minimization

True solution’s actual cost could be lower than BSSF.

actual

cost

B(s) islower

bound

BSSF

Low

High

Maximization ?


B b for maximization
B&B for Maximization

Minimization

True solution’s actual cost could be lower than BSSF.

actual

cost

B(s) islower

bound

BSSF

Low

High

Maximization

True solution’s actual cost could be higher than BSSF.

actual

cost

B(s) isupper

bound

BSSF

B(s) must always be optimistic.


Completeness and optimality
Completeness and Optimality

Important properties of search:

  • Completeness:

    • Is your search guaranteed to find a solution when one exists?

  • Optimality:

    • Does your search find the optimal (i.e., minimum or maximum cost) solution?

    • Does your search only prune sub-optimal states?


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