Con$nuous Op$miza$on: The "Right" Language for Fast Graph Algorithms? Aleksander Mądry

Transcription

1 Con$nuous Op$miza$on: The "Right" Language for Fast Graph Algorithms? Aleksander Mądry

2 We have been studying graph algorithms for a while now Tradi$onal view: Graphs are combinatorial objects graph algorithms should be combinatorial too ( combinatorial = talks about paths, trees, cuts, par@@ons, matchings, ) Conven$onal wisdom: combinatorial = fast There are notable excep@ons (spectral graph theory!) but this is the dominant mindset But: Is this mindset (and conven@onal wisdom) actually correct? OGen: combinatorial = ad hoc/hard to analyze (We are stuck on many central graph problems because of that) Is there a bejer way to approach graph algorithms?

3 BeJer(?) approach: Augment our combinatorial understanding of graphs with tools and concepts of Combinatorial methods (trees, paths, matchings, ) Convex opt. primi$ves (gradient- descent, interior- point methods, ) Linear- algebraic tools (eigenvalues, electrical flows, linear systems, ) Our plan for today: Illustrate this theme on an example of a single problem Problem: Maximum flow

4 Maximum flow problem Input: Directed graph G, integer capaci$es u e, source s and sink t s t Think: arcs = roads capaci@es = # of lanes s/t = origin/des@na@on Task: Find a feasible s- t flow of max value (Think: Es@mate the max possible rate of traffic from s to t)

5 Maximum flow problem value = net flow out of s Input: Directed graph G, integer capaci$es u e, source s and sink t s t Think: arcs = roads capaci@es = # of lanes s/t = origin/des@na@on Max flow value Here, value = 7 F*=10 no overflow on arcs: 0 f(e) u(e) no leaks at all v s,t Task: Find a feasible s- t flow of max value (Think: Es@mate the max possible rate of traffic from s to t)

6 Maximum flow problem Input: Directed graph G, integer capaci$es u e, source s and sink t s t Think: arcs = roads capaci@es = # of lanes s/t = origin/des@na@on Max flow value Here, value = 7 F*=10 Task: Find a feasible s- t flow of max value (Think: Es@mate the max possible rate of traffic from s to t)

7 Why is this a good problem to study? Max flow is a fundamental op@miza@on problem Extensively studied since 1930s (classic textbook problem ) Surprisingly diverse set of applica$ons Very influen$al in development of (graph) algorithms Transporta$on (Route planning) Graph par$$oning (Clustering) Scheduling, Assignment problems Connec$vity Analysis Max Flow Computer Vision (Image segmenta@on)

8 What is known about Max Flow? A LOT of previous work

9 What is known about Max Flow? A (very) rough history outline [Dantzig 51] [Ford Fulkerson 56] [Dinitz 70] O(mn 2 U) O(mn U) O(mn 2 ) [Dinitz 70] [Edmonds Karp 72] O(m 2 n) [Dinitz 73] [Edmonds Karp 72] O(m 2 log U) [Dinitz 73] [Gabow 85] O(mn log U) [Goldberg Rao 98] Õ(m min(m 1/2,n 2/3 ) log U) [Lee Sidford 14] Õ(mn 1/2 log U) Our focus: Sparse graph (m=o(n)) and unit- capacity (U=1) regime It is a good benchmark for combinatorial graph algorithms Already captures interes@ng problems, e.g., bipar$te matching (n = # of ver@ces, m = # of arcs, U = max capacity, Õ() hides polylogs)

10 What is known about Max Flow? A (very) rough history outline [Dantzig 51] [Ford Fulkerson 56] [Dinitz 70] O(n 3 ) O(n 2 ) O(n 3 ) [Dinitz 70] [Edmonds Karp 72] O(n 3 ) [Dinitz 73] [Edmonds Karp 72] Õ(n 2 ) [Dinitz 73] [Gabow 85] Õ(n 2 ) [Goldberg Rao 98] Õ(n 3/2 ) [Lee Sidford 14] Õ(n 3/2 ) Our focus: Sparse graph (m=o(n)) and unit- capacity (U=1) regime It is a good benchmark for combinatorial graph algorithms Already captures interes@ng problems, e.g., bipar$te matching (n = # of ver@ces, m = # of arcs, U = max capacity, Õ() hides polylogs)

11 What is known about Max Flow? Emerging barrier: Ω(n 3/2 ) [Even Tarjan 75, Karzanov 73]: Achieved this bound for U=1 ago Last 40 years: Matching this bound in increasingly more general seengs, but no improvement This indicates a fundamental limita@on of our techniques Our goal: A new approach finally breaking this barrier (n = # of ver@ces, m = # of arcs, U = max capacity, Õ() hides polylogs)

12 Breaking the Ω(n 3/2 ) barrier Undirected graphs and approx. answers (Ω(n 3/2 ) barrier s@ll holds here) [M 10]: Crude approx. of max flow value in close to [CKMST 11]: (1- ε)- approx. to max flow in Õ(n 4/3 ε - 3 [LSR 13, S 13, KLOS 14, P 14]: (1- ε)- approx. in Õ(nε - 2 But: What about the directed and exact seeng? [M 13]: Exact Õ(n 10/7 )=Õ(n 1.43 alg. (n = # of ver@ces, Õ() hides polylog factors)

13 Previous approach

14 Augmen$ng paths framework [Ford Fulkerson 56] Basic idea: Repeatedly find s- t paths in the residual graph s t

15 Augmen$ng paths framework [Ford Fulkerson 56] Basic idea: Repeatedly find s- t paths in the residual graph s t

16 Augmen$ng paths framework [Ford Fulkerson 56] Basic idea: Repeatedly find s- t paths in the residual graph s t

17 Augmen$ng paths framework [Ford Fulkerson 56] Basic idea: Repeatedly find s- t paths in the residual graph s t

18 Augmen$ng paths framework [Ford Fulkerson 56] Basic idea: Repeatedly find s- t paths in the residual graph Advantage: Simple, purely combinatorial and greedy (flow is built path- by- path) Problem: Very difficult to analyze s Naïve implementa$on: Unclear O(n 2 how to get ( n augmen@ng a further paths speed- up + via to this find route each path) Sophis$cated implementa$on and arguments: O(n 3/2 [Karzanov 73] [Even Tarjan 75] t

19 Beyond augmen$ng paths

20 New approach: Bring linear- algebraic techniques into play Idea: Probe the global flow structure of the graph by solving linear systems How to relate flow structure to linear algebra? (And why should it even help?) Key object: Electrical flows

21 Electrical flows (Take I) Input: Undirected graph G, resistances r e, source s and sink t s t Recipe for elec. flow: 1) Treat edges as resistors

22 Electrical flows (Take I) Input: Undirected graph G, resistances r e, source s and sink t s t resistance r e Recipe for elec. flow: 1) Treat edges as resistors 2) Connect a bajery to s and t

23 Electrical flows (Take I) Input: Undirected graph G, resistances r e, source s and sink t s t resistance r e Recipe for elec. flow: 1) Treat edges as resistors 2) Connect a bajery to s and t

24 Electrical flows (Take II) Input: Undirected graph G, resistances r e, source s and sink t Principle of least energy Electrical flow of value F: The unique minimizer of the energy E(f) = Σ e r e f(e) 2 among all s- t flows f of value F Electrical flows = l 2 - minimiza@on

25 How to compute an electrical flow? L x = b Electrical flow computa@on Solving linear systems in graph Laplacian (Laplacian = a key object in spectral graph theory) Result: Electrical flow is a nearly- linear $me primi@ve [ST 04, KMP 10, KMP 11, KOSZ 13, LS 13, CKPPR 14] How to employ it?

26 From electrical flows to undirected max flow [CKMST 11]

27 Approx. undirected max flow via electrical flows Assume: F * known (via binary search) s Treat edges as resistors of resistance 1 t

28 Approx. undirected max flow via electrical flows Assume: F * known (via binary search) s Treat edges as resistors of resistance 1 Compute electrical flow of value F * t

29 Approx. undirected max flow via electrical flows Assume: F * known (via binary search) Treat edges as resistors of resistance 1 Compute electrical flow of value F * (This flow has no leaks, but can overflow some edges) 3/4 3/4 s 3/ /4 t

30 Approx. undirected max flow via electrical flows Assume: F * known (via binary search) 3/4 s Treat edges as resistors of resistance 1 Compute electrical flow of value F * (This flow has no leaks, but can overflow some edges) To fix that: Increase resistances on the overflowing edges Repeat (hope: it doesn t happen too ooen) 1.5 3/4 3/4 3/4 Surprisingly: This approach can be made work! t

31 Approx. undirected max flow via electrical flows Assume: F * known (via binary search) 3/4 s Treat edges as resistors of resistance 1 Compute electrical flow of value F * (This flow has no leaks, but can overflow some edges) To fix that: Increase resistances on the overflowing edges Repeat (hope: it doesn t happen too ooen) At the end: Take an average of all the flows as the final answer 1.5 3/4 3/4 t 3/4 Evolu$on of resistances: Based on Mul@plica@ve Weight Update method [FS 97, PST 95, AHK 05]

32 Bounding the running $me Each runs in How many do we need? Can show: # of itera@ons worst- case overflow ρ Think: ρ measures the electrical vs. max flow discrepancy Key ques$on: If f E = elect. flow of value F * wrt all r e =1 What is ρ = max e f E (e)? Claim: ρ m 1/2 =O(n 1/2 ) Proof: Suffices to show that E(f E ) m

33 Bounding the running $me Each runs in How many do we need? Can show: # of itera@ons worst- case overflow ρ Think: ρ measures the electrical vs. max flow discrepancy Key ques$on: If f E = elect. flow of value F * wrt all r e =1 What is ρ = max e f E (e)? Claim: ρ m 1/2 =O(n 1/2 ) Proof: Suffices to show that E(f E ) = Σ e r e f E (e) 2 = Σ e f E (e) 2 m Note: if f * is the max flow (of value F * ) then E(f * ) = Σ e r e f * (e) 2 = Σ e f * (e) 2 Σ e 1 m

34 Bounding the running $me Each runs in How many do we need? Can show: # of itera@ons worst- case overflow ρ Think: ρ measures the electrical vs. max flow discrepancy Key ques$on: If f E = elect. flow of value F * wrt all r e =1 What is ρ = max e f E (e)? Claim: ρ m 1/2 =O(n 1/2 ) Proof: Suffices to show that E(f E ) = Σ e r e f E (e) 2 = Σ e f E (e) 2 m Note: if f * is the max flow (of value F * ) then This E(f * ) gives = Σ e ran e f * Õ(nρε (e) 2 = - 3 Σ e ) = f * (e) Õ(n 2 3/2 Σε e m But, as f E minimizes (1- ε)- approx energy among algorithm all the flows of value F * E(f E ) E(f * ) m

35 Breaking the Ω(n 3/2 ) $me bound Problem: The m 1/2 =O(n 1/2 ) itera@ons bound we just proved n ½ paths with n ½ ver@ces each s t one edge

36 Breaking the Ω(n 3/2 ) $me bound Problem: The m 1/2 =O(n 1/2 ) itera@ons bound we just proved Max flow: n ½ paths with n ½ ver@ces each F* n ½ s t one edge

37 Breaking the Ω(n 3/2 ) $me bound Problem: The m 1/2 =O(n 1/2 ) itera@ons bound we just proved Elect. flow: s t n 1/2 But: It is so only in presence of edges that (single- handily) contribute most of the energy of f E (Recall: We showed ρ 2 = max e f E (e) 2 Σ e f E (e) 2 = E(f E ) m)

38 Breaking the Ω(n 3/2 ) $me bound Problem: The m 1/2 =O(n 1/2 ) itera@ons bound we just proved Elect. flow: s t n 1/2 Key idea: Perturb the graph by removing such high- energy edges whenever they emerge

39 Breaking the Ω(n 3/2 ) $me bound Problem: The m 1/2 =O(n 1/2 ) itera@ons bound we just proved s t Key idea: Perturb the graph by removing such high- energy edges whenever they emerge

40 Breaking the Ω(n 3/2 ) $me bound Problem: The m 1/2 =O(n 1/2 ) itera@ons bound we just proved s t Key idea: Perturb the graph by removing such high- energy edges whenever they emerge

41 Breaking the Ω(n 3/2 ) $me bound Problem: The m 1/2 =O(n 1/2 ) itera@ons bound we just proved s t Careful energy- based argument gives the desired Õ(n 4/3 ε - 3 algorithm

42 From electrical flows to exact directed max flow [M 13]

43 Why the progress on approx. undirected max flow does not apply to the exact directed case? Temp$ng answer: Directed graphs are just different (for one, electrical flow is an undirected But: exact directed max flow reduces to exact undirected We need a case more powerful So, it is all about geeng sufficiently intermediary good approxima@on s t MWU rou@ne Key obstacle: Gradient descent methods (like MWU) are inherently unable to deliver good enough accuracy

44 (Path- following) Interior- point method (IPM) [Dikin 67, Karmarkar 84, ] A powerful framework for solving general LPs (and more) LP: min c T x s.t. Ax = b x 0 Idea: Take care of hard constraints by adding a barrier to the objec@ve easy constraints (use projec@on) hard constraints

45 (Path- following) Interior- point method (IPM) [Dikin 67, Karmarkar 84, ] A powerful framework for solving general LPs (and more) LP(μ): min c T x - μ Σ i log x i s.t. Ax = b x 0 Idea: Take care of hard constraints by adding a barrier to the objec@ve Observe: The barrier term enforces x 0 implicitly Furthermore: for large μ, LP(μ) is easy to solve and LP(μ) original LP, as μ 0 + Path- following rou$ne: Start with (near- )op@mal solu@on to LP(μ) for large μ>0 Gradually reduce μ (via Newton s method) while maintaining the (near- )op@mal solu@on to current LP(μ)

46 central path = op@mal solu@ons to LP(μ) c for all μ>0 OPT (0,0) x c P A P A = {x Ax = b} x c analy@c center Path- following rou$ne: Start with (near- )op@mal solu@on to LP(μ) for large μ>0 Gradually reduce μ (via Newton s method) while maintaining the (near- )op@mal solu@on to current LP(μ)

47 Can we use IPM to get a faster max flow alg.? Conven$onal wisdom: This will be too slow! Each Newton's step = solving a linear system O(n ω )=O(n (prohibi@ve!) But: When solving flow problems only [DS 08] Fundamental ques$on: What is the number of itera@ons? [Renegar 88]: O(m 1/2 log ε - 1 ) Unfortunately: This gives only an Õ(m 3/2 algorithm Improve the O(m 1/2 ) bound? Although believed to be very subop@mal, its improvement is a major challenge?

48 [M 13]: An improved O(m 3/7 ) itera@ons bound for unit- capacity max flow interior- point method Observa$on: IPM is solving max flow using electrical flows too! A simple energy- based argument recovers the O(m 1/2 ) bound Lack of high- energy edges bejer than O(m 1/2 ) convergence Problem: Removal of such high- energy edges is too dras@c Instead: Apply a careful perturba$on + precondi$oning of the LP (This not only changes the current solu@on but also the central path) Most of these elements seems broadly applicable (and new) In par$cular: We can solve every LP faster, but only if we perturb it Will this lead to breaking the Ω(m 1/2 ) convergence barrier for all LPs?

49 Conclusions and the Bigger Picture

50 Maximum Flows and Electrical Flows s t Interior- point method Elect. flows + IPMs A powerful new approach to max flow Can this lead to a nearly- linear $me algorithm for the exact directed max flow? We seem to have the cri@cal mass of ideas Elect. flows = next genera@on of spectral tools? BeJer spectral graph par@@oning, Algorithmic grasp of random walks,

51 Grand challenge: Can we make algorithmic graph theory run in nearly- New recipe : Fast alg. for combinatorial problems via linear- algebraic tools + con$nuous opt. methods How about applying this framework to other graph problems that got stuck at O(n 3/2 )? (min- cost flow, general matchings, nega$ve- lengths shortest path ) Second- order/ipm- like methods: the next fron@er for fast (graph) algorithms?

52 Should opt. be the language of graph alg.? Combinatorial approaches work great when they do work, but they are ooen ad hoc and hard to work with/analyze opt. gives us a much more principled approach Phrase your graph problem as a (convex) op@miza@on ques@on Derive proper@es of the objec@ve fun. needed for fast convergence Crao the objec@ve func@on accordingly Apply a generic (and well- implemented) off- the- shelf opt. alg. Only place where combinatorial/problem- specific insights are needed All the algorithms I discussed (and their subsequent improvements) can be derived in this way (even if it might be not apparent) Intriguing possibility: Can the classic graph algorithms be derived in this way too?

53 Bridging the Combinatorial and the Con$nuous paths, trees, matchings, data structures matrices, eigenvalues, linear systems, gradients, convex sets Powerful approach: the interplay of the two worlds Some other early success stories of this approach: Spectral graph theory aka the eigenvalue Fast SDD/Laplacian system solvers Graph random spanning tree this is just the beginning!

54 Thank you Ques$ons?