2 COLORING. Given G, nd the minimum number of colors to color G. Given graph G and positive integer k, is X(G) k?

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1 2 COLORING OPTIMIZATION PROBLEM Given G, nd the minimum number of colors to color G. (X(G)?) DECISION PROBLEM Given graph G and positive integer k, is X(G) k? EQUIVALENCE OF OPTIMAIZTION AND DE- CISION PROBLEMS X(G) = CHROMATIC NUMBER OF G 1

2 2.1 Edge Coloring CHROMATIC INDEX = Chromatic Index = = highest degree delta(g) VIZING'S THEOREM Chromatic index of a graph G is either (G) or (G) +1 3 CLIQUE OPTIMIZATION Given a graph, nd the size of the maximum clique in G. 2

3 CLIQUE DECISION Given G and integer k, does G have a k-clique? CLIQUE SEARCH Given G, nd a maximum clique of G. 4 SAT (SATISFIABILITY) C 1 = x 1 + x 2 + x 3 C 2 = x ; 1 + x ; 2 C 3 = x ; 3 + x 3 + x 4 C 4 = x 2 + x ; 4 Satisfying Truth Assignment x 1 = 1 x 2 = 0 x 3 = 0 x 4 = 0 3 C 1 & C 2 & C 3 & C 4

4 C = x + x 1 C = x + x C = x + x C = x + x C = x + x C = x + x x x x No satisfying truth assignment x x Variables: x 1 x 2 x 3 Literals: x 1 x ; 1 x 2 x ; 2 x 3 x ; 3 Clauses: C 1 C 2 C 5 SAT: Given n variables and m clauses over these variables, is there a satisfying truth assignment? 3SAT: All clauses have exactly 3 literals. 2SAT: Can be solved in polynomial time. 4

5 5 P and NP P: class of decision problems which have polynomially bounded algorithms Polynomially Bounded An algorithm whose worst-case time complexity w(n) p(n) n: input size p: a polynomial as n 2 + 2n + 5 5

6 6 NON DETERMINISTIC ALGORITHM Polynomial Time Veriability: We wish to bring together all such problems for which their candidate solutions can be veried to be correct solutions in polynomial time. 6.1 Graph Coloring 1. Guess a coloring C : V! f1 2 3 kg C(w) is the color of node w 2. Verify that the guess is correct for all fu vg 2 E c(u) 6= c(v) Step 1 O(n) Step 2 O(m) Total O(n + m) 6

7 ! Non Deterministic Algorithm will be polynomial time if verication is polynomial. 6.2 Denitions Problem : Coloring Instance I: a graph G and numberofcolorsk Domain of Pi D pi No Instance Yes Instances Y Pi No Instances D ; Y I 62 Y 6.3 Polynomially Bounded Non-Deterministic algorithm is whose w(n) is polynomial for all yes instances. 7

8 6.4 NP: A class of decision problems which have polynomially bounded non-deterministic algorithms. NP contains those problems whose guesses are polynomial time veriable. Coloring 2 NP Clique 2 NP SAT 2 NP guess: a truth assignment T: fx 1 x 2 x n g! f0 1g check: verify that each clause is satised for all c 2 fc 1 c 2 c m g c has at-least one true literal. 8 time analysis guess! O(n) check! O(nm) total O(nm) is bounded by a polynomial in length of

9 input O(nm) Clique 2 NP Sorting 2 NP, but no guess needed 9

10 6.5 Polynomial Reduction Algorithm for Pi2 x an input for Pi1 T(x) an input for Pi2 Algo for Pi Yes or No Pi1 proportional to Pi2 T: 1) polynomial time transformation 2) answer preserving eg. SAT K-Clique C C C... C k T Graph G=(V,E) K-clique Algorithm Yes No or x x... x 1 2 n input for SAT T: V = f< i> j isaliteralinc i g 10

11 E = f< i> < j>g j i 6= j & 6= ; g Clique size k = k number of clauses T: polynomial in k and n. SAT length O(kn logn) time for T: kn nodes and k 2 n 2 =2 edges total O(k 2 n 2 logn) If 1 2 and 2 2 P then 1 2 P 6.6 NP-Complete (NPC) A problem is NP-complete if 2 NP and for every other problem 0 2 NP 0 If 2 NPC and 2 P then P = NP 11

12 7 Cook's Theorem SAT 2 NPC It can be shown that i) graph coloring 2 NP ii) SAT coloring! graph coloring 2 NPC Similarly K-Clique, HC, TSP etc. arealsonpc. Def. NP-hard: is NP-hard if for all problem 0 2 NP 0 : 12

13 8 Restriction of NPC Problems eg. coloring 2-coloring 2 P 3-coloring 2 NPC coloring of planar graph using 4 colors takes linear time. 3-coloring of planar graphs is NPC 13

14 9 Approximation Algorithms Polynomial-time algorithm for an NP-Complete (or NP-hard) problem which do not guarantee the optimal solution, but would generally give one that is close to optimal 14

15 9.1 Bin Packing Problem Given: n objects to be placed in bins of capacity Leach. Object i requires l i units of bin capacity. Objective: determine the minimum number of bins needed to accommodate all n objects. eg. Let L = 10, l 1 = 5 l 4 = 7 l 2 = 6 l 5 = 5 l 3 = 3 l 6 = 4 Theorem Bin packing problem is NP complete when formulated as a decision problem. As an optimization problem bin packing is NP-hard Approximation Algorithm for Bin Packing: 1. First Fit (FF) 15

16 - Label bins as 1, 2, 3,... - Objects are considered for packing in the order 1, 2, 3,... -Pack object i in bin j where j is the least index such that bin j can contain object i. 2. Best Fit (BF) Same as FF, except that when object i is to be packed, nd out that bin which after accommodating object i will have the least amount of space left. 3. First Fit Decreasing (FFD) reorder objects so that l i l i+1 1 i n then use FF. 4. Best Fit Decreasing (BFD) reorder objects as above and then use BF. Th. Packing generated by either FF or BF uses no more than 17 OP T + 2 bins. 10 That by either FFD or BFD uses no more than 11 OP T bins: 16

17 =

18 10 REFERENCES NP-Complete Theory, Application, Examples, etc Computer and Intractability: A guide to the Theory of NP-Completeness by Michael R. Garey and David S. Johnson Publisher: W. H. Freeman