Backtracking

Backtracking
Sum of Subsets
and
Knapsack

Backtracking 2
Backtracking
• Two versions of backtracking algorithms
– Solution needs only to be feasible (satisfy
problem’s constraints)
• sum of subsets
– Solution needs also to be optimal
– knapsack

Backtracking 3
The backtracking method
• A given problem has a set of constraints and
possibly an objective function
• The solution optimizes an objective function,
and/or is feasible.
• We can represent the solution space for the
problem using a state space tree
– The root of the tree represents 0 choices,
– Nodes at depth 1 represent first choice
– Nodes at depth 2 represent the second choice,
etc.
– In this tree a path from a root to a leaf
represents a candidate solution

Backtracking 4
Sum of subsets
• Problem: Given n positive integers w1, ... wn and a
positive integer S. Find all subsets of w1, ... wn that
sum to S.
• Example:
n=3, S=6, and w1=2, w2=4, w3=6
• Solutions:
{2,4} and {6}

Backtracking 5
Sum of subsets
• We will assume a binary state space tree.
• The nodes at depth 1 are for including (yes, no)
item 1, the nodes at depth 2 are for item 2, etc.
• The left branch includes wi, and the right branch
excludes wi.
• The nodes contain the sum of the weights
included so far

Backtracking 6
Sum of subset Problem:
State SpaceTree for 3 items
w1 = 2, w2 = 4, w3 = 6 and S = 6
i1
i2
i3
yes no
0
0
0
0
2
2
2
6
612 8
4
410 6
yes
yes
no
no
no
nonono
The sum of the included integers is stored at the node.
yes
yes yesyes

Backtracking 7
A Depth First Search solution
• Problems can be solved using depth first search
of the (implicit) state space tree.
• Each node will save its depth and its (possibly
partial) current solution
• DFS can check whether node v is a leaf.
– If it is a leaf then check if the current solution
satisfies the constraints
– Code can be added to find the optimal solution

Backtracking 8
A DFS solution
• Such a DFS algorithm will be very slow.
• It does not check for every solution state (node)
whether a solution has been reached, or whether
a partial solution can lead to a feasible solution
• Is there a more efficient solution?

Backtracking 9
Backtracking
• Definition: We call a node nonpromising if it
cannot lead to a feasible (or optimal) solution,
otherwise it is promising
• Main idea: Backtracking consists of doing a
DFS of the state space tree, checking whether
each node is promising and if the node is
nonpromising backtracking to the node’s parent

Backtracking 10
Backtracking
• The state space tree consisting of expanded
nodes only is called the pruned state space tree
• The following slide shows the pruned state space
tree for the sum of subsets example
• There are only 15 nodes in the pruned state space
tree
• The full state space tree has 31 nodes

Backtracking 11
A Pruned State Space Tree (find all solutions)
w1 = 3, w2 = 4, w3 = 5, w4 = 6; S = 13
0
0
0
3
3
3
7
712 8
4
49
5
3
4 40
0
0
5 50 0 0
06
13 7
Sum of subsets problem

Backtracking 12
Backtracking algorithm
void checknode (node v) {
node u
if (promising ( v ))
if (aSolutionAt( v ))
write the solution
else //expand the node
for ( each child u of v )
checknode ( u )

Backtracking 13
Checknode
• Checknode uses the functions:
– promising(v) which checks that the partial solution
represented by v can lead to the required solution
– aSolutionAt(v) which checks whether the partial
solution represented by node v solves the
problem.

Backtracking 14
Sum of subsets – when is a node “promising”?
• Consider a node at depth i
• weightSoFar = weight of node, i.e., sum of numbers
included in partial solution node represents
• totalPossibleLeft = weight of the remaining items i+1
to n (for a node at depth i)
• A node at depth i is non-promising
if (weightSoFar + totalPossibleLeft < S )
or (weightSoFar + w[i+1] > S )
• To be able to use this “promising function” the wi
must be sorted in non-decreasing order

Backtracking 15
A Pruned State Space Tree
w1 = 3, w2 = 4, w3 = 5, w4 = 6; S = 13
0
0
0
3
3
3
7
712 8
4
49
5
3
4 40
0
0
5 50 0 0
06
13 7 - backtrack
1
2
3
4 5
6 7
8
109
11
12
15
14
13
Nodes numbered in “call” order

Backtracking 16
sumOfSubsets ( i, weightSoFar, totalPossibleLeft )
1) if (promising ( i )) //may lead to solution
2) then if ( weightSoFar == S )
3) then print include[ 1 ] to include[ i ] //found solution
4) else //expand the node when weightSoFar < S
5) include [ i + 1 ] = "yes” //try including
6) sumOfSubsets ( i + 1,
weightSoFar + w[i + 1],
totalPossibleLeft - w[i + 1] )
7) include [ i + 1 ] = "no” //try excluding
8) sumOfSubsets ( i + 1, weightSoFar ,
totalPossibleLeft - w[i + 1] )
boolean promising (i )
1) return ( weightSoFar + totalPossibleLeft ≥ S) &&
( weightSoFar == S || weightSoFar + w[i + 1] ≤ S )
Prints all solutions!
Initial call sumOfSubsets(0, 0, )∑=
n
i
iw
1

Backtracking 17
Backtracking for optimization problems
• To deal with optimization we compute:
best - value of best solution achieved so far
value(v) - the value of the solution at node v
– Modify promising(v)
• Best is initialized to a value that is equal to a candidate
solution or worse than any possible solution.
• Best is updated to value(v) if the solution at v is “better”
• By “better” we mean:
– larger in the case of maximization and
– smaller in the case of minimization

Backtracking 18
Modifying promising
• A node is promising when
– it is feasible and can lead to a feasible solution
and
– “there is a chance that a better solution than
best can be achieved by expanding it”
• Otherwise it is nonpromising
• A bound on the best solution that can be
achieved by expanding the node is computed and
compared to best
• If the bound > best for maximization, (< best for
minimization) the node is promising
How is
it determined?

Backtracking 19
Modifying promising for
Maximization Problems
• For a maximization problem the bound is an upper
bound,
– the largest possible solution that can be
achieved by expanding the node is less or
equal to the upper bound
• If upper bound > best so far, a better solution
may be found by expanding the node and the
feasible node is promising

Backtracking 20
Modifying promising for
Minimization Problems
• For minimization the bound is a lower bound,
– the smallest possible solution that can be
achieved by expanding the node is less or
equal to the lower bound
• If lower bound < best a better solution may be
found and the feasible node is promising

Backtracking 21
Template for backtracking in the case of
optimization problems.
Procedure checknode (node v ) {
node u ;
if ( value(v) is better than best )
best = value(v);
if (promising (v) )
for (each child u of v)
checknode (u );
}
• best is the best value so
far and is initialized to a
value that is equal or
worse than any possible
solution.
• value(v) is the value of
the solution at the node.

Backtracking 22
Notation for knapsack
• We use maxprofit to denote best
• profit(v) to denote value(v)

Backtracking 23
The state space tree for knapsack
• Each node v will include 3 values:
– profit (v) = sum of profits of all items included in
the knapsack (on a path from root to v)
– weight (v)= the sum of the weights of all items
included in the knapsack (on a path from root to v)
– upperBound(v). upperBound(v) is greater or equal
to the maximum benefit that can be found by
expanding the whole subtree of the state space
tree with root v.
• The nodes are numbered in the order of expansion

Backtracking 24
Promising nodes for 0/1 knapsack
• Node v is promising if weight(v) < C, and
upperBound(v)>maxprofit
• Otherwise it is not promising
• Note that when weight(v) = C, or maxprofit =
upperbound(v) the node is non promising

Backtracking 25
Main idea for upper bound
• Theorem: The optimal profit for 0/1 knapsack ≤
optimal profit for KWF
• Proof:
• Clearly the optimal solution to 0/1 knapsack is a
possible solution to KWF. So the optimal profit of KWF
is greater or equal to that of 0/1 knapsack
• Main idea: KWF can be used for computing the upper
bounds

Backtracking 26
Computing the upper bound for 0/1 knapsack
• Given node v at depth i.
• UpperBound(v) =
KWF2(i+1, weight(v), profit(v), w, p, C, n)
• KWF2 requires that the items be ordered by non
increasing pi / wi, so if we arrange the items in this
order before applying the backtracking algorithm,
KWF2 will pick the remaining items in the required
order.

Backtracking 27
KWF2(i, weight, profit, w, p, C, n)
1. bound = profit
2. for j=i to n
3. x[j]=0 //initialize variables to 0
4. while (weight<C)&& (i<=n) //not “full”and more items
5. if weight+w[i]<=C //room for next item
6. x[i]=1 //item i is added to knapsack
7. weight=weight+w[i]; bound = bound +p[i]
8. else
9. x[i]=(C-weight)/w[i] //fraction of i added to knapsack
10. weight=C; bound = bound + p[i]*x[i]
11. i=i+1 // next item
12. return bound
• KWF2 is in O(n) (assuming items sorted before applying
backtracking)

Backtracking 28
C++ version
• The arrays w, p, include and bestset have size
n+1.
• Location 0 is not used
• include contains the current solution
• bestset the best solution so far

Backtracking 29
Before calling Knapsack
numbest=0; //number of items considered
maxprofit=0;
knapsack(0,0,0);
cout << maxprofit;
for (i=1; i<= numbest; i++)
cout << bestset[i]; //the best solution
• maxprofit is initialized to $0, which is the worst profit
that can be achieved with positive pis
• In Knapsack - before determining if node v is
promising, maxprofit and bestset are updated

Backtracking 30
knapsack(i, profit, weight)
if ( weight <= C && profit > maxprofit)
// save better solution
maxprofit=profit //save new profit
numbest= i; bestset = include//save solution
if promising(i)
include [i + 1] = “ yes”
knapsack(i+1, profit+p[i+1], weight+ w[i+1])
include[i+1] = “no”
knapsack(i+1,profit,weight)

Backtracking 31
Promising(i)
promising(i)
//Cannot get a solution by expanding node
if weight >= C return false
//Compute upper bound
bound = KWF2(i+1, weight, profit, w, p, C, n)
return (bound>maxprofit)

Backtracking 32
Example from Neapolitan & Naimipour
• Suppose n = 4, W = 16, and we have the following:
i pi wi pi / wi
1 $40 2 $20
2 $30 5 $6
3 $50 10 $5
4 $10 5 $2
• Note the the items are in the correct order needed by
KWF

Backtracking 33
maxprofit = $0 (n = 4, C = 16 )
Node 1
a) profit = $ 0
weight = 0
b) bound = profit + p1 + p2 + (C - 7 ) * p3 / w3
= $0 + $40 + $30 + (16 -7) X $50/10 =$115
c) 1 is promising because its weight =0 < C = 16
and its bound $115 > 0 the value of maxprofit.
The calculation for node 1

Backtracking 34
Item 1 with profit $40 and weight 2 is included
maxprofit = $40
a) profit = $40
weight = 2
b) bound = profit + p2 + (C - 7) X p3 / w3
= $40 + $30 + (16 -7) X $50/10 =$115
c) 2 is promising because its weight =2 < C = 16
and its bound $115 > $40 the value of maxprofit.

Backtracking 35
Item 1 with profit $40 and weight 2 is not included
At this point maxprofit=$90 and is not changed
a) profit = $0
weight = 0
b) bound = profit + p2 + p3+ (C - 15) X p4 / w4
= $0 + $30 +$50+ (16 -15) X $10/5 =$82
c) 13 is nonpromising because its bound $82 < $90
the value of maxprofit.

Backtracking 36
1
13
$0
0
$115
$40
2
$115
$0
0
$82
Item 1 [$40, 2]
B
82<90Item 2 [$30, 5]
$70
7
$115
$120
17
2
3
4
F
17>16
Item 3 [$50, 10]
Item 4 [$10, 5]
$40
2
$98
8
$70
7
$80
5
$90
12
$98
9
$40
2
$50
12
B
50<90$80
12
$80
6
$70
7
$70
7
N N
$100
17
10 $90
12
$90
11
maxprofit = 90
profit
weight
bound
Example
F - not feasible
N - not optimal
B- cannot lead to
best solution
Optimal
maxprofit =0
maxprofit =40
maxprofit =70
maxprofit =80
F
17>16

Backtracking

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