Notes

Skiena Chapter 7

This chapter covers the basic ideas of searching for solutions for combinatorial/permutation problems.

Three principal techniques covered:

• Random sampling (Monte Carlo method)
• Local search
• Simulated annealing

Combinatorial problems and solutions

Backtracking:

• Backtracking as example of combinatorial search (e.g., with N Queens problem you are searching among a space of possible configurations for a solution that satisfies the given constraints)
• Basic skeleton given for a backtracking method:
backtrack_dfs(A,k):
if A = (a1, a2, ..., ak) is solution:
process solution
else:
k = k + 1
compute Sk
while Sk != null set, do:
ak = element of Sk
Sk = Sk - ak
backtrack_dfs(A,k)


Or, to express in a more terse way:

• Base case: found a solution
• Recursive case:
• For each choice in a set of choices:
• Make a choice
• Explore the consequences
• Unmake the choice
• Move on to the next choice

A few notes on his program:

• new array each time, inside each method, so set of possible solutions is not passed around - copy is created so free to modify/not restore
• Stubs used fror method calls like processing solution, making move, unmaking move - this enables application of the same function to different problems

When solving, the main question is, how do you represent a set?

Example:

• Checking membership of an element in a set
• Can use bit vector - an array of booleans - indicating whether an element is a member of a particular set or not
• The order in which sets are constructed depends on the set S construction procedure, called above

Generating Permutations

Example: Construct all possible subsets

• Each element doubles the number of subsets that can be created
• Number of subsets with 1 element in the set: 2 subsets (empty/full)
• Number of subsets with 2 elements in the set: 4 subsets
• Number of subsets with n elements in the set: 2^n subsets

Constructing permutations:

• Permutations do not allow for duplicates
• If we fix a_1, we have n-1 candidates for the second position, n-2 for the third, etc. Total is:

${\displaystyle n!=\prod _{j=1}^{n}j}$

• utilize this to construct solution representations:
• element i of an array can be any eleent that has not already appearedd
• complicated bit vector solution - basically, it says, who's in this permutation - then it adds items that aren't already in the solution to the new permutations

More Examples

Shortest path:

• Again, a bit complicated with the logic details, but basically:
• Start at node S
• Move to next node ni if it is on the graph and not part of hte solution already
• Repeat

Sudoku puzzle solution:

• backtracking solution
• elements of the solution are the open squares, each of which has coordinates (i,j) and a set of possible digits that can go in that square 1-9
• We know the set of feasible digits are those that do not appear in row i or column j or the local 3x3 sector
• Soltion vector: one integer per position
• To store coordinate (i,j), use a separate array of point (x,y) structs
• Also have a board struct - stores a matrix of board contents as int[][], and a free count (open nuber of squares), and a move array of points
• to pick the next candidate:
• Pick the next open square to fill
• identify candidate solutions for that square
• IMPORTANT: These two routines have a HUGE impact on the runtime!!!
• Then, given a coordinate x and y and a board, if possible, increase the number of andidates
• Make a move funtion
• Unmake a move function (both of these modify the move array)
• Solution check method - are there any open squares on the board?
• process solution: print board, set flag FINISHED to true

Two BIG improvements can be made here:

• Improving the selection of the next square - by carefully picking the next square to solve, we can reduce the search space and tackle the easiest problems first
• Improving the possible values that we are considering

Improving the next square selection procedure: two options

• Local count - disallows candidates that appear in row i, row j (this is the "plain vanilla" default approach)
• Look ahead - check if there are any other squares where making our choice would lead to ZERO candidates; if so, eliminate choice as a possible solution

Using the worst of both to solve a sudoku puzzle can take up to 1M function calls, while the best of both can solve with 50 or so.

Heuristic search

3 techniques covered here are:

• Random sampling (Monte Carlo method)
• Local search
• Simulated annealing

These three techniques have some common components that are required:

• Solution space representation - deciding how to represent the problem (e.g., when solving Traveling Salesperson Problem, deciding which nodes to visit, which have been visited)
• Cost function - a way to rank the quality of each solution

Random sampling

Random sampling/Monte Carlo:

• Solution space must be uniformly sampled
• This can be subtle/tricky
• See selction 14.4-14.7 in Skiena on generating permutations and combinations, etc.
• Basic traveling salesperson problem randomized solution:
for i in Nsamples:
generate random solution
calculate cost of solution
if this costs less than best cost:
keep this solution


When does random sampling work well?

• High proportion of acceptable solutions (e.g., prime numbers)
• No coherence to solution space (no way to gauge closeness to a solution)

This solution approach works terribly on the traveling salesperson problem, which has a sparse solution space.

Implementing Random Sampling

There are subtleties to how random sampling should be implemented - how the solution space should be sampled.

For example: suppose we have a graph with vertices, and we are selecting random pairs for vertex swaps:

For a set of vertices ${\displaystyle {1,\dots ,n}}$ we are selecting randomly from the ${\displaystyle {\binom {n}{2}}}$ possible unordered pairs

Solution

• We cannot say,
i = rand_int(1, n-1)
j = rand_int(i+1,n)


This introduces bias in the random generator!

Possibility of selecting a random pair that is arbitrary, say ${\displaystyle (p,q)}$, is given by:

${\displaystyle P(p)={\dfrac {1}{n-1}}\qquad P(q)={\dfrac {1}{n-1}}\qquad P(p,q)={\dfrac {1}{(n-1)^{2}}}}$

The last term is the probability of getting the pair (p,q).

But now consider the random pair ${\displaystyle (n-1,n)}$: in this case the probability of j being n is 100%.

${\displaystyle P(n-1)={\dfrac {1}{n-1}}\qquad P(n)=1\qquad P(n-1,n)={\dfrac {1}{n-1}}}$

which is different!

The real solution is to make both random selections unbiased, and throw out duplicate selections:

do
i = random(1,n)
j = random(1,n)
while(i==j)
swap(i,j)


Local Search

Local search methods:

• sometimes called greedy methods
• search for locally optimal solutions
• This can cause problems later, if the locally optimal solution is not a part of the globally optimal solution.
• Similar to a perturbation method, looking for small, limited-size local steps that will lead to a better solution
• Can think of it like binary search in a convex space - a step in one direction or the other will always take you toward your solution

When to use local search, and why:

• Convex problem spaces lend themselves well to local search for solutions
• Cost of incremental changes to an existing solution can also be much cheaper relative to looking for brand new solutions
• If we have a large search space and a limited time budget, local search is the way to go

Simulated Annealing

Inspiration for simulated annealing from thermodynamics

Think about the search through solution space as a thermodynamic process, trying to reach minimum energy

The higher the temperature, the more random motions there are, the easier it is to transition to another state

Simulated annealing sets an initial temperature and a cooling schedule, and evaluates the probability of a random jump to a nearby solution as a function of the temperature and the cost difference between the two possible states.

More formally: we have a transition probability ${\displaystyle P(s_{i},s_{j},T)}$ from state i to state j at some given temperature, which is given by an exponential decay function:

${\displaystyle P(s_{i},s_{j},T)=\exp {\left({\dfrac {s_{i}-s_{j}}{k_{B}T}}\right)}}$

where ${\displaystyle k_{B}}$ is the Boltzmann constant, which is the gas constant divided by Avagadro's number:

${\displaystyle k_{B}={\dfrac {R}{N_{A}}}}$

In the case of simulated annealing, we replace the difference ${\displaystyle s_{i}-s_{j}}$ with the cost function C(s) associated with each of the two states, and we replace the constant k with an arbitrary constant.

${\displaystyle P(s_{i},s_{j},T)=\exp {\left({\dfrac {C(s_{i})-C(s_{j})}{kT}}\right)}}$

Initially we start at a high temperature, where large jumps are more likely; as time goes on, we gradually decrease the temperature, allowing the solution to settle into a low-cost state.

For applying simulated annealing to combinatorial optimization, we use the cost function as the energy function.

Pseudocode:

def simulated_annealing():
create initial solution
initialize temperature
while solution still changing:
for i in number of iterations:
generate random transition s to s_j
if C(s) > C(s_j):
s = s_j
else if exp( (C(s)-C(s_j))/(k*T) ) > random(0,1):
s = s_j
reduce temperature
return s


Cooling schedule:

• Regulates likelihood of selecting (locally) bad solutions
• Depends on a couple of parameters:
• initial temperature
• temperature decrement
• number of iterations inside T step
• acceptance criteria for state
• stopping criteria

Example problems where simulated annealing works well:

• Maximum cut (finding partitioning of graphs that cuts the most edges)
• Independent set (largest subgraphs in G with no edges containing both endpoints in S)
• Circuit board placement - minimizing aspect ratio, minimizing total wire length (rectangular modules r1, r2, ..., rn with dimensions Hi x Li and wires wij connecting the modules)

Skiena End of Chapter Questions

Skiena Chapter 7 Problem 14 - String Permutations - write a function to find all permutations of the letters in a particular string. For example, "HELLO".

Examples and Applications

Project Euler provides fantastic examples of combinatoric problems that often have astronomical problem spaces that can be cleverly traversed much faster with a few tricks.

Project Euler Problem 501 is an excellent example of this.

Project Euler Problem 502 is another excellent combinatoric search question.

Related Pages

Notes from Knuth's Art of Computer Programming related to combinatorics and permutations:

Combinatorics Problems

Collected combinatorics problems across this wiki.

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