Balanced Check Partitions

Balanced Check Partitions - PCPs and Inapproxiability CIS...

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PCPs and Inapproxiability CIS 6930 November 5, 2009 Lecture Balanced Connected Partitions Lecturer: Dr. My T. Thai Scribe: Yilin Shen 1 Problem Definition Definition 1 (Max Balanced Connected q -Partition ( BCP q )) Given a connected graph G = ( V,E ) with a weight function w : V Z + and q 2 be a positive integer. For X V , let w ( X ) denote the sum of the weights of the vertices in X . The BCP q problem on G is to find a q -partition P = ( V 1 ,V 2 ,...,V q ) of V such that G [ V i ] is connected (1 i q ) and P maximizes min { w ( V i ) : 1 i q } . Definition 2 (Exact Cover by 3-Sets (X3C)) Given a set X with | X | = 3 q and a fam- ily C of 3-element subsets of X , | C | = 3 q , where each element of X appears in exactly 3 sets of C , decide whether C contains an exact cover for X , that is, a subcollection C C such that each element of X occurs in exactly one member of C . 2 Related works The simpler unweighted version of BCP q is the special case of BCP q in which all vertices have weight 1 (denoted by 1- BCP q ): For every q 2 , the problem 1- BCP q is NP -hard (even for bipartite graphs); When the input graph has a higher connectivity, we have: Let G be a q - connected graph with n vertices, q 2 , and let n 1 ,n 2 ,...,n q be positive natural numbers such that n 1 + n 2 + ... + n q = n . Then G has a connected q -partition ( V 1 ,V 2 ,...,V q ) such that | V i | = n i for i = 1 , 2 ,...,q . The more general weighted case: BCP q is polynomially solvable only for ladders and for trees; BCP q restricted to grids G m × n with n 3 is already NP -hard; BCP 2 is NP -hard on connected graphs, bipartite graphs, and graphs with at least one block containing (log | V | ) articulation points and complete graphs. Balanced Connected Partitions-1

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3 Hardness of Approximation Theorem 3 The decision version of BCP 2 is NP -complete in the strong sense for 2- connected graphs. Proof: We will show the reduction from X 3 C problem to BCP 2 problem The Construction: Given an instance ( X,C ) of X 3 C , let G = ( V,E ) be the graph with vertex set V = X C a,b and edge set E = u 3 q j =1 [ C j x i | x i C j C j a C j b ] . Clearly, G can be constructed in polynomial time in the size of ( X,C ) . It is also not difficult to see that G is 2-connected. w(a)=9q^2+2q w(b)=3q w(C_j)=1 w(x_i)=3q Figure 1 : Example: Graph obtained by the reduction for the instance ( X,C ) , where C = { C 1 ,C 2 ,...,C 6 } , C 1 = { x 3 ,x 4 ,x 5 } , C 2 = { x 1 ,x 2 ,x 3 } , C 3 = { x 1 ,x 3 ,x 6 } , C 4 = { x 1 ,x 4 ,x 6 } , C 5 = { x 2 ,x 5 ,x 6 } and C 6 = { x 2 ,x 4 ,x 5 } Define a weight function w : V Z + as follows: w ( a ) = 9 q 2 + 2 q ; w ( b ) = 3 q ; w ( x i ) = 3 q for i = 1 ,..., 3 q ; and w ( C j ) = 1 for j = 1 ,..., 3 q . Note that w ( V ) = 2(9 q 2 + 4 q ) . We will prove that
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This note was uploaded on 05/20/2011 for the course CIS 6930 taught by Professor Staff during the Spring '08 term at University of Florida.

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Balanced Check Partitions - PCPs and Inapproxiability CIS...

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