Combinatorial Optimization

1
Combinatorial Optimization

• Chapter 8
• Algorithms Complexity

2
Solving combinatorial problems

• Of course many of the solution methods for
  combinatorial problems are very time consuming of
  you have to do them by hand.
• Without computers combinatorial optimization
  wouldnt have advanced so much.
• Computers solve problems by algorithms Precise
  and universally understood sequences of
  instructions that solve any instance of
  rigorously defined computational problems.
• The mathematical equivalent of an algorithm is a
  Turing machine.

3
Time bounds

• The elementary steps of algorithms are assumed to
  take unit time.
• Elementary operations are ,-.,-,,v,....
• The number of steps of an algorithm depend on the
  input.
• The complexity of an algorithm for an input of
  size s, is the maximum number of steps it takes,
  where the maximum is over all inputs over size s.

4
Definitions for time bounds

• Definition 8.1 Let f(n), g(n) be functions from
  the positive integers to the positive reals.
• We write f(n) O(g(n)) if there exists a
  constant c gt 0 such that, for large enough n,
  f(n) c g(n).
• We write f(n) T(g(n)) if there exists a
  constants c,c gt 0 such that, for large enough n,
  cg(n) f(n) cg(n).

5
The size of an instance

• In a combinatorial problem, the input describes
  combinatorial objects, such as integers, sets,
  graphs, et cetera.
• The formal description of the input is called the
  encoding, for an instance as a string on a Turing
  machine tape.

6
Encoding

• The size of an encoding of an integer B, where 2i
  B 2i1 is i1. (where i is also an integer.)
• Encoding a graph
• adjacency matrix aij, where aij 1 if (vi,vj) e
  E, 0 otherwise. O(V2).
• adjacency list, For each node v e V, a list of
  nodes adjacent to it. O(E).
• Sometimes an O(E2) algorithm is better than a
  O(V3) algorithm.

7
Complexity of the max flow algorithm

• The scan procedure can examine each arc (x,y) at
  most twice, once when scanning for x, and once
  when scanning for y. Scan takes at most O(A)
  time.
• When t is labelled, the path needs to be traced
  back to do the increments. The path can consist
  of at most V edges. This each iteration takes
  time O(AV) O(A). When the capacities are
  integral, every flow increment must be integral.
  Hence the flow increases by at least 1 per step.
  But then the number of steps is bounded from
  above by Sx,yeAb(x,y). Thus the complexity is
  O(Sx,yeAb(x,y) A).
• Question Does this upper bound imply that the
  algorithm is polynomial in the size of the input?
• Answer No, input is roughly O(log maxx,yeA
  b(x,y) A).
• We shall see in the next chapter that this upper
  bound is attainable

8
Polynomial algorithms

• Definition An algorithm is considered a
  polynomial algorithm if its computation time is
  bounded from above by a polynomial in the size of
  the input. Thus, let fA(n) denote the computation
  time of algorithm A, as a function of the input
  size n, then A is called polynomial if
  fA(n)O(p(n)) for some polynomial function p(n).
• Definition Algorithms which are not polynomial
  algorithms are exponential algorithms.

9
Linear programming

• The Simplex Algorithm (1947) is not a polynomial
  algorithm
• Khachyans Ellipsoid method (1979) is a
  polynomial algorithm
• Karmarkars method (1983) is a polynomial
  algorithm

10
Rest of chapter

• Skip 8.6, 8.7

11
Exercises

• 4a
• 4c
• 4d
• 4e
• 4f