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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 kclique? 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 worstcase 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 NonDeterministic algorithm is whose w(n) is polynomial for all yes instances.
7 6.4 NP: A class of decision problems which have polynomially bounded nondeterministic 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 atleast 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 KClique T Graph G=(V,E) Kclique 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 NPComplete (NPC) A problem is NPcomplete 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 KClique, HC, TSP etc. are also NPC. Def. NPhard: is NPhard if for all problem 0 2 NP 0 : 12 8 Restriction of NPC Problems eg. coloring
2coloring 2 P 3coloring 2 NPC coloring of planar graph using 4 colors takes linear time. 3coloring of planar graphs is NPC 13 9 Approximation Algorithms
Polynomialtime algorithm for an NPComplete (or NPhard) 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 NPhard 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
NPComplete Theory, Application, Examples, etc Computer and Intractability: A guide to the Theory of NPCompleteness
by Michael R. Garey and David S. Johnson Publisher: W. H. Freeman 1979 18 ...
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This note was uploaded on 04/21/2009 for the course CS 94.503 taught by Professor K.d. during the Spring '09 term at UMass Lowell.
 Spring '09
 K.D.
 Algorithms

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