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Unformatted text preview: 15750 — Graduate Algorithms — Spring 2009Miller and Dinitz and TangwongsanAssignment 5 Solutions1Something about Laplacian(25 pts.)The Laplacian of a graph encodes a number of interesting properties of the graph. In this problem,we will examine some (random) facts about graph Laplacians.a)Positive Semidefiniteness.Ann×nreal symmetric matrixAispositive semidefiniteif forallx∈Rn,x>Ax≥0. We know a lot of nice properties of such matrices from the theory oflinear algebra. For instance, it is known that the following statements are equivalent: 1) alleigenvalues ofAare nonnegative, 2) for allx∈Rn,x>Ax≥0, and 3) there exists a matrixBsuch thatBTB=A. Indeed, an example of positive semidefinite matrices is the Laplacianof a undirected graph.Using any of the equivalent definitions above, prove that ifG= (V,E) is a simple undirectedgraph with weights given byw:V×V→R+∪ {}, thenL(G) is positive semidefinite. Weadopt the convention thatw(i,j) = 0 if (i,j)6∈E.Solution:LetA= [ai,j]n×n=L(G), andx∈Rnbe given. Consider that theith row ofAx,denoted by(Ax)i, is∑nj=1ai,j·xj, sox>Ax=nXi=1xi·(Ax)i=nXi=1nXj=1ai,jxixj.Now sinceA=L(G)is symmetric andai,i=∑j6=iw(i,j) =∑j6=iw(j,i), we havex>Ax=nXi=1ai,ix2i+ 2Xi<jai,jxixj=nXi=1Xj6=iw(i,j)x2i2Xi<jw(i,j)xixj=Xi<jw(i,j)(x2i2xixj+x2j) =Xi<jw(i,j)(xixj)2≥,which proves thatL(G)is positive semidefinite.b)MaxCut.In theMaxCutproblem, we are given a weighted undirected graphG=(V,E,w), and we are interested in finding a partition ofVintoSandS=V\Ssuchthat∑(i,j)∈S×Sw(i,j) is maximized. This corresponds to maximizing the number of edgescrossing the cut in the unweighted version. LetMaxCut(G) denote the value of the best cutonG. That is,MaxCut(G) := maxS⊆VX(i,j)∈S×Sw(i,j).Assume that all edge weights are nonnegative. Show thatMaxCut(G) =maxx∈{1,1}n14x>L(G)x.15750 HW 5 Solutions2Solution:For a cut(S,V\S), defineval(S,V\S) =X(i,j)∈S×(V\S)w(i,j).In part (a), we established thatx>L(G)x=∑i<jw(i,j)(xixj)2for anyx∈Rn. Specifically,ifx∈ {1,1}n, we havex>L(G)x= 4Xi<jxi6=xjw(i,j),which suggests the following natural bijection between cuts and±1vectors: theith componentof the vectorx∈ {1,1}nis1if and only ifibelongs to the setSin the cut(S,V\S). As such,ifxis the vector corresponding to(S,V\S), then14x>L(G)x=Xi<jxi6=xjw(i,j) =val(S,V\S)Since every cut has a corresponding±1vector and vice versa, we conclude1that MaxCut(G) =maxx∈{1,1}n14x>L(G)x.2RandomizedstConnectivity(25 pts.)Suppose we’re given an undirected graphG= (V,E) and two nodessandtinG, and we want todetermine if there is a path connectingsandt. As we have seen in class, this is easily accomplishedinO(V+E) using DFS or BFS. The space usage of such algorithms, however, isO(V)....
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This document was uploaded on 11/03/2009.
 Spring '09
 Algorithms

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