hw5solns - 15-750 — Graduate Algorithms — Spring...

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Unformatted text preview: 15-750 — 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 Semi-definiteness.Ann×nreal symmetric matrixAispositive semi-definiteif 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 non-negative, 2) for allx∈Rn,x>Ax≥0, and 3) there exists a matrixBsuch thatBTB=A. Indeed, an example of positive semi-definite 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 semi-definite. 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 thei-th 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)x2i-2Xi<jw(i,j)xixj=Xi<jw(i,j)(x2i-2xixj+x2j) =Xi<jw(i,j)(xi-xj)2≥,which proves thatL(G)is positive semi-definite.b)Max-Cut.In theMax-Cutproblem, 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. LetMax-Cut(G) denote the value of the best cutonG. That is,Max-Cut(G) := maxS⊆VX(i,j)∈S×Sw(i,j).Assume that all edge weights are non-negative. Show thatMax-Cut(G) =maxx∈{-1,1}n14x>L(G)x.15-750 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)(xi-xj)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: thei-th 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 Max-Cut(G) =maxx∈{-1,1}n14x>L(G)x.2Randomizeds-tConnectivity(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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hw5solns - 15-750 — Graduate Algorithms — Spring...

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