APPROXIMATION ALGORITHMS FOR GEOMETRIC PROBLEMS
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1 APPROXIMATION ALGORITHMS FOR GEOMETRIC PROBLEMS Subhas C. Nandy Advanced Computing and Microelectronics Unit Indian Statistical Institute Kolkata 70010, India.
2 Organization Introduction Approximation Algorithms for Intersection Graphs Approximation Algorithm for Art Gallery Problem Approximation Algorithm for Some Important Geometric Optimization Problems
3 Introduction Criteria for a good algorithm for a computatinal problem Fast execution time, Requirement of small extra space in addition to the space required for storing the input, and Easy to code.
4 Introduction Criteria for a good algorithm for a computatinal problem Fast execution time, Requirement of small extra space in addition to the space required for storing the input, and Easy to code. Measures of execution time Actual CPU time for running the program, Asymptotic running time.
5 Introduction Assymptotic Running time Worst case time complexity. Average case time complexity.
6 Introduction Assymptotic Running time Worst case time complexity. Average case time complexity. Worst case running time An upper bound on the running time of the algorithm for any input.
7 Introduction Assymptotic Running time Worst case time complexity. Average case time complexity. Worst case running time An upper bound on the running time of the algorithm for any input. Average case running time The average over the time required to run the algorithm for all possible inputs, where the input is assumed to follow some distribution (known in advance).
8 Introduction Assymptotic Running time Worst case time complexity. Average case time complexity. Worst case running time An upper bound on the running time of the algorithm for any input. Average case running time The average over the time required to run the algorithm for all possible inputs, where the input is assumed to follow some distribution (known in advance). The time complexity is measured as a function of the input size, or the output size, or both.
9 Tractibility versus Intractibility A problem is said to be tractable if there exists an algorithm for that problem whose worst case time complexity is a polynomial function in n (the input size). Otherwise, the problem is said to be intractable.
10 Tractibility versus Intractibility A problem is said to be tractable if there exists an algorithm for that problem whose worst case time complexity is a polynomial function in n (the input size). Otherwise, the problem is said to be intractable. Complexity Classes P: A decision problem that can be solved in polynomial time. NP: A decision problem that, for an input, can be verified in polynomial time.
11 NP-Completeness An important question Whether P = NP or P NP? NP-Complete Class: A class of decision problems in NP such that if one of them can be solved in polynomial time, all other problems can also be solved in polynomial time. NP NPC P
12 Proof for NP-Completeness For the given problem, Step 1: Design a non-deterministic polynomial time algorithm for the problem, and Step : Reduce a known NP-complete problem to the given problem in polynomial time.
13 NP-Complete Problems Circuit satisfiability, 3-SAT (satisfiability where the expression is in CNF, and each clause is of length at most 3), Clique decision problem, Vertex Cover decision, Travelling salesman problem, Knapsack problem, Bin packing Art gallery problem to name only a few.
14 Handling the NP-Hard Problems Brute-force algorithm Design a clever enumeration strategy, so that it guarantees the optimality of the solution, but no guarantee on the running time.
15 Handling the NP-Hard Problems Brute-force algorithm Design a clever enumeration strategy, so that it guarantees the optimality of the solution, but no guarantee on the running time. Heuristic Develop an intuitive algorithm, such that it guarantees to run in polynomial time, but no guarantee on the quality of the solution.
16 Approximation Algorithm Approximation Algorithm Guaranteed to run in polynomial time, and guaranteed to produce a high quality solution.
17 Approximation Algorithm Approximation Algorithm Guaranteed to run in polynomial time, and guaranteed to produce a high quality solution. But, what is a high quality solution?
18 Approximation Algorithm Approximation Algorithm Guaranteed to run in polynomial time, and guaranteed to produce a high quality solution. But, what is a high quality solution? Is the solution = OPT + α, or α OPT, or (1 + ɛ) OPT where, OPT the optimum solution, α a constant, ɛ a very small constant.
19 Approximation Algorithm Note: Designing approximation algorithm for a problem having polynomial time algorithm is also relevant if the time complexity of the problem is very high.
20 Approximation Algorithm Note: Designing approximation algorithm for a problem having polynomial time algorithm is also relevant if the time complexity of the problem is very high. Difficulty: One needs to prove that the solution is close to optimum, without knowing the optimum solution.
21 Approximation Algorithm Absolute Approximation Algorithm An algorithm A for a problem P that produces a solution Q such that Q OPT α, for a constant α.
22 Approximation Algorithm Absolute Approximation Algorithm An algorithm A for a problem P that produces a solution Q such that Q OPT α, for a constant α. Example: For a planar graph, testing whether the graph is 3 colorable is NP-complete. But, coloring the nodes of the graph with 5 colors needs O(n ) time. Thus, we have a -absolute approximation algorithm for the planar graph coloring problem
23 Approximation Algorithm f (n)-approximation Algorithm Let P be a problem of size n. An algorithm A for a problem P is said to be an f (n)-approximation algorithm if and only if for every instance I of P, A(I ) OPT /OPT f (n). It is assumed that OPT 0. Note: f (n) can be a constant also.
24 Approximation Algorithm f (n)-approximation Algorithm Let P be a problem of size n. An algorithm A for a problem P is said to be an f (n)-approximation algorithm if and only if for every instance I of P, A(I ) OPT /OPT f (n). It is assumed that OPT 0. Note: f (n) can be a constant also. Performance ratio: f (n) = A(I )/OPT in case P is a minimization problem OPT /A(I ) in case P is a maximization problem
25 Vertex Cover Problem Definition Given a graph G = (V, E), a vertex cover S is a set of nodes S V such that at least one end-point of every edge e E is incident on a member of S.
26 Vertex Cover Problem Definition Given a graph G = (V, E), a vertex cover S is a set of nodes S V such that at least one end-point of every edge e E is incident on a member of S. Vertex Cover Problem Given a graph G = (V, E), compute a vertex cover of minimum size.
27 Vertex Cover Problem Definition Given a graph G = (V, E), a vertex cover S is a set of nodes S V such that at least one end-point of every edge e E is incident on a member of S. Vertex Cover Problem Given a graph G = (V, E), compute a vertex cover of minimum size. Status: NP-hard.
28 A Greedy Algorithm Find a vertex having maximum degree. Choose it in the vertex cover Delete that vertex and all its adjacent edges from the graph Repeat the above three steps until all the edges of the graph are removed
29 A Greedy Algorithm Find a vertex having maximum degree. Choose it in the vertex cover Delete that vertex and all its adjacent edges from the graph Repeat the above three steps until all the edges of the graph are removed An example:
30 A Greedy Algorithm A generalized Example k! vertices of degree k k! vertices k of degree k k! k 1 vertices of degree k 1 k! vertices of degree SOL = k!( 1 k + 1 k k ) = k! log k
31 A Greedy Algorithm A generalized Example k! vertices of degree k k! vertices k of degree k k! k 1 vertices of degree k 1 k! vertices of degree SOL = k!( 1 k + 1 k k ) = k! log k Optimum Solution OPT = k!
32 Inference: The greedy algorithm is not a constant factor approximation algorithm. A Greedy Algorithm A generalized Example k! vertices of degree k k! vertices k of degree k k! k 1 vertices of degree k 1 k! vertices of degree SOL = k!( 1 k + 1 k k ) = k! log k Optimum Solution OPT = k!
33 Constant Factor Approximation Algorithm Step 1: Choose an edge e = (u, v) E Step : Update the vertex cover S by S {u, v} Step 3: Remove the vertices u and v from V. Remove all the edges incident at u and v from E. Step 3: Repeat Steps 1 and until all the edges in E are removed.
34 Constant Factor Approximation Algorithm Theorem Step 1: Choose an edge e = (u, v) E Step : Update the vertex cover S by S {u, v} Step 3: Remove the vertices u and v from V. Remove all the edges incident at u and v from E. Step 3: Repeat Steps 1 and until all the edges in E are removed. Approximation factor =. Time Complexity = O( E ).
35 Minimum Weight Steiner Tree Problem Problem: Given: An edge weighted undirected graph G = (V, E), with vertices classified into two groups, namely terminal A and non-terminal B, where A B = V. v 1 v 3 v u u u 4 u v 4 v 5
36 Minimum Weight Steiner Tree Problem Problem: Given: An edge weighted undirected graph G = (V, E), with vertices classified into two groups, namely terminal A and non-terminal B, where A B = V. v 1 v 3 v u u u 4 u v v 4 v 5 v Objective: To compute a tree T that contains all the vertices of A, and the total weight of the edges of the tree T is minimum. v 3 u u 4 3 v 4 v 5
37 Minimum Weight Steiner Tree Problem Problem: Given: An edge weighted undirected graph G = (V, E), with vertices classified into two groups, namely terminal A and non-terminal B, where A B = V. v 1 v 3 v u u u 4 u v v 4 v 5 v Objective: To compute a tree T that contains all the vertices of A, and the total weight of the edges of the tree T is minimum. v 3 u u 4 3 v 4 v 5 Status The decision version of the problem is NP-complete.
38 Approximation Algorithm for Steiner Tree Problem Input: A graph G = (V, E, w) and a terminal set L V. Output: A Steiner Tree
39 Approximation Algorithm for Steiner Tree Problem Input: A graph G = (V, E, w) and a terminal set L V. Output: A Steiner Tree Step 1 Consider the metric closure G L = (L, E L, ω) on the terminal set L. [G L - A complete graph with node set L. For each u, v L cost of the edge (u, v) = shortest path cost from u to v in G.
40 Approximation Algorithm for Steiner Tree Problem Input: A graph G = (V, E, w) and a terminal set L V. Output: A Steiner Tree Step 1 Consider the metric closure G L = (L, E L, ω) on the terminal set L. [G L - A complete graph with node set L. For each u, v L cost of the edge (u, v) = shortest path cost from u to v in G. Step find a minimum spanning tree (MST) T L on G L.
41 Approximation Algorithm for Steiner Tree Problem Input: A graph G = (V, E, w) and a terminal set L V. Output: A Steiner Tree Step 1 Consider the metric closure G L = (L, E L, ω) on the terminal set L. [G L - A complete graph with node set L. For each u, v L cost of the edge (u, v) = shortest path cost from u to v in G. Step find a minimum spanning tree (MST) T L on G L. Step 3 set T =.
42 Approximation Algorithm for Steiner Tree Problem Input: A graph G = (V, E, w) and a terminal set L V. Output: A Steiner Tree Step 1 Consider the metric closure G L = (L, E L, ω) on the terminal set L. [G L - A complete graph with node set L. For each u, v L cost of the edge (u, v) = shortest path cost from u to v in G. Step find a minimum spanning tree (MST) T L on G L. Step 3 set T =. Step 4 for each edge (u, v) E L do Step 4a find a shortest path Π from u to v in G. Step 4b if Π contains less than two vertices in T, then add Π to T. else let x and y be first and last vertices on Π that are also in T. add sub-paths u to x and y to v in T.
43 Approximation Algorithm for Steiner Tree Problem Input: A graph G = (V, E, w) and a terminal set L V. Output: A Steiner Tree Step 1 Consider the metric closure G L = (L, E L, ω) on the terminal set L. [G L - A complete graph with node set L. For each u, v L cost of the edge (u, v) = shortest path cost from u to v in G. Step find a minimum spanning tree (MST) T L on G L. Step 3 set T =. Step 4 for each edge (u, v) E L do Step 4a find a shortest path Π from u to v in G. Step 4b if Π contains less than two vertices in T, then add Π to T. else let x and y be first and last vertices on Π that are also in T. add sub-paths u to x and y to v in T. Step 5 output T
44 Demonstration of the Algorithm v1 v u3 9 u1 1 5 u4 v3 4 u 5 3 v5 v4 Given Graph
45 Demonstration of the Algorithm u3 v1 9 u1 v 1 5 u4 v v 5 u v3 v4 Given Graph v5 v3 7 v4 9 v5 Metric Closure G L = (L, E L, ω)
46 Demonstration of the Algorithm u3 v1 9 u1 v 1 5 u4 v v 5 v v 5 u v3 v4 Given Graph v5 v3 7 v4 9 v5 Metric Closure G L = (L, E L, ω) v3 v4 v5 MST on G L = (L, E L, ω)
47 Demonstration of the Algorithm u3 v1 9 u1 v 1 5 u4 v v 5 v v 5 u v3 v4 Given Graph v5 v3 7 v4 9 v5 Metric Closure G L = (L, E L, ω) v3 v4 v5 MST on G L = (L, E L, ω) v1 u1 1 u 3 v4 Expansion of edge (v 1, v 4 )
48 Demonstration of the Algorithm u3 v1 9 u1 v 1 5 u4 v v 5 v v 5 u v3 v4 Given Graph v5 v3 7 v4 9 v5 Metric Closure G L = (L, E L, ω) v3 v4 v5 MST on G L = (L, E L, ω) v1 v1 v u1 1 u1 1 3 u v3 4 u 3 v4 Expansion of edge (v 1, v 4 ) v4 Expansion of edge (v, v 3 )
49 Demonstration of the Algorithm u3 v1 9 u1 v 1 5 u4 v v 5 v v 5 u v3 v4 Given Graph v5 v3 7 v4 9 v5 Metric Closure G L = (L, E L, ω) v3 v4 v5 MST on G L = (L, E L, ω) v1 v1 v v1 v u1 1 3 u v3 u1 4 1 u 3 v3 u u 3 v5 v4 Expansion of edge (v 1, v 4 ) v4 Expansion of edge (v, v 3 ) v4 Expansion of edge (v, v 5 )
50 Analysis Theorem MST based Steiner Tree algorithm produces a -factor approximation solution for the Steiner Minimal Tree problem on an undirected weighted graph in O( V L ) time.
51 Analysis Theorem MST based Steiner Tree algorithm produces a -factor approximation solution for the Steiner Minimal Tree problem on an undirected weighted graph in O( V L ) time. Proof: Let T be a Steiner Tree of minimum cost, and T be the Steiner Tree obtained as output of our algorithm. Since at most all shortest paths of T L are inserted in T, we have ω(t ) ω(t L )
52 Analysis of Approximation Factor Important points:
53 Analysis of Approximation Factor Important points: 1. If X is an Eulerian Tour by doubling each edge of T, then ω(x ) = ω(t )
54 Analysis of Approximation Factor Important points: 1. If X is an Eulerian Tour by doubling each edge of T, then ω(x ) = ω(t ). If Y is the TSP Tour (Minimum Hamiltonian Circuit) on G L, then ω(x ) ω(y ) ω(t L )
55 Analysis of Approximation Factor Important points: 1. If X is an Eulerian Tour by doubling each edge of T, then ω(x ) = ω(t ). If Y is the TSP Tour (Minimum Hamiltonian Circuit) on G L, then ω(x ) ω(y ) ω(t L ) 3. ω(t ) ω(t L )
56 Analysis of Approximation Factor Important points: 1. If X is an Eulerian Tour by doubling each edge of T, then ω(x ) = ω(t ). If Y is the TSP Tour (Minimum Hamiltonian Circuit) on G L, then ω(x ) ω(y ) ω(t L ) 3. ω(t ) ω(t L ) Thus, we have the final result w(t ) ω(t )
57 Analysis of Approximation Factor Important points: 1. If X is an Eulerian Tour by doubling each edge of T, then ω(x ) = ω(t ). If Y is the TSP Tour (Minimum Hamiltonian Circuit) on G L, then ω(x ) ω(y ) ω(t L ) 3. ω(t ) ω(t L ) Thus, we have the final result w(t ) ω(t ) Time complexity: O( V L ) Follows from the time complexity of computing G L.
58 Demonstration of an worst case example Given Graph Our Solution Optimum Solution
59 Improving Approximation Factor Can we have an Eulerian Tour without doubling the edges: YES How Replace Step of the earlier algorithm by the following three steps: Step.1: Identify the odd degree vertices V in the minimum spanning tree T L. Step.: Find a set of edges M that corresponds to the minimum maximal matching of the graph V. Step.3: Compute Eulerian tour in the graph G = (V, T L M).
60 Demonstration u3 v1 9 u1 v 1 5 u4 v v 5 v v 5 u v3 v4 Given Graph v5 v3 7 v4 9 v5 Metric Closure G L = (L, E L, ω) v3 v4 v5 MST on G L = (L, E L, ω)
61 Demonstration u3 v1 9 u1 v 1 5 u4 v v 5 v v 5 u v3 v4 Given Graph v5 v3 7 v4 9 v5 Metric Closure G L = (L, E L, ω) v3 v4 v5 MST on G L = (L, E L, ω) v1 4 v v3 7 Minimum maximal matching M in the graph induced by odd degree vertices v4 9 v5
62 Demonstration u3 v1 9 u1 v 1 5 u4 v v 5 v v 5 u v3 v4 Given Graph v5 v3 7 v4 9 v5 Metric Closure G L = (L, E L, ω) v3 v4 v5 MST on G L = (L, E L, ω) v1 4 v v1 4 v v3 7 Minimum maximal matching M in the graph induced by odd degree vertices v4 9 v5 v3 7 v4 v5 The graph G = (V, T L M)
63 Analysis Theorem Thus we have an 3 approximation of the Steiner tree problem.
64 Analysis Theorem Thus we have an 3 approximation of the Steiner tree problem. Proof: τ: Optimal TSP tour in G L. τ Tour in G obtained by short-cutting τ. cost(τ ) cost(τ) (* by triangles inequality *) τ consists of two perfect matchings on V each consisting of alternate edges of τ Total cost in the eulerian tour due to the edges in τ OPT, and Total cost in the eulerian tour due to the edges of M OPT
65 Available Results on Steiner Tree approximation algorithm Best known approximation result for edge weighted graph: 1.39 (* An LP based algorithm *) J. Byrka, F Grandoni, L Rothvoβ and L. Sanita, STOC 010. for a complete graph of edge weight 1 or : 1.5 M. Bern and P. Plassman, IPL, Vol. 3, 199. for node weighted graph: where k is the number of terminal nodes. P. Klein and R. Ravi, Technical Report, for node weighted planar graph: E. D. Demaine, M. T. Hajiaghayi and P. N. Klein. log k O(1)
66 Set Cover Problem Input: A finite set X, and a set of subsets F = {S 1, S,... S k } of X such that every element of X belongs to at least one subset of F. Thus, X = S i F S i
67 Set Cover Problem Input: A finite set X, and a set of subsets F = {S 1, S,... S k } of X such that every element of X belongs to at least one subset of F. Thus, X = S i F S i Objective: Choose minimum number of subsets C from F to cover all the elements of X, i.e., X = S C S.
68 Set Cover Problem Input: A finite set X, and a set of subsets F = {S 1, S,... S k } of X such that every element of X belongs to at least one subset of F. Thus, X = S i F S i Objective: Choose minimum number of subsets C from F to cover all the elements of X, i.e., X = S C S. Usefulness Many practical problems can be formulated as a set cover problem. Example: Art gallery Problem, Hitting Set Problem, etc.
69 Algorithm Greedy-Set-Cover(X, F) U X (* U contains uncovered elements *) C (* Set cover *) while U do select an S F that maximizes S U U U S C C S endwhile end. S 1 S 5 S 6 S 4 S 3 S
70 Algorithm Greedy-Set-Cover(X, F) U X (* U contains uncovered elements *) C (* Set cover *) while U do select an S F that maximizes S U U U S C C S endwhile end. S 1 S 5 S 6 S 4 S 3 S Here the smallest set cover is {S 3, S 4, S 5 }
71 Algorithm Greedy-Set-Cover(X, F) U X (* U contains uncovered elements *) C (* Set cover *) while U do select an S F that maximizes S U U U S C C S endwhile end. S 1 S 5 S 6 S 4 S 3 S Here the smallest set cover is {S 3, S 4, S 5 } But Greedy algorithm produces {S 4, S 1, S 3, S 5 }.
72 Approximation Factor Theorem Greedy set cover has a ratio bound H(max{ S : S F})
73 Approximation Factor Theorem Greedy set cover has a ratio bound H(max{ S : S F}) c x = cost allocated to element x X. Each element is assigned cost only once. If x is covered for the first time by S i, then c x = 1 S i (S 1 +S +...+S i 1 ) Let the algorithm finds a solution C. The cost of the solution is C = x X c x. Since the optimum solution C also covers X, we have C = x X c x S C x S c x
74 Approximation Factor An Useful Result x S c x H( S ), where H(n) = n i=1 1 i = ln n Thus C S C H( S ) C H(max{ S : S F})
75 Approximation Factor Proof of the result: x S c x H( S ) Consider a set S F. Let u 0 = S, and u i = S (S 1 S... S i ). (* the number of elements in S that remains uncovered after S 1, S,... S i are already selected by the algorithm.*)
76 Approximation Factor Proof of the result: x S c x H( S ) Consider a set S F. Let u 0 = S, and u i = S (S 1 S... S i ). (* the number of elements in S that remains uncovered after S 1, S,... S i are already selected by the algorithm.*) Let k be the least index such that u k = 0 (* all elements in S are covered by S 1, S,..., S k *)
77 Approximation Factor Proof of the result: x S c x H( S ) Consider a set S F. Let u 0 = S, and u i = S (S 1 S... S i ). (* the number of elements in S that remains uncovered after S 1, S,... S i are already selected by the algorithm.*) Let k be the least index such that u k = 0 (* all elements in S are covered by S 1, S,..., S k *) Thus, u i 1 u i, and (u i 1 u i ) elements of S are covered for the first time by S i. Thus, x S c x = k i=1 (u i 1 u i ) 1 S i (S 1 S... S i 1 )
78 Approximation Factor In the i-th step, S i is the better choice than S by our greedy algorithm Thus, S i (S 1 S S i 1 ) S (S 1 S S i 1 ) u i 1 x S c x k i=1 (u i 1 u i ) 1 u i 1 Another Important Result For any integer a and b, where a < b, we have H(b) H(a) = b i=a+1 1 i (b a) 1 b Thus, x S c x k i=1 (H(u i 1) H(u i )) = H(u 0 ) H(u k ) = H(u 0 ) = H( S )
79 Polynomial Time Approximation Scheme (PTAS) Definition An algorithm A for a problem P is said to be a polynomial time approximation scheme (PTAS) if given an instance Π of P, the algorithm A returns an ɛ-approximation result in time polynomial on the length of the input Π, where (the polynomial function depends on ɛ) Note that, this function may be of the form 1 ɛ n or n 1 ɛ.
80 Polynomial Time Approximation Scheme (PTAS) Definition An algorithm A for a problem P is said to be a polynomial time approximation scheme (PTAS) if given an instance Π of P, the algorithm A returns an ɛ-approximation result in time polynomial on the length of the input Π, where (the polynomial function depends on ɛ) Note that, this function may be of the form 1 ɛ n or n 1 ɛ. Definition An PTAS algorithm A for a problem P is said to be a fully polynomial time approximation scheme (FPTAS) if given an instance Π of P, the algorithm A returns an ɛ-approximation result in time polynomial in the length of the input Π, and 1 ɛ.
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