Generalized Sparsest Cut and Embeddings of Negative-Type Metrics Shuchi Chawla, Anupam Gupta, Harald Räcke Carnegie Mellon University 1/25/05

Sparsest Cut and Embeddings of Negative-type Metrics Finding Bottlenecks Find the cut across which demand exceeds capacity by the largest factor capacity of cut links demand across cut Sparsity of a cut = Sparsest cut Capacity = 2.1 units Demand = 3 units 1 Sparsity of the cut = 0.7 10 0.1 Sparsest Cut and Embeddings of Negative-type Metrics

The Generalized Sparsest Cut Problem The givens: a graph G=(V,E) capacities on edges c(e) demands on pairs of vertices D(x,y) Sparsity of a cut S  V, (S) = (S)c(e) xS, yS D(x,y) Sparsity of graph G, (G) = minSV (S) Our result: an O(log¾n)-approximation for (G) V S\V Sparsest Cut and Embeddings of Negative-type Metrics

Sparsest Cut and Embeddings of Negative-type Metrics What’s known Uniform-demands – a special case D(x,y) = 1 for all x  y O(log n)-approx [Leighton Rao’88] based on LP-rounding Cannot do better than O(log n) using the LP O(log n)-approx [Arora Rao Vazirani’04] based on an SDP relaxation General case O(log n)-approx [Linial London Rabinovich’95 Aumann Rabani’98] based on LP-rounding and low-distortion embeddings Our result: O(log¾n)-approx Extends [ARV04] using the same SDP Sparsest Cut and Embeddings of Negative-type Metrics

Sparsest Cut and Embeddings of Negative-type Metrics A metrics perspective Given set S, define a “cut” metric S(x,y) = 1 if x and y on different sides of cut (S, V-S) 0 otherwise (S) = e c(e) S(e) x,y D(x,y) S(x,y) Finding sparsest cut  minimizing above function over all metrics Typical technique: Minimize over class ℳ of metrics, with ℳ  ℓ1, and embed into ℓ1 (d) = e c(e) d(e) x,y D(x,y) d(x,y) NP-hard ℓ1 cut Sparsest Cut and Embeddings of Negative-type Metrics

Sparsest Cut and Embeddings of Negative-type Metrics A metrics perspective (d) = e c(e) d(e) x,y D(x,y) d(x,y) Finding sparsest cut  minimizing a(d) over metrics Lemma: Minimize over a class ℳ to obtain d + have -distortion embedding from d into  -approx for sparsest cut ℓ1 ℓ1 When ℳ = all metrics, obtain O(log n) approximation [Linial London Rabinovich ’95, Aumann Rabani ’98] Cannot do any better [Leighton Rao ’88] Sparsest Cut and Embeddings of Negative-type Metrics

Squared-Euclidean, or ℓ2-metrics A metrics perspective (d) = e c(e) d(e) x,y D(x,y) d(x,y) Finding sparsest cut  minimizing a(d) over metrics Lemma: Minimize over a class ℳ to obtain d + have -avg-distortion embedding from d into  -approx for “uniform-demands” sparsest cut ℓ1 Squared-Euclidean, or ℓ2-metrics 2 ℓ1 ℳ = “negative-type” metrics  O(log n) approx [Arora Rao Vazirani ’04] Question: Can we obtain O(log n) for generalized sparsest cut, or an O(log n) distortion embedding from into ℓ2 2 ℓ1 Sparsest Cut and Embeddings of Negative-type Metrics

Arora et al.’s O(log n)-approx ℓ2 2 Solve an SDP relaxation to get the best representation Key Theorem: Let d be a “well-spread-out” metric. Then  m – an embedding from d into a line, such that, - for all pairs (x,y), m(x,y)  d(x,y) - for a constant fraction of (x,y), m(x,y)  1 ⁄O(log n) d(x,y) The general case – issues Well-spreading does not hold Constant fraction is not enough Want low distortion for every demand pair. ℓ2 2 For a const. fraction of (x,y), d(x,y) > const.  diameter Implies an avg. distortion of O(log n) Sparsest Cut and Embeddings of Negative-type Metrics

1. Ensuring well-spreading Divide pairs into groups based on distances Di = { (x,y) : 2i  d(x,y)  2i+1 } At most O(log n) groups Each group by itself is well-spread, by definition Embed each group individually distortion O(log n) contracting embedding into a line for each (assume for now) “Glue” the embeddings appropriately Sparsest Cut and Embeddings of Negative-type Metrics

Sparsest Cut and Embeddings of Negative-type Metrics Gluing the groups Start with an a = O(log n) embedding for each scale A naïve gluing concatenate all the embeddings and renormalize by dividing by O(log n) Distortion O(alog n) = O(log n) A better gluing lemma “measured-descent” by Krauthgamer, Lee, Mendel & Naor (2004) (Recall the previous talk by James Lee) Gives distortion O(a log n)  distortion O(log¾n) Sparsest Cut and Embeddings of Negative-type Metrics

2. Average to worst-case distortion Arora et al.’s guarantee – a constant fraction of pairs embed with low distortion We want – every pair should embed with low distortion Idea: Re-embed pairs that have high distortion Problem: Increases the number of embeddings, implying a larger distortion A “re-weighting” solution: Don’t ignore low-distortion pairs completely – keep them around and reduce their importance Sparsest Cut and Embeddings of Negative-type Metrics

Weighting-and-watching Initialize weight = 1 for each pair Apply ARV to weighted instance For pairs with low-distortion, decrease weights by factor of 2 For other pairs, do nothing Repeat until total weight < 1/k Total weight decreases by constant factor every time O(log k) iterations Each individual weight decreases from 1 to 1/k Each pair contributes to W(log k) iterations Implies low distortion for every pair A constant fraction of the weight is embed with low distortion Sparsest Cut and Embeddings of Negative-type Metrics

Sparsest Cut and Embeddings of Negative-type Metrics Summarizing… Start with a solution to the SDP For every distance scale Use [ARV04] to embed points into line Use re-weighting to obtain good worst-case distortion Combine distance scales using measured-descent In practice Write another SDP to find best embedding into Use J-L to embed into and then into a cut-metric ℓ2 ℓ2 ℓ1 Sparsest Cut and Embeddings of Negative-type Metrics

Sparsest Cut and Embeddings of Negative-type Metrics Recent developements Arora, Lee & Naor obtained an O(log n log log n) approximation for sparsest cut The improvement lies in a better concatenation technique Nearly optimal embedding from into Evidence for hardness Khot & Vishnoi: W(log log log n) integrality gap for the SDP l.b. for embedding into Chawla, Krauthgamer, Kumar, Rabani & Sivakumar: W(log log n) hardness based on “Unique Games Conjecture” Evidence that constant factor approximation is not possible Other approximations using similar SDP relaxations Feige, Hajiaghayi & Lee: O(log n) approx for min-wt. vertex cuts ℓ1 ℓ2 ℓ2 2 ℓ1 Sparsest Cut and Embeddings of Negative-type Metrics

Sparsest Cut and Embeddings of Negative-type Metrics Open Problems Beating the [ALN05] O(log n log log n) approximation Can the SDP give a better bound? Exploring flow-based techniques Closing the gap between hardness and approximation Other applications of SDP with triangle inequalities Other partitioning problems Directed versions? SDP/LP don’t seem to work Sparsest Cut and Embeddings of Negative-type Metrics

Questions?