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On Partitioning Graphs via Single Commodity Flows Lorenzo Orecchia UC Berkeley Leonard J. Schulman Caltech Umesh V. Vazirani UC Berkeley Nisheeth K. Vishnoi IBM Delhi – work done while visiting UC Berkeley STOC 2008 , Victoria The SPARSEST CUT problem Given a graph G=(V,E) and partition (S, S ) | E(S, S ) | Expansion of (S, S ) = min{| S |,| S |} E(S, S ) S S The SPARSEST CUT problem Given a graph G=(V,E) and partition (S, S ) | E(S, S ) | Expansion of (S, S ) = min{| S |,| S |} SPARSEST CUT: find (S, S ) with minimum expansion !(G). Applications: Divide-and-Conquer, Image Segmentation, VLSI design, Clustering. Theoretical Importance: Metric Embeddings, Spectral Methods. The SPARSEST CUT problem Given a graph G=(V,E) and partition (S, S ) | E(S, S ) | Expansion of (S, S ) = min{| S |,| S |} SPARSEST CUT: find (S, S ) with minimum expansion !(G). Applications: Divide-and-Conquer, Image Segmentation, VLSI design, Clustering. Theoretical Importance: Metric Embeddings, Spectral Methods. The SPARSEST CUT problem is NP-hard. Approximation Algorithms for SPARSEST CUT Algorithm Output Expansion ** Running Time * & d2n # ** O$$ 2 !! % ' " Spectral 2 d! Leigthon-Rao ! log n ~ 2 O(n ) ! log n [ ARV ] poly(n) ~ [ AHK ] O(n2 ) ARV ~ *All graphs have been sparsified to O(n) edges. ** For a d-regular graph G. Approximation Algorithms for SPARSEST CUT Algorithm Output Expansion ** Running Time * & d2n # ** O$$ 2 !! % ' " Spectral 2 d! Leigthon-Rao ! log n ~ 2 O(n ) ! log n [ ARV ] poly(n) ~ [ AHK ] O(n2 ) ARV IN PRACTICE: Too slow for massive data sets. Spectral and heuristics like METIS used instead. ~ *All graphs have been sparsified to O(n) edges. ** For a d-regular graph G. Fast Approximation Algorithms Algorithm Output Expansion ** Running Time * & d2n # ** O$$ 2 !! % ' " Spectral 2 d! Leigthon-Rao ! log n ~ 2 O(n ) ! log n [ ARV ] poly(n) ~ [ AHK ] O(n2 ) ARV KRV !(log n) 2 ~ O(n3 / 2 ) CUT-MATCHING GAME: FRAMEWORK FOR COMPUTING APPROX USING s-t MAXFLOW COMPUTATIONS ~ *All graphs have been sparsified to O(n) edges. ** For a d-regular graph G. Fast Approximation Algorithms Algorithm Output Expansion Leigthon-Rao ! log n Running Time ~ 2 O(n ) ARV ! log n [ ARV ] poly(n) ~ [ AHK ] O(n2 ) KRV !(log n)2 ~ O(n3 / 2 ) ! log n ~ 3/2 O(n ) AK ~ *All graphs have been sparsified to O(n) edges. ** For a d-regular graph G. Our Contribution Algorithm Output Expansion Running Time KRV !(log n)2 ~ O(n3 / 2 ) ! log n ~ O(n3 / 2 ) ! log n ~ O(n3 / 2 ) AK THIS PAPER IN KRV CUT-MATCHING GAME FRAMEWORK Our Contribution Algorithm Output Expansion Running Time KRV !(log n)2 ~ O(n3 / 2 ) ! log n ~ O(n3 / 2 ) ! log n ~ O(n3 / 2 ) AK THIS PAPER IN KRV CUT-MATCHING GAME FRAMEWORK LOWER BOUND No better approx than Ω((log n)1/2) in KRV framework. Our Contribution Algorithm Output Expansion KRV !(log n)2 ~ 3/2 O(n ) ! log n ~ 3/2 O(n ) ! log n ~ 3/2 O(n ) AK THIS PAPER Running Time LOWER BOUND No better approx than Ω((log n)1/2) in KRV framework. Best integrality gap for SDP is Ω(loglog(n)). Our Contribution Algorithm Output Expansion KRV !(log n)2 ~ 3/2 O(n ) ! log n ~ 3/2 O(n ) ! log n ~ 3/2 O(n ) AK THIS PAPER Running Time LOWER BOUND No better approx than Ω((log n)1/2) in KRV framework. Best integrality gap for SDP is Ω(loglog(n)). CUT-MATCHING RIGHT ABSTRACTION? The KRV Cut-Matching Game CUT PLAYER H0 MATCHING PLAYER The KRV Cut-Matching Game CUT PLAYER H0 (S1, S1 ) 50-50 Cut MATCHING PLAYER The KRV Cut-Matching Game CUT PLAYER H0 MATCHING PLAYER (S1, S1 ) M1 50-50 Cut Perfect Matching The KRV Cut-Matching Game CUT PLAYER H1 MATCHING PLAYER (S1, S1 ) M1 50-50 Cut Perfect Matching The KRV Cut-Matching Game CUT PLAYER H1 MATCHING PLAYER (S1, S1 ) M1 50-50 Cut Perfect Matching (S 2 , S2 ) 50-50 Cut The KRV Cut-Matching Game CUT PLAYER H1 (S1, S1 ) 50-50 Cut MATCHING PLAYER M1 Perfect Matching (S 2 , S2 ) M1 50-50 Cut Perfect Matching The KRV Cut-Matching Game CUT PLAYER H2 MATCHING PLAYER (S1, S1 ) M1 50-50 Cut Perfect Matching (S 2 , S2 ) M1 50-50 Cut Perfect Matching … The KRV Cut-Matching Game CUT PLAYER H2 MATCHING PLAYER (S1, S1 ) M1 50-50 Cut Perfect Matching (S 2 , S2 ) M1 50-50 Cut Perfect Matching … Go until time T when 1 "(HT ) ! 4 GOAL: GOAL: Minimize T Maximize T The KRV Cut-Matching Game CUT PLAYER H2 MATCHING PLAYER (S1, S1 ) M1 50-50 Cut Perfect Matching (S 2 , S2 ) M1 50-50 Cut Perfect Matching … Go until time T when 1 "(HT ) ! 4 GOAL: GOAL: Minimize T Maximize T KRV: there exists a cut strategy achieving T = O((log n)2). The KRV Cut-Matching Game Runs in time c(n) per iteration CUT PLAYER STRATEGY T = t(n) The KRV Cut-Matching Game Runs in time c(n) per iteration Running time: t(n)· (Tmaxflow + c(n)) CUT PLAYER STRATEGY APPROXIMATION ALGORITHM T = t(n) Approx Ratio: t(n) The KRV Cut-Matching Game Runs in time c(n) per iteration CUT PLAYER STRATEGY Running time: t(n)· (Tmaxflow + c(n)) Time to compute s-t maxflow in G ~ O(n3 / 2 ) APPROXIMATION ALGORITHM T = t(n) Approx Ratio: t(n) The KRV Cut-Matching Game Runs in time c(n) per iteration CUT PLAYER STRATEGY Running time: t(n)· (Tmaxflow + c(n)) APPROXIMATION ALGORITHM T = t(n) Approx Ratio: t(n) ~ KRV strategy has c(n) = O(n) and t(n)= O((log n)2). ~ O(n3 / 2 ) TOTAL RUNNING TIME: ~ 3/2 O(n ) Our Version of the Cut-Matching Game • MODIFIED GAME 1.No Stopping Condition 2.Value of Game is !(HT ) T Our Version of the Cut-Matching Game • MODIFIED GAME 1.No Stopping Condition 2.Value of Game is • STILL YIELDS APPROX ALGORITHM !(HT ) T Approximation Ratio = !(HT ) T Our Version of the Cut-Matching Game • MODIFIED GAME 1.No Stopping Condition 2.Value of Game is • STILL YIELDS APPROX ALGORITHM !(HT ) T Approximation Ratio = !(HT ) T • RESULTS CUT STRATEGY: MATCHING STRATEGY: & 1 # ((HT ) '(log n) !! = = '$$ 2 T O(log n) % log n " & 1 # '(HT ) ! = O$ $ log n ! T % " Cut Strategies: Finding Cuts Quickly After t iterations, Ht = { M1, M2, …, Mt }. = +1 charge = –1 charge Random assignment of charge Cut Strategies: Finding Cuts Quickly After t iterations, Ht = { M1, M2, …, Mt }. = +1 charge Random assignment of charge = –1 charge (x+y)/2 x Mix the charges along the matchings { M1, M2, …, Mt} y (x+y)/2 Cut Strategies: Finding Cuts Quickly After t iterations, Ht = { M1, M2, …, Mt }. = +1 charge = –1 charge Random assignment of charge Cut Strategies: Finding Cuts Quickly After t iterations, Ht = { M1, M2, …, Mt }. = +1 charge = –1 charge Random assignment of charge Cut Strategies: Finding Cuts Quickly After t iterations, Ht = { M1, M2, …, Mt }. = +1 charge Random assignment of charge = –1 charge Mix the charges along the matchings { M1, M2, …, Mt } Cut Strategies: Finding Cuts Quickly After t iterations, Ht = { M1, M2, …, Mt }. = +1 charge Random assignment of charge = –1 charge If cut is small, unbalance remains. Mix the charges along the matchings { M1, M2, …, Mt } Cut Strategies: Finding Cuts Quickly Order the vertices according to the final charge present and cut in half. S n/2 S n/2 The KRV mixing walk KRV-walk . At round t: & I + Mt '1 #& I + Mt '2 # & I + M1 # P( t ) = $ !$ !...$ ! % 2 "% 2 " % 2 " Lazy random walk traversing matchings in order. 0 0 M1 M2 1 0 The KRV mixing walk KRV-walk . At round t: & I + Mt '1 #& I + Mt '2 # & I + M1 # P( t ) = $ !$ !...$ ! % 2 "% 2 " % 2 " Lazy random walk traversing matchings in order. 0 0 M1 Averaging along M1 M2 1/2 1/2 The KRV mixing walk KRV-walk . At round t: & I + Mt '1 #& I + Mt '2 # & I + M1 # P( t ) = $ !$ !...$ ! % 2 "% 2 " % 2 " Lazy random walk traversing matchings in order. 1/4 1/4 M1 Averaging along M2 M2 1/4 1/4 Sketch of KRV Analysis 1. Mixing of P(t) measured by potential function 2 F "t =|| P( t ) ! J / n || 2. If P(t) mixes well, Ht has good expansion. Possible to embed Kn in Ht. 3. Potential Reduction at every iteration #t = #t "1 " L(Mt ) ! P( t ) Mixing due to matching Mt Decomposition possible as KRV walks matchings in order. 4. Cut-finding procedure reduces potential by a fixed factor & 1 # !! (t = (t '1$$1 ' % log n " Yields expander in O((log n)2) rounds Why KRV cannot do better Recall: Approximation is !(HT ) T Why KRV cannot do better Recall: Approximation is Can KRV get better than O(1) expansion? Why KRV cannot do better Recall: Approximation is S V-S SUPPOSE: Can KRV get better than O(1) expansion? Walk on M1, M2, …, Mk mixes perfectly on S and V-S and no edge cross (S,V-S) Why KRV cannot do better Recall: Mk+1 Approximation is Now walk mixes perfectly. Expansion is O(1). S V-S SUPPOSE: Can KRV get better than O(1) expansion? Walk on M1, M2, …, Mk mixes perfectly on S and V-S and no edge cross (S,V-S) Our Cut Strategy: a Different Walk IDEA: use lazy natural random walk I M1 + M2 + ... + Mt !1 P( t ) = + 2 2( t ! 1) ADVANTAGES: – – Eliminates bad case: possible to get better expansion. Better handle on expansion through mixing by Cheeger’s Inequality. CHALLENGE: – Impossible to decompose potential as in KRV. #t = #t "1 " L(Mt ) ! P( t ) Additional matching modifies all steps of walk. Our Cut Strategy: a Different Walk IDEA: use lazy natural random walk & I M1 + M2 + ... + Mt '1 # !! P( t ) = $$ + 2( t ' 1) %2 " d ADVANTAGES: – – Eliminates bad case: possible to get better expansion. Better handle on expansion through mixing by Cheeger’s Inequality. CHALLENGE: – Impossible to decompose potential as in KRV. #t = #t "1 " L(Mt ) ! P( t ) Additional matching modifies all steps of walk. Modified Walk and Matrix Inequalities CHALLENGE: Impossible to decompose potential as in KRV. Additional matching modifies all steps of walk. SOLUTION: Use round-robin walk close to natural walk: Ni = d 1 I+ Mi d +1 d +1 P( t ) = (N1N2 ...Nt !1Nt !1Nt !2 ...N1 ) Apply matrix inequality: t ( ABA ) ! A tB t A t d Modified Walk and Matrix Inequalities CHALLENGE: Impossible to decompose potential as in KRV. Additional matching modifies all steps of walk. SOLUTION: Use round-robin walk close to natural walk: Ni = d 1 I+ Mi d +1 d +1 P( t ) = (N1N2 ...Nt !1Nt !1Nt !2 ...N1 ) Apply matrix inequality: t ( ABA ) ! A tB t A t #t = #t "1 " L(Mt ) ! P( t ) d Modified Walk and Matrix Inequalities CHALLENGE: Impossible to decompose potential as in KRV. Additional matching modifies all steps of walk. SOLUTION: Use round-robin walk close to natural walk: Ni = d 1 I+ Mi d +1 d +1 P( t ) = (N1N2 ...Nt !1Nt !1Nt !2 ...N1 ) Apply matrix inequality: t ( ABA ) ! A tB t A t #t = #t "1 " L(Mt ) ! P( t ) d Modified Walk and Matrix Inequalities SOLUTION: Use round-robin walk close to natural. Apply matrix inequality. Yields same potential reduction as KRV. But our walk is better related to expansion: In O((log n)2) rounds, conductance (1\log n) by Cheeger. Modified Walk and Matrix Inequalities SOLUTION: Use round-robin walk close to natural. Apply matrix inequality. Yields same potential reduction as KRV. But our walk is better related to expansion: In O((log n)2) rounds, conductance (1\log n) by Cheeger. Ω(log n) expansion in O((log n)2) rounds. Modified Walk and Matrix Inequalities SOLUTION: Use round-robin walk close to natural. Apply matrix inequality. Yields same potential reduction as KRV. But our walk is better related to expansion: In O((log n)2) rounds, conductance (1\log n) by Cheeger. Ω(log n) expansion in O((log n)2) rounds. TIME: only polylog factors worse than KRV Lower Bound Matching player yielding & 1 # '(HT ) ! = O$ $ log n ! T % " against any Cut player. No better approximation than O((log n)1/2) in KRV Cut-Matching game Lower Bound Idea A NAÏVE MATCHING PLAYER: Fix a cut (S,V-S). Keep it as sparse as possible. S Lower Bound Idea A NAÏVE MATCHING PLAYER: Fix a cut (S,V-S). Keep it sparse. Cut player plays… S Lower Bound Idea A NAÏVE MATCHING PLAYER: Fix a cut (S,V-S). Keep it sparse. Cut player plays… S GAME OVER Lower Bound Idea A NAÏVE MATCHING PLAYER: Fix a cut (S,V-S). Keep it sparse. Cut player plays… S GAME OVER IDEA: hedge over many cuts Lower Bound Idea THE REAL PLAYER - AT START: Matching player selects log(n) ‘orthogonal’ 50-50 cuts in V. Orthogonal 50-50 cuts Minimum correlation Cut player cannot ‘kill’ many cuts at once THE REAL PLAYER - THROUGHOUT THE GAME: Matching player adds matchings to minimize average expansion. Main Lemma 8 50-50 cut (S,V-S), d Hd = {! 1,+1} Main Lemma 8 50-50 cut (S,V-S), 9 a perfect matching M, s.t. d Hd = {! 1,+1} Main Lemma 8 50-50 cut (S,V-S), 9 a perfect matching M, s.t. "| u ! v | 1 ( u,v )#M d Hd = {! 1,+1} ( ) =O d Conclusion and Open Problems POWER OF CUT-MATCHING GAME: Simple yet powerful framework for SPARSEST CUT. OPEN QUESTION: Can we use Cut-Matching to get fast (log n)1/2 approximation?