NP-Complete Problems (Fun part) - PowerPoint PPT Presentation

About This Presentation
Title:

NP-Complete Problems (Fun part)

Description:

... distance between vi and vj,, find a shortest circuit that visits each city ... the cities are points in the plane and the distance between two points is ... – PowerPoint PPT presentation

Number of Views:34
Avg rating:3.0/5.0
Slides: 62
Provided by: csCit
Category:

less

Transcript and Presenter's Notes

Title: NP-Complete Problems (Fun part)


1
NP-Complete Problems (Fun part)
  • Polynomial time vs exponential time
  • Polynomial O(nk), where n is the input size
    (e.g., number of nodes in a graph, the length of
    strings , etc) of our problem and k is a constant
    (e.g., k1, 2, 3, etc).
  • Exponential time 2n or nn.
  • n 2, 10, 20, 30
  • 2n 4 1024 1 million 1000 million
  • Suppose our computer can solve a problem of size
    k (i.e., compute 2k operations) in a
    hour/week/month. If the new computer is 10 times
    faster than ours, then the new computer can solve
    the problem of size k3. The improvement is very
    little.

2
Story
  • All algorithms we have studied so far are
    polynomial time algorithms.
  • Facts people have not yet found any polynomial
    time algorithms for some famous problems, (e.g.,
    Hamilton Circuit, longest simple path, Steiner
    trees).
  • Question Do there exist polynomial time
    algorithms for those famous problems.
  • Answer No body knows.

3
Story
  • Research topic Prove that polynomial time
    algorithms do not exist for those famous
    problems, e.g., Hamilton circuit problem.
  • You can get Turing award if you can give the
    proof.
  • In order to answer the above question, people
    define two classes of problems, P class and NP
    class.
  • To answer if P?NP, a rich area, NP-completeness
    theory is developed.

4
Class P and Class NP
  • Class P contains those problems that are solvable
    in polynomial time.
  • They are problems that can be solved in O(nk)
    time, where n is the input size and k is a
    constant.
  • Class NP consists of those problem that are
    verifiable in polynomial time.
  • What we mean here is that if we were somehow
    given a solution, then we can verify that the
    solution is correct in time polynomial in the
    size of the input to the problem.
  • Example Hamilton Circuit given (v1, v2, , vn),
    we can test if (vi, v i1) is an edge in G and
    (vn, v1) is an edge in G.

5
Class P and Class NP
  • Based on definitions, P?NP.
  • If we can design an polynomial time algorithm for
    problem A, then problem A is in P.
  • However, if we have not been able to design a
    polynomial time algorithm for problem A, then
    there are two possibilities
  • polynomial time algorithm does not exists for
    problem A and
  • we are too dump.

6
NP-Complete
  • A problem is NP-complete if it is in NP and is
    as hard as any problem in NP.
  • If an NP-complete problem can be solved
    polynomial time, then any problem in class NP
    can be solved in polynomial time.
  • NPC problems are the hardest in class NP.
  • The first NPC problem is Satisfiability probelm
  • Proved by Cook in 1971 and obtains the Turing
    Award for this work
  • Definition of Satisfiability problem Input A
    boolean formula f(x1, x2, xn), where xi is a
    boolean variable (either 0 or 1). Question

7
Boolean formula
  • A boolean formula f(x1, x2, xn), where xi are
    boolean variables (either 0 or 1), contains
    boolean variables and boolean operations AND, OR
    and NOT .
  • Clause variables and their negations are
    connected with OR operation, e.g., (x1 OR NOTx2
    OR x5)
  • Conjunctive normal form of boolean formula
  • contains m clauses connected with AND
    operation.
  • Example
  • (x1 OR NOT x2) AND (x1 OR NOT x3 OR x6) AND
    (x2 OR x6) AND (NOT x3 OR x5).
  • Here we have four clauses.

8
Satisfiability problem
  • Input conjunctive normal form with n variables,
    x1, x2, , xn.
  • Problem find an assignment of x1, x2, , xn
    (setting each xi to be 0 or 1) such that the
    formula is true (satisfied).
  • Example conjunctive normal form is
  • (x1 OR NOTx2) AND (NOT x1 OR x3).
  • The formula is true for assignment
  • x11, x20, x31.

9
Satisfiability problem
  • Theorem Satisfiability problem is NP-complete.
  • It is the first NP-complete problem.
  • S. A. Cook in 1971
  • Won Turing prize for this work.
  • Significance
  • If Satisfiability problem can be solved in
    polynomial time, then ALL problems in class NP
    can be solved in polynomial time.
  • If you want to solve P?NP, then you should work
    on NPC problems such as satisfiability problem.
  • We can use the first NPC problem, Satisfiability
    problem, to show that other problems are also
    NP-complete.

10
How to show that a problem is NPC?
  • To show that problem A is NP-complete, we can
  • First find a problem B that has been proved to be
    NP-complete.
  • Show that if Problem A can be solved in
    polynomial time, then problem B can also be
    solved in polynomial time.
  • Remarks Since A NPC problem is the hardest in
    class NP, problem A is also the hardest
  • Example We know that Hamilton circuit problem
    is NP-complete. (See a research paper.) We want
    to show that TSP problem is NP-complete.
  • .

11
Hamilton circuit and Traveling Salesman problem
  • Hamilton circuit a circle uses every vertex of
    the graph exactly once except for the last
    vertex, which duplicates the first vertex.
  • It was shown to be NP-complete.
  • Traveling Salesman problem (TSP) Input Vv1,
    v2, ..., vn be a set of nodes (cities) in a
    graph and d(vi, vj) the distance between vi
    and vj,, find a shortest circuit that visits
    each city exactly once.
  • (weighted version of Hamilton circuit)
  • Theorem 1 TSP is NP-complete.

12
Proof of Theorem 1.
  • Proof Given any graph G(V, E) (input of
    Hamilton circuit problem), we can construct an
    input for the TSP problem as follows
  • For each edge (vi, vj) in E, we assign a
    distance d(vi, vj) 1.
  • The new problem becomes that is there a shortest
    circuit that visits each city (node) exactly
    once?
  • The length of the shortest circuit is V since
    we need V edges to form the circuit and the
    length of each edge is 1.
  • Note The reduction (transformation) is from
    Hamilton circuit (known to be NPC) to the new
    problem TSP (that we want to prove it to be NPC).

13
Some basic NP-complete problems
  • 3-Satisfiability Each clause contains at most
    three variavles or their negations.
  • Vertex Cover Given a graph G(V, E), find a
    subset V of V such that for each edge (u, v) in
    E, at least one of u and v is in V and the size
    of V is minimized.
  • Hamilton Circuit (definition was given before)
  • History Satisfiability?3-Satisfiability?vertex
    cover?Hamilton circuit.
  • Those proofs are very hard.
  • Karp proves the first few NPC problems and
    obtains Turing award.

14
Approximation Algorithms
  • Concepts
  • Steiner Minimum Tree
  • TSP
  • Knapsack
  • Vertex Cover

15
Concepts of Approximation Algorithms
  • Optimization Problem
  • An optimization problem ? consists of the
    following three parts
  • A set D? of instances
  • for each instance I?D?, a set S?(I) of candidate
    solutions for I and
  • a function C that assigns to each solution
    s(I)?S?(I) a positive number C(s(I)), called the
    solution value (or the cost of the solution).

16
  • Maximization
  • An optimal solution is a candidate solution
    s(I)?S?(I) such that C(s(I))C(s'(I)) for any
    s'(I)?S?(I).
  • Minimization
  • An optimal solution is a candidate solution
    s(I)?S?(I) such that C(s(I))C(s'(I)) for any
    s'(I)?S?(I).

17
Approximation Algorithm
  • An algorithm A is an approximation algorithm for
    ? if, given any instance I?D?, it finds a
    candidate solution s(I)?S?(I).
  • How good an approximation algorithm is?
  • We use performance ratio to measure the
    "goodness" of an approximation algorithm.

18
Performance ratio
  • For minimization problem, the performance ratio
    of algorithm A is defined as a number r such that
    for any instance I of the problem,
  • where OPT(I) is the value of the optimal solution
    for instance I and A(I) is the value of the
    solution returned by algorithm A on instance I.

19
Performance ratio
  • For maximization problem, the performance ratio
    of algorithm A is defined as a number r such that
    for any instance I of the problem,
  • OPT(I)
  • A(I)
  • is at most r, where OPT(I) is the value of
    the optimal solution for instance I and A(I) is
    the value of the solution returned by algorithm A
    on instance I.

20
Simplified Knapsack Problem
  • Given a finite set U of items, a size s(u)?Z, a
    capacity Bmaxs(u)u?U, find a subset U'?U such
    that and such that the above summation is as
    large as possible. (It is NP-hard.)

21
Ratio-2 Algorithm
  • Sort u's based on s(u)'s in increasing order.
  • Select the smallest remaining u until no more u
    can be added.
  • Compare the total value of selected items with
    the item with the largest size, and select the
    larger one.
  • Theorem The algorithm has performance ratio 2.

22
Proof
  • Case 1 the total of selected items 0.5B (got
    it!)
  • Case 2 the total of selected items lt 0.5B.
  • In this case, the size of the largest remaining
    item gt0.5B. (Otherwise, we can add it in.)
  • Selecting the largest item gives ratio-2.

23
The 0-1 Knapsack problem
  • The 0-1 knapsack problem
  • N items, where the i-th item is worth vi dollars
    and weight wi pounds.
  • vi and wi are integers.
  • A thief can carry at most W (integer) pounds.
  • How to take as valuable a load as possible.
  • An item cannot be divided into pieces.
  • The fractional knapsack problem
  • The same setting, but the thief can take
    fractions of items.

24
Ratio-2 Algorithm
  • Delete the items I with wigtW.
  • Sort items in decreasing order based on vi/wi.
  • Select the first k items item 1, item 2, , item
    k such that
  • w1w2, wk ?W and w1w2, wk w
    k1gtW.
  • 4. Compare vk1 with v1v2vk and select the
    larger one.
  • Theorem The algorithm has performance ratio 2.

25
Proof of ratio 2
  • C(opt) the cost of optimum solution
  • C(fopt) the optimal cost of the fractional
    version.
  • C(opt)?C(fopt).
  • v1v2vk v k1gt C(fopt).
  • So, either v1v2vk gt0.5 C(fopt)?c(opt)
  • or v k1 gt0.5 C(fopt)?c(opt).
  • Since the algorithm choose the larger one from
    v1v2vk and v k1
  • We know that the cost of the solution obtained by
    the algorithm is at least 0.5 C(fopt)?c(opt).

26
Steiner Minimum Tree
  • Steiner minimum tree in the plane
  • Input a set of points R (regular points) in the
    plane.
  • Output a tree with smallest weight which
    contains all the nodes in R.
  • Weight weight on an edge connecting two points
    (x1,y1) and (x2,y2) in the plane is defined as
    the Euclidean distance

27
  • Example Dark points are regular points.

28
Triangle inequality
  • Key for our approximation algorithm.
  • For any three points in the plane, we have
  • dist(a, c ) dist(a, b) dist(b, c).
  • Examples

c
5
4
a
b
3
29
Approximation algorithm(Steiner minimum tree in
the plane)
  • Compute a minimum spanning tree for R as the
    approximation solution for the Steiner minimum
    tree problem.
  • How good the algorithm is? (in terms of the
    quality of the solutions)
  • Theorem The performance ratio of the
    approximation algorithm is 2.

30
Proof
  • We want to show that for any instance (input) I,
    A(I)/OPT(I) r (r1), where A(I) is the cost of
    the solution obtained from our spanning tree
    algorithm, and OPT(I) is the cost of an optimal
    solution.

31
  • Assume that T is the optimal solution for
    instance I. Consider a traversal of T.
  • Each edge in T is visited at most twice. Thus,
    the total weight of the traversal is at most
    twice of the weight of T, i.e.,
  • w(traversal)2w(T)2OPT(I). .........(1)

32
  • Based on the traversal, we can get a spanning
    tree ST as follows (Directly connect two nodes
    in R based on the visited order of the traversal.)

From triangle inequality, w(ST)w(traversal)
2OPT(I). ..........(2)
33
  • Inequality(2) says that the cost of the spanning
    tree ST is less than or equal to the cost of an
    optimal solution.
  • So, if we can compute ST, then we can get a
    solution with cost2OPT(I).(Great! But finding
    ST may also be very hard, since ST is obtained
    from the optimal solution T, which we do not
    know.)
  • We can find a minimum spanning tree MT for R in
    polynomial time.
  • By definition of MST, w(MT) w(ST) 2OPT(I).
  • Therefore, the performance ratio is 2.

34
Story
  • The method was known long time ago. The
    performance ratio was conjectured to be
  • Du and Hwang proved that the conjecture is true.

35
Graph Steiner minimum tree
  • Input a graph G(V,E), a weight w(e) for each
    e?E, and a subset R?V.
  • Output a tree with minimum weight which contains
    all the nodes in R.
  • The nodes in R are called regular points. Note
    that, the Steiner minimum tree could contain some
    nodes in V-R and the nodes in V-R are called
    Steiner points.

36
  • Example Let G be shown in Figure a. Ra,b,c.
    The Steiner minimum tree T(a,d),(b,d),(c,d)
    which is shown in Figure b.
  • Theorem Graph Steiner minimum tree problem is
    NP-complete.

37
Approximation algorithm(Graph Steiner minimum
tree)
  1. For each pair of nodes u and v in R, compute the
    shortest path from u to v and assign the cost of
    the shortest path from u to v as the length of
    edge (u, v). (a complete graph is given)
  2. Compute a minimum spanning tree for the modified
    complete graph.
  3. Include the nodes in the shortest paths used.

38
  • Theorem The performance ratio of this algorithm
    is 2.
  • Proof
  • We only have to prove that Triangle Inequality
    holds. If
  • dist(a,c)gtdist(a,b)dist(b,c) ......(3)
  • then we modify the path from a to c like
  • a?b?c
  • Thus, (3) is impossible.

39
  • Example II-1

g
a
e
c
d
f
b
The given graph
40
  • Example II-2

e-c-g /7
g /3
e /4
a
c
d
f/ 2
e /3
b
f-c-g/5
Modified complete graph
41
  • Example II-3

g/3
a
c
d
f /2
e /3
b
The minimum spanning tree
42
  • Example II-4

g
2
1
2
a
e
c
d
1
1
f
b
1
The approximate Steiner tree
43
Approximation Algorithm for TSP with triangle
inequality
  • Assumption the triangle inequality holds. That
    is, d (a, c) d (a, b) d (b, c).
  • This condition is met, for example, whenever the
    cities are points in the plane and the distance
    between two points is the Euclidean distance.

44
  • The above assumption is very strong, since we
    assume that the underline graph is a complete
    graph. Example II does not satisfy the
    assumption. If we want to use the cost of the
    shortest path between two vertices as the cost,
    we have to repeat some nodes.
  • Theorem TSP with triangle inequality is also
    NP-hard.

45
Ratio 2 Algorithm
  • Algorithm A
  • Compute a minimum spanning tree algorithm (Figure
    a)
  • Visit all the cities by traversing twice around
    the tree. This visits some cities more than once.
    (Figure b)
  • Shortcut the tour by going directly to the next
    unvisited city. (Figure c)

46
  • Example

47
Proof of Ratio 2
  1. The cost of a minimum spanning tree cost(t), is
    not greater than opt(TSP), the cost of an optimal
    TSP. (Why? n-1 edges in a spanning tree. n edges
    in TSP. Delete one edge in TSP, we get a spanning
    tree. Minimum spanning tree has the smallest
    cost.)
  2. The cost of the TSP produced by our algorithm is
    less than 2cost(T) and thus is less than
    2opt(TSP).

48
  • Another description of Algorithm A
  • Compute a minimum spanning tree algorithm.
  • Convert the spanning tree into an Eulerian graph
    by doubling each edge of the tree.
  • Find an Eulerian tour of the resulting graph.
  • Convert the Eulerian tour into traveling salesman
    tour by using shortcuts.
  • Review An Eulerian graph is a graph in which
    every vertex has even degree.

49
Ratio 1.5 Algorithm
  • Modify Step 2 as follows
  • Let V'a1,a2,...,a2k be the set of vertices
    with odd degree. (There are even number of odd
    degree vertices)
  • Find a minimum weight matching for V' in the
    induced graph.
  • The cost of the minimum match is at most 0.5
    times of that of TSP. (Why?)

50
  • Adding the match to the spanning tree, we get an
    Eulerian graph.
  • The cost of the Euler circuit is less than
    1.5opt(TSP).
  • The shortcut (i.e., the TSP) produced in step 4
    has a cost not greater than that of the Euler
    circuit produced in step 3.
  • So, the TSP produced by our algorithm has a
    cost1.5opt(TSP).

51
Non-approximatility for TSP without triangle
inequality
  • Theorem For any xgt0, TSP problem without
    triangle inequality cannot be approximated within
    ratio x.

52
  • Example
  • Solid edges are the original edges in the graph
    G.
  • Dashed edges are added edges.
  • OPTn-1 if a Hamilton circuit exists. Otherwise,
    OPTgt2xn.
  • Since ratiox, if our solutionlt2xn, Hamilton
    circuit exists.

53
Proof
  • The basic idea
  • Show that if there is an approximation algorithm
    A with ratio x for TSP problem without triangle
    inequality, then we can use the approximation
    algorithm A for TSP to solve the Hamilton
    Circuit problem, which is NP-hard.
  • Since NP-hard problems do not have any polynomial
    time algorithms, the ratio x approximation
    algorithm can not exist.

54
  1. Given an instance of Hamilton Circuit problem
    G(V,E), where there are n nodes in V, we assign
    weight 1 to each edge in E. (That is, the
    distance between any pair of vertices in G is 1
    if there is an edge connecting them.)
  2. Add edges of cost 2xn to G such that the
    resulting graph G'(V,E?E') is a complete graph.

55
  • 3. Now, we use the approximation algorithm A for
    TSP to find a TSP.
  • 4. Now, we want to show that if the cost of the
    solution obtained from A is not greater than
    1xn, then there is a Hamilton circuit in G,
    otherwise, there is no Hamilton Circuit in G.

56
  • If A gives a solution of cost not greater than
    1xn, then no added edge (weight 2xn) is used.
  • Thus, G has a Hamilton circuit.
  • If the cost of the solution given by A is greater
    than 1xn, then at least one added edge is used.
  • If there is a circuit in G, the cost of an
    optimal solution is n. The approximation
    algorithm A should give a solution with cost
    xn.
  • There is no Hamilton circuit in G.

57
Vertex Cover Problem
  • Given a graph G(V, E), find V'?V such that for
    each edge (u, v)?E at least one of u and v
    belongs to V.
  • V' is called vertex cover.
  • The problem is NP-hard.

58
Ratio-2 Approximation Algorithm
  • Based on maximal matching
  • A match M is a set of edges in E such that no two
    edges in M incident upon the same node in V.
  • Maximal matching A matching E' is maximal if
    every remaining edge in E-E' has an endpoint in
    common with some member of E'.

59
  • Computation of maximal match
  • Add edge to E' until no longer possible.
  • Facts
  • The 2E' nodes in E' form a cover.
  • The optimal vertex cover contains at least E'
    nodes.
  • From 1 and 2, the ratio is 2.

60
Assignment
  • Question 3. A graph is given as follows.

61
  • Use the ratio 1.5 algorithm to compute a TSP.
    (When do shortcutting, if there is no edge, add
    an edge with cost that is equal to the cost of
    the corresponding path.)
Write a Comment
User Comments (0)
About PowerShow.com