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FAST ALGORITHMS FOR SURFACE EMBEDDED GRAPHS

VIA HOMOLOGY The Final Exam of

Kyle J. Fox

Doctoral committee: Jeff Erickson, Chair Chandra Chekuri Derek Hoiem David Eppstein, University of California, Irvine

Graph Algorithms

• Major target for algorithms research

• Provide natural abstractions for practical problems

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Not Efficient

• Many problems are NP-hard

• Many polynomial time algorithms are quadratic or worse

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Planarity• A natural assumption for many problem domains

• Speeds up computation

• Polynomial time → near-linear • NP-hard → polynomial time

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Beyond Planarity• Many planar graph algorithms generalize

• Bounded genus surface graphs

• Minor free graphs

• Generalizations are sparse and have small well-balanced separators

• But that’s often not enough!

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Surface Graphs• Drawn on 2-manifolds with boundary

• Vertices map to points, edges to internally disjoint curves

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Flows and Cuts

• Network flow algorithms for planar graphs do not generalize easily

• First results for max flows/min cuts on surfaces in 2009 [Chambers, Erickson, and Nayyeri STOC/SoCG ’09]

• Algorithms either have high dependency on g or require very complicated subroutines

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Thesis: Homology Helps• Many algorithms for surface embedded graphs can

be improved by using homology

• Homology was the key tool in the Chambers et al. flow/cut papers

• Homology characterizes curves by whether their difference bounds a weighted sum of faces

• Intuitively, how certain cycles or flows travel around features of the surface

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The Dissertation

• Algorithms for combinatorial optimization problems related to maximum flows and minimum cuts

• Assume graphs with n vertices and O(n) edges on orientable surfaces of genus g = O(n)

• All algorithms use results in homology theory

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Results1. A gO(g)n log log n time algorithm for global

minimum cut [Erickson, F, and Nayyeri SODA ’12]

2. Algorithms for shortest non-trivial cycles tailored for undirected and directed graphs [F SODA ’13]

3. A 2O(g)n² time algorithm for counting minimum s,t-cuts [Chambers, F, and Nayyeri SoCG ’13]

4. Preliminary work on computing maximum s,t-flows [Erickson and F ‘13]

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Global Minimum Cut• Remove smallest weight set of edges to separate

graph into non-empty components

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• gO(g)n log log n time algorithm matches best result for planar graphs when g = O(1) [Łącki and Sankowski ’11]

Dual Graph

u*

v*

f* g*f g

u

v

• vertices ⇄ faces • edges ⇄ edges • faces ⇄ vertices

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Separating Subgraphs• Find a minimum weight set of cycles in the dual

graph that separate the faces

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• Reduce problem to two cases

1. some minimum weight subgraph bounds a disk 2. all minimum weight subgraphs decomposable

into topologically non-trivial cycles






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Non-trivial Cycles• Want short cycles that are topologically non-trivial

• Particularly non-contractible or non-separating

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Contractible &

Separating

Non-contractible &

Separating

Non-contractible &

Non-separating

Non-trivial Running TimesWhen Directed? Running Time Citation

Before Undirected gO(g)n log log n [Italiano et al. ’11]

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Now Undirected 2O(g)n log log n [F ’13]

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Before Directed

Non-separating: O(g² n log n) Non-contractible: O(gO(g) n log n)

[Erickson ’11]

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Before Directed

Non-separating: O(g² n log n) Non-contractible: O(gO(g) n log n)

[Erickson ’11]

Now Directed Non-contractible: O(g³ n log n) [F ’13]

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Covering Spaces• Spaces that map down to the input

• Non-trivial cycles gain structure in the right cover

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… …






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Counting Cuts• Input gives vertices s and t

• An s,t-cut is a subset of vertices containing s but not t

• Want to count s,t-cuts that minimize weight of edges leaving cut

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Counting Cuts Example

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Counting Cuts Results• 2O(g)n² time algorithm for counting minimum s,t-cuts

matches the best results for planar graphs when g = O(1) [Bezáková and Friedlander ’12]

• Count forward t,s-cuts in a directed acyclic graph

• Edges leaving forward t,s-cuts are dual to directed cycles in a particular cohomology class

• Afterward, sample minimum cuts in O(n log n) time per sample

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Maximum Flow• Classic non-trivial polynomial time algorithm • Planar case considered since the seminal maximum

flow/minimum cut paper [Ford and Fulkerson ’56]

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Figure 2From Harris and Ross [1955]: Schematic diagram of the railway network of the Western So-viet Union and Eastern European countries, with a maximum flow of value 163,000 tons fromRussia to Eastern Europe, and a cut of capacity 163,000 tons indicated as ‘The bottleneck’.

The max-flow min-cut theorem

In the RAND Report of 19 November 1954, Ford and Fulkerson [1954] gave (next to definingthe maximum flow problem and suggesting the simplex method for it) the max-flow min-cut theorem for undirected graphs, saying that the maximum flow value is equal to theminimum capacity of a cut separating source and terminal. Their proof is not constructive,but for planar graphs, with source and sink on the outer boundary, they give a polynomial-time, constructive method. In a report of 26 May 1955, Robacker [1955a] showed that themax-flow min-cut theorem can be derived also from the vertex-disjoint version of Menger’stheorem.

As for the directed case, Ford and Fulkerson [1955] observed that the max-flow min-cuttheorem holds also for directed graphs. Dantzig and Fulkerson [1955] showed, by extendingthe results of Dantzig [1951a] on integer solutions for the transportation problem to the

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[Harris and Ross ’55]

Flow in the Plane

• General sparse graphs: O(n² / log n) time [Orlin ’13]

• For undirected graphs: O(n log log n) time [Italiano et al. ’11]

• For directed graphs: O(n log n) time [Borradaile and Klein ’09]

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Flow in Surfaces• For integer capacities summing to C:

O(g8n log²n log²C) [Chambers, Erickson, and Nayyeri ’09]

• For arbitrary real capacities: gO(g)n3/2 [Chambers, Erickson, and Nayyeri ’09]

• Both algorithms are very complicated and probably outperformed by more general quadratic time algorithms

• Is there a clean O(gO(1) n log n) time algorithm?

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A New Algorithm• A true generalization of Borradaile and Klein’s

algorithm

• Uses duality between minimum cost flow problems in the primal and dual graphs

• Provably runs in strongly polynomial time

• May actually run in O(gO(1) n log n) time

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Chains and Circulations• Each (oriented) edge uv has two darts u→v and

v→u

• A [0,1,2]-chain assigns real values to the vertices, oriented edges, or faces.

• Flow is a synonym for 1-chain

• A flow f is a circulation if for each vertex v,∑uv f (uv) - ∑vu f (vw) = 0

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Capacities and Feasibility• For flow f and edge uv, we say

f (u→v) = f (uv) and f (v→u) = -f (uv)

• A capacity function c assigns real values to the darts

• Flow f is feasible if for each dart u→v, we havef (u→v) ≤ c(u→v)

• f induces residual capacity function cf, where cf (u→v) = c(u→v) - f (u→v)

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Maximum s,t-flow• An s,t-flow has ∑uv f (uv) - ∑vu f (vu) = 0 for each

vertex v except s and t

• Value of an s,t-flow is ∑s→v f (s→v)

• Given capacity function c, want to find a feasible s,t-flow of maximum value

• Value equal to minimum capacity of darts leaving any s,t-cut [Ford and Fulkerson ’56]

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Borradaile and Klein

• Algorithm of Borradaile and Klein removes residual clockwise cycles, and then sends flow along leftmost residual s,t-paths

• But what does leftmost mean in surfaces with genus?

• Erickson [’10] offers a different interpretation

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Borradaile and Klein(according to Erickson)

A planar graph with capacity function c has a feasible s,t-flow matching the value of s,t-flow f if and only if there are no negative length dual cycles wrt cf.

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[Erickson ‘10]

s

t

C o

s*

t*

• Negative length cycles are over-saturated s,t-cuts[Venkatesan ‘83]

Parametric Shortest Paths• Compute a dual shortest path tree from any dual

vertex o incident to t using c for the dart lengths

• Continuously increase flow from s to t along any path, changing residual capacities

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[Erickson ‘10]

Parametric Shortest Paths• Shortest path tree pivots at critical moments…

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[Erickson ‘10]

Parametric Shortest Paths• Shortest path tree pivots at critical moments…

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[Erickson ‘10]

• …until a directed cycle is formed

Parametric Shortest Paths• The directed cycle is dual to a minimum s,t-cut

• Each pivot can be performed in O(log n) time

• O(n) pivots (not obvious), so O(n log n) overall

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[Erickson ‘10]

… on Surfaces• The flow you send from s to t affects how the

shortest path tree changes

• A flow f may induce negative cycles in the dual residual graph even if there are feasible flows of the same value [Chambers, Erickson, and Nayyeri ’09]

• And it may take Ω(n²) pivots to reach a negative cycle [Erickson ’10]

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Boundary Circulations• The boundary of a 2-chain α is the flow ∂α where ∂α(uv) = α(right(uv)) - α(left(uv))

• Boundary circulations are the boundaries of 2-chains

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72 5

Homology• Two flows f and f’ are homologous if f - f’ is a

boundary circulation

• Forms an equivalence relation between flows

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Coflows and Capacities• Flows, circulations, etc. are called coflows,

cocirculations, etc. in the dual

• Let Θ be a coflow and c be a capacity function. Define c’ on the edges so that - c’(uv) = c(u→v) if Θ(uv) ≥ 0 - c’(uv) = -c(v→u) otherwise

• The capacity of Θ wrt c is the dot product ⟨Θ, c’⟩

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Homological Max Flow/Min Cut

• Extends a theorem of Chambers, Erickson, and Nayyeri [’09]

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Theorem: Let c be a capacity function and Θ be a coflow. The maximum cost ⟨Φ, Θ⟩ of any feasible circulation Φ is equal to the minimum capacity of any coflow cohomologous to Θ.

Min Capacity Coflows

• Fix a circulation Φ and capacity function c. If coflow Θ is minimum capacity for its cohomology class wrt c then Θ is minimum capacity wrt cΦ

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Back to the Plane• Find a minimum cost coflow

• Continuously increase flow from s to t along any path, changing residual capacities

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1

11

1

1

12

23

3

4

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Back to the Plane• Minimum cost cocirculation pivots at critical

moments…

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1

11

1

1

12

23

3

4

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11

1

12

3

214

5

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1

• …by pushing coflow around s,t-cuts

Optimal Circulations and Coflows

• Fix a coflow Θ and capacity function c. Circulation Φ maximizes ⟨Φ, Θ⟩ and Θ is minimum capacity for its cohomology class wrt c if and only if Φ(u→v) = c(u→v) for every dart u→v such that Θ(u→v) > 0

• Φ saturates the darts carrying coflow

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Generalizations• Extend parametric minimum capacity coflows to

arbitrary surfaces and cohomology classes

• Minimum cost coflow is resilient to circulations; can push flow from s to t along any path

• Maintain feasible flow saturating darts with positive coflow

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1 2

s

t

coflow

Augmentations• D: edges carrying non-zero coflow

• Cannot send primal flow through edges in D if we want to keep darts saturated

• Push flow through G \ D until there is an s,t-cut S with saturated outgoing darts C in G \ D

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1 2

s

t 1 2

s

t

outgoing darts C in G \ Dcoflow edges D

Pivots• Some edges from C + D bound s,t-cut S in G

• Push coflow around coboundary of S until an edge goes empty

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1 2

s

t

1

3

s

t

coboundary of s,t-cut S in G

Pivots• Pushed coflow around a coboundary so the new

coflow is in the same cohomology class

• Every dart carrying positive coflow is saturated, so coflow is still minimum cost

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1 2

s

t

1

3

s

t

Termination

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• Process ends when no edge can go empty

• We’ve saturated a minimum s,t-cut in G and computed a maximum s,t-flow

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s

t

1

3

s

t

1

saturated s,t-cut

Efficient Pivots• Initial coflow runs along a shortest path tree from o

• o is the only “source” so edges with dual coflow stay connected

• Both D and G \ D have a simple structure, so O(g log n + g²) time per pivot

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s

t

1 1

11 3

3

7o

Time AnalysisTheorem: Let N be the number of pivots. Our algorithm computes a maximum s,t-flow inO(n log n + gN(g + log n)) time.

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• Ideally, N = O(gO(1)n), even for arbitrary cohomology classes

• Right now, can prove N = O(n²)

Winding Numbers• For s,t-path π, let f π be the flow that sends 1 unit on

each dart of π

• The winding number of coflow Θ wrt π is ⟨f π, Θ⟩

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1 2

s

t

winding number = 2

More Winding

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1 2

s

t

1

3

s

t

• The winding number of a fixed path π wrt each minimum cost coflow Θ increases by at least 1 with each pivot

winding number = 2 winding number = 3

Means Less Winding

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1 2

s

t

• Each iteration’s augmenting paths avoid edges used for the minimum cost coflow

• The winding number of each augmenting path π wrt the original minimum cost coflow Θ₀ decreases by at least 1 with each pivot

winding number = 0 winding number = -1

1 2

s

t

Bounding Pivots

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s

t

1 1

11 3

3

7o

• In the shortest path coflow Θ₀, Θ₀(u→v) ≤ O(n)

• So lowest winding number possible for any augmenting path is -O(n²)

• First augmenting paths have winding number 0 wrt Θ₀

Bounding Pivots

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• So after O(n²) pivots, there can never be another augmenting path avoiding edges with coflow

Theorem: Our algorithm computes a maximum s,t-flow in O(gn²(g + log n)) time.

Next Steps in Flows

• Prove O(gO(1)n) bound on number of pivots

• If it applies to arbitrary coflows, there are likely applications to minimum cost flow, multiple-source multiple-sink maximum flow, …

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What Comes Next?

• Postdoc at ICERM at Brown University participating in the Spring 2014 program Network Science and Graph Algorithms

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Thank you!

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