Effective-Resistance-Reducing Flows, Spectrally Thin Trees, and ATSP Nima Anari UC Berkeley Shayan Oveis Gharan Univ of Washington History of Graph Sparsification 2 Spectral Sparsifier Spielman-Teng04 Cut Sparsifier Benczur-Karger96
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Presentation Transcript
01
Effective-Resistance-Reducing Flows, Spectrally Thin Trees, and ATSP Nima Anari
UC Berkeley Shayan Oveis Gharan
Univ of Washington<br>
02
History of Graph Sparsification 2 Spectral Sparsifier [Spielman-Teng’04] Cut Sparsifier
[Benczur-Karger’96] Spectrally Thin Tree Proof of Kadison-Singer
[Marcus-Spielman-Srivastava’13] Asymmetric TSP
[O-Anari’14] Laplacian Solvers
[Spielman-Teng’04] SS by Effective Resistance [Spielman-Srivastava’09] Linear size SS [Batson-Spielman-Srivastava’10] This Talk<br>
03
Asymmetric TSP (ATSP) 3<br>
04
Linear Programming Relaxation [Held-Karp’72] 4 Integrality Gap:<br>
Main Result 6 For any cost function, the integrality gap of the LP relaxation is polyloglog(n).<br>
07
Plan of the Talk 7 ATSP Thin Spanning Tree Thin Basis Problem Method of Interlacing Polys<br>
08
Thin Spanning Trees 8 Kn 2/n-thin tree Application: f(n)/k-thin trees imply 5f(n)-approx for ATSP<br>
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Main Result 9 For any cost function, the integrality gap of the LP Relaxation is polyloglog(n).<br>
10
10 In Pursuit of Thin Trees<br>
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Main Ingredients 11 Interlacing Polynomials and
thin basis problem Effective resistance reduction
by Convex Programming<br>
12
A General Framework for L1 Opt Problems 12<br>
13
We can use convex duality to analyze Q(.) of the optimum. 13 Has exp. many constraints but can be solved using ellipsoid Since Q(.) is convex the program is convex<br>
14
14<br>
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15<br>
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Plan of the Talk 16 ATSP Thin Spanning Tree Thin Basis Problem Method of Interlacing Polys Min Cost
Flow Eff Resist
Reduction<br>
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Thin Basis Prob [Marcus-Spielman-Srivastava’13] 17 d Linearly independent set of vectors<br>
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A Weaker Suff Cond for Thin Basis Problem 18 ………………………………………<br>
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Proof of Thin Basis Thm 19<br>
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Method of Interlacing Polynomials 20<br>
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Summary/Future Directions Main Ingredients:
Effective Resistance Reduction
Thin Basis Problem
Future Directions:
Algorithmic proof of [MSS’13].
Existence of C/k thin trees
Subsequent work: Svensson’s 27-app algorithm for graph ATSP. 21<br>
22
Previous Works: Randomized Rounding 22<br>
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Main Result 23 For any cost function, the integrality gap of the LP Relaxation is polyloglog(n).<br>
24
Previous Works: Randomized Rounding 24<br>
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Graph Laplacian 25 E.g.,<br>
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A Necessary Condition for Spectral Thinness 26 where<br>
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A k-con Graph with no Spectrally Thin Tree 27 n/k vertices k edges k edges<br>
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A Sufficient Condition for Spectral Thinness 28<br>
29
Spectrally Thin Trees (Summary) 29 k-edge
connectivity O(1/k)-combinatorial thin tree O(1/k)-spectrally thin tree [MSS13] ?<br>
30
Our Approach 30<br>
31
Main Idea 31 Symmetrize L2 structure of G
while preserving its L1 structure<br>
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An Example 32 n/k vertices<br>
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An Observation 33<br>
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Main Idea 34 D+G has a spectrally thin tree and
any spectrally thin tree of G+D is (comb) thin in G. Bypasses Spectral Thinness Barrier.<br>
35
An Impossibility Theorem 35<br>
36
Proof Overview 36 D is not a graph<br>
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Proof Overview 37<br>
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A Weaker Goal: Satisfying Degree Cuts 38<br>
39
A Convex Program for Optimum D 39<br>
40
Main Result 40 For any cost function, the integrality gap of the LP Relaxation is polyloglog(n).<br>
41
Conclusion Main Idea:
Symmetrize L2 structure of G while preserving its L1 structure
