CSC21 - Metric Embeddings Lecture 8: Sparsest Cut and Embedding to Notes taken by Nilesh ansal revised by Hamed Hatami Summary: Sparsest Cut (SC) is an important problem ith various applications, including those in VLSI layout design, packet routing in distributed netorking, and clustering ut since sparsest cut is NP-hard, e need to find approximate algorithms Solution to uniform Multi Commodity Flo (MCF) problem using Linear Programming (LP) can be used to approximate SC by in polynomial time We then discuss, Poincaré inequalities for metrics, hich can be used to find loer bounds for distortion for embedding a metric to This discussion is further continued, and e define -gonal inequalities and hypermetrics 1 Sparsest Cut Definition 11 Flux of a graph is defined as,!#" $ %& ')(+*-,/ '0 1# *2 3 6 2 here 89;:<2 The cut hich minimizes the flux is knon as the minimum quotient separator Computing minimum quotient separator is NP-complete Definition 12 Sparsity of a graph >?@ is defined as, " $ %& ')(+* 6 2 CD The cut hich minimizes the sparsity is knon as the sparsest cut (SC), hich is NP-hard to compute Remark 13 Sparsity and flux of a graph are closely related! "FE "FEHG! " I Lecture Notes for a course given by vner Magen, Dept of Computer Sciecne, University of Toronto 1
_ 11 pproximate Solutions to Sparsest Cut Lemma 1 Solving sparsest cut is equivalent to solving minimize ONP RSUTV subject to ON * WRUTDX [ ZY Proof If \ ' represents the metric corresponding to the cut, e can rite, 6 >2 V ONP \ ' WRSTD ^ \ ' WRSTD H_ and therefore, $ %& 6 >2 ONP \ ' RSTD ' V H$ %& ' \ ' RSUTV H_ Recall that C metrics are linear combinations of cut metrics, and therefore cut metrics are extreme rays of C From the lemma proved in the last lecture, ratio in the equation above is minimized at one of the extreme rays of the cone Therefore, 6 >2 ONP KfL $`%& ' Ca H$ %& b Nced g_ Since this is invariant to scaling, ithout loss of generality, e can assume that the sum X C to If e relax our requirement from is a metric by adding i#j>klnm triangle inequalities, e can solve this problemhy in polynomial time using Linear Programming (LP) The relaxed LP to solve is minimize subject to ONPo WRSTD KfL ON * WRUTD?pX RSTDrqgs, and WRSTD tta@ru RSTD E WRSvx TSvy (1) Theorem 1 There exists an W t a? approximate algorithm for the sparsest cut problem Theorem 1 is due to [LLR9 but it originally appeared in [LR88 2
Y Proof Equation 1 can be solved using LP to get a solution (hich is a metric) Using the ourgain s theorem [ou8, e can find an embedding )z [y}/ @ y k of {z to Y ith distortion No can be expressed as a linear combination of cut metrics ' N < ' \ ', here ˆ is a collection of cuts Since is in the cone of cut metrics, From ourgain s theorem, ut, Therefore, $ %t& ' N ONP ONP \ ' RSTD \ ' RSTD $ %t& ' N ONP ONP \ ' RSTD \ ' RSTD H_ -z bs $`%& ONP is metric z E ONP? E ONP H_ E ONP VŠKfL ŠKfL KfL -z KfL z KfL Dz z E ONP \ ' $`%& ' K \ ' WRSTD WRUTD _ E?-$`%& ' _ ONP \ ' WRSTD \ ' WRSTD 12 Non-Uniform Sparsest Cut What e have discussed till no can be generalized to the case of non-uniform sparsest cut, here e have to minimize 0 KfL \ ' RSTD +Œ KfL \ ' WRSTD For the problem of uniform sparsest cut, X if RST Œ X alays 2 Multi Commodity Flo Problem, and 0 otherise; and In a Multi Commodity Flo (MCF) problem, there are ŽqX commodities, each ith its on source K, sink K and demand K The aim is to simultaneously route all the commodities from their source to sink in a ay that total amount of commodity passing through an edge is not more than the capacity of the edge In our analysis e ill only discuss a special kind of MCF that e call a uniform multi-commodity flo 3
E E problem In this special case, all edges have capacity X, and demand K is same for all the commodities Hence the problem statement in the uniform multi-commodity flo problem is to ship simultaneously maximum amount of commodity beteen each pair of vertices Remark 21 Uniform multi-commodity flo problem forms the dual to the approximate sparsest cut problem presented in Equation (1) 21 Uniform Sparsest Cut units from each vertex in to, the total flo across the cut ill Since the number of edges carrying this load is 6 >, the maximal flo If e ant to ship be t beteen each pair is bounded by 6 >2 ny feasible solution to uniform MCF must therefore have the solution to sparsest cut problem, " ;$ %& ')(+* ", here " P 'V, ' is '{ / '{ While E " is neces- sary, it is not alays sufficient for a uniform MCF to have a flo of size Since MCF Therefore, forms dual to approximate y [ ", from Theorem 1, " E W t a? E " E W t a? No e ill prove that W t a? is a tight bound by providing an example here " q W t a Consider a constant degree expander graph ith degree š We ant to ship units of commodity beteen every pair of vertices The contribution to total load from flo beteen to vertices and œ is at least " eœv, here " W eœv is the length of shortest path Hence total load is at least " WRSTD Since for H_ a constant degree graph, a large fraction of pair of vertices are in distance asymptotically, _ x " WRUTD?y
Y Y Y Since total number of edges is +šž G ith capacity 1 each, t a E +š G Xa Therefore Since ; P 'V, ' k k Ÿ '0,/ ') q for every F g in expander graphs, 6 > CV q X 6 2? $ %& The difference in the solution to MCF and SC in this case is Remark 22 Sparsest Cut is NP-hard MCF is solvable using linear programming Solution to MCF is ithin W t a? to the solution of the sparsest cut Hence e can use MCF to find approximation to sparsest cut 3 Loer ound for Embedding to X To compute the loer bounds for distortion hen embedding to, e ill first construct an inequality that holds for, and then use this inequality to say something about distortion hile embedding to The inequality e ill construct falls under the general class of Poincaré inequalities, and is of form,! D K+ LV qh D K+ LV (2) We need to determine! to ill be at least, a metric such that Equation (2) holds true Distortion for embedding KfL! RSUTV RSTD if Equation (2) holds true for all C metrics Since linear metrics can be expressed as a linear combination of cut metrics, for every [, ' N ' \ ' Thus Equation (2) ill hold true if the equation belo holds true for all ˆ Let be a graph ith determine! and can be to set,! KfL X if RSUT and KfL! \ ' RSTD q X KfL \ ' WRSTD K ª K '! q K ',L KfL,L separated by,l separated by (3) " as the metric induced by it, then one possible attempt to, a constant and s otherise ( is the edge set), y
In this case, for a cut, and separated by '! separated by ' Since this must hold true for all, e can set 6 >? a CD $ %& ' P 'V, ' '0 / 'v for the Poincaré inequality in Equation 2 to hold true Therefore, the minimum distortion for embedding a metric in [ is at least,! RSTD WRUTD $ %& ' P 'V, ' '0 / '{ WRUTD For a constant degree expander ith nodes and degree š this becomes, «W k +š ; O because in a constant degree graph, a constant fraction of asymptotically pair of vertices have length Theorem 31 constant degree expander graph requires distortion W t a for embedding to 31 -Gonal Inequalities Let be an -dimensional integral vector, holds true if e set 1,! K L y and KfL > K L ±< Y k, such that K K X Equation (2) 1 For a real number ², ³ ²nµ> ² if ² ¹ º and º otherise ³²[µu»Ž ¼³²[µW ¾½`² Examples, ³À yµ> Á Â, ³W½ yµ> 6 Žº, ³W½ÃOµW ĺ, ³>½ÃOµU»6 ŽÃ 6
E E s Ë Õ Õ l Õ Ó Ó l Ó K K This can proved by proving the Equation (3) for all separated by '! separated by ' Remark 32 For all separated by ' @ > K L S Ä > K L ± separated by ' K L Åx K N ' Å K N ' K Æ ÇÈ K Æ ÇÈ X Ë L N 3 ' LOÉÊ X LON ' LyÉÊ because here ËÌ KWN ' K X K is an integer Y k, such that K K X, > K L Kx LD q > K L ± K+ LV () is a valid inequality This inequality is knon as -gonal inequality, ith K Example 33 Let K px, L X and SÍx X and all other ÏÎÐ s Equation can be ritten as, K LV K+ LV Ñ K Í K+ Í F L Í Í LV E s) ie, K+ LV E Kx Í Í LV hich is the ell knon triangle inequality Example 3 Consider a vector oò exïxaïxa X X[ [, it must satisfy, Y¼ ÔÓ Thus if a metric l l () Õ Ó Consider the bipartite graph Ö Ï, l ith metric For this graph metric, LHS of Equation () is 8, hile RHS is 6 Hence Ö Ï, l V can eø Ùnot be isometrically embedded to [, and the distortion must be at least ÚÛ is a hyper- Remark 3 If a metric metric Therefore, C is a hypermetric satisfies the all the -gonal inequalities, then is in
1 2 3 Figure 1: Ö O, l bipartite graph [ hypermetrics all metrics Figure 2: schematic diagram shoing that all metrics are, hich in turn are hypermetrics References [ou8 ourgain On lipschitz embedding of finite metric spaces in hilbert space Israel ournal of Mathematics, 198 [LLR9 Nathan Linial, Eran London, and Yuri Rabinovich The geometry of graphs and some of its algorithmic applications In 3th nnual Symposium on Foundations of Computer Science, pages 91, Santa Fe, Ne Mexico, 20 22 November 199 IEEE [LR88 Tom Leighton and Satish Rao n approximate max-flo min-cut theorem for uniform multicommodity flo problems ith applications to approxima- 8
tion algorithms In 29th nnual Symposium on Foundations of Computer Science, pages 22 31, White Plains, Ne York, 2 26 October 1988 IEEE 9