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# NP - 2 COLORING OPTIMIZATION PROBLEM Given G nd the minimum...

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Unformatted text preview: 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 DECISION PROBLEMS X (G) = CHROMATIC NUMBER OF G 1 2.1 Edge Coloring 1 CHROMATIC INDEX 3 2 =3 2 1 2 Chromatic Index = 5 1 1 2 3 1 5 1 2 3 = 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 CLIQUE DECISION CLIQUE SEARCH Given G and integer k, does G have a k-clique? Given G, nd a maximum clique of G. 4 SAT (SATISFIABILITY) C1 = x1 + x2 + x3 C2 = x; + x; 1 2 C3 = x; + x3 + x4 3 C4 = x2 + x; 4 Satisfying Truth Assignment 3 x1 = 1 x2 = 0 x3 = 0 x4 = 0 C1 & C2 & C3 & C4 C 1= x + x 1 1 __ __ C2 = x + x 1 3 C 3= x + x3 2 __ __ x 1 x 2 x 0 0 3 1 0 1 1 x x C 4= x + x2 1 __ __ C 5= x + x3 2 C6 = x + x 1 3 2 3 4 5 No satisfying truth assignment Variables: x1 x2 x3 Literals: x1 x; x2 x; x3 x; 1 2 3 Clauses: C1 C2 C5 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 P and NP P: class of decision problems which have polynomially bounded algorithms An algorithm whose worst-case time complexity w(n) p(n) n: input size p: a polynomial as n2 + 2n + 5 Polynomially Bounded 5 6 NON DETERMINISTIC ALGORITHM Polynomial Time Veri ability: We wish to bring together all such problems for which their candidate solutions can be veri ed 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 veri cation is polynomial. 6.2 De nitions ! Non Deterministic Algorithm will be polynomial time if Problem : Coloring Instance I: a graph G and number of colors k Domain of Pi Dpi 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 6.4 NP: A class of decision problems which have polynomially bounded non-deterministic algorithms. NP contains those problems whose guesses are polynomial time veri able. Coloring 2 NP Clique 2 NP SAT 2 NP guess: a truth assignment T: fx1 x2 xng ! f0 1g check: verify that each clause is satis ed for all c 2 fc1 c2 cmg c has at-least one true literal. time analysis guess ! O(n) check ! O(nm) 8 total O(nm) is bounded by a polynomial in length of input O(nm) Clique 2 NP Sorting 2 NP, but no guess needed 9 6.5 Polynomial Reduction Algorithm for Pi2 x an input for Pi1 T(x) an input for Pi2 Algo for Pi Yes No or Pi1 proportional to Pi2 T: 1) polynomial time transformation 2) answer preserving eg. SAT C1 C2 C3 . . . Ck x1 x2 . . . xn K-Clique T Graph G=(V,E) K-clique Algorithm Yes No or input for SAT T: V = f< 10 i> j is a literal in cig Clique size k = k number of clauses T: polynomial in k and n. E = f< i> < j >g j i 6= j & 6= ;g length O(kn logn) time for T: kn nodes and k2n2=2 edges total O(k2 n2 logn) If 1 2 SAT and 2 2 P then 1 2P 6.6 NP-Complete (NPC) A problem is NP-complete if other problem 2 NP 0 0 2 NP and for every If 2 NPC and 2 P then P = NP 11 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. are also NPC. Def. NP-hard: is NP-hard if for all problem 0 2 NP 0 : 12 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 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 9.1 Bin Packing Problem Given: n objects to be placed in bins of capacity L each. Object i requires li units of bin capacity. Objective: determine the minimum number of bins needed to accommodate all n objects. eg. Let L = 10, l1 = 5 l4 = 7 l2 = 6 l5 = 5 l3 = 3 l6 = 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 - 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 li then use FF. li+1 1 i n 4. Best Fit Decreasing (BFD) reorder objects as above and then use BF. 17 than 10 OPT + 2 bins. That by either FFD or BFD uses no more than 4 bins: Th. Packing generated by either FF or BF uses no more 11 9 OPT + 16 11 9 = 9 9 + 2 9 17 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 1979 18 ...
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