[PPT] - Fast distributed approximation algorithms for vertex cover and set PowerPoint Presentation

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Fast distributed approximation algorithms for vertex cover and set cover in anonymous networks

Jukka Suomela

Helsinki Institute for Information Technology HIIT University of Helsinki, Finland Braunschweig, 29 November 2010 Joint work with Matti Åstrand

2789 1378 3 2

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Vertex cover problem

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Vertex cover C

for a graph G:

Subset C of nodes

that “covers” all edges of the graph

Each edge has at least
ne endpoint in C
Can we find a small

vertex cover?

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Vertex cover problem

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Classical NP-hard
ptimisation problem
Simple 2-approximation

algorithm: endpoints

f a maximal matching
No polynomial-time

algorithm with approximation factor 1.999 known

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Research question

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Distributed approximation

algorithms for vertex cover

Find a small vertex cover in

any communication network

Best possible approximation ratio
As fast as possible: running time independent of n
Weakest possible models:

no randomness, no unique node identifiers

Let’s first define the models...

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Distributed algorithms

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Communication graph G
Node = computer
Edge = communication

Edge = link

Computers exchange

messages and finally decide whether they are in vertex cover C

“Local output”, 0 or 1

G

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Distributed algorithms

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All nodes are identical,

run the same algorithm

We can choose

the algorithm

An adversary chooses

the structure of G

Our algorithm must

produce a valid vertex cover in any graph G G

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Model 1: Unique identifiers

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The “standard model”
Node identifiers

are a subset of 1, 2, ..., poly(n)

Subset chosen

by adversary 6 1 3 2

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Model 2: Port-numbering model

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No unique identifiers
A node of degree d can

refer to its neighbours by integers 1, 2, ..., d

Port-numbering chosen

by adversary

1 1 1 1 2 2 2 3 1 1 2 2 2 3 1 1

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Model 3: Broadcast model

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No identifiers,

no port numbers

A node has to send

the same message to each neighbour

A node does not know

which message was received from which neighbour (multiset)

Send “A” Send “B” Receives: 2 × “A” 1 × “B” Send “A”

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Deterministic distributed algorithms for vertex cover

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Guaranteed approximation ratios?
E.g., 2-approximation of minimum vertex cover =

at most 2 times as large as the smallest vertex cover

Fast?
Time = number of communication rounds
n = number of nodes
Δ = maximum degree
In weak models of distributed computing?

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Deterministic distributed algorithms for vertex cover: approximation ratios

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Time lower upper lower upper lower upper O(n) f(Δ) + polylog(n) f(Δ) + O(log* n) f(Δ) 1 Broadca model Broadcast model Po number Port numbering Unique identif Unique identifiers

Trivial algorithm

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Deterministic distributed algorithms for vertex cover: approximation ratios

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Time lower upper lower upper lower upper O(n) f(Δ) + polylog(n) f(Δ) + O(log* n) f(Δ) 1 2 2 Broadca model Broadcast model Po number Port numbering Unique identif Unique identifiers

Maximal matching

(Panconesi & Rizzi 2001)

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Deterministic distributed algorithms for vertex cover: approximation ratios

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Time lower upper lower upper lower upper O(n) f(Δ) + polylog(n) f(Δ) + O(log* n) f(Δ) 2 1 2 2 2 Broadca model Broadcast model Po number Port numbering Unique identif Unique identifiers

Near-maximal edge packing

(Khuller et al. 1994)

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Deterministic distributed algorithms for vertex cover: approximation ratios

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Time lower upper lower upper lower upper O(n) f(Δ) + polylog(n) f(Δ) + O(log* n) f(Δ) 2 1 2 2 2 + ε 2 2 + ε 2 + ε Broadca model Broadcast model Po number Port numbering Unique identif Unique identifiers

Deterministic LP rounding

(Kuhn et al. 2006)

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Deterministic distributed algorithms for vertex cover: approximation ratios

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Time lower upper lower upper lower upper O(n) f(Δ) + polylog(n) f(Δ) + O(log* n) f(Δ) 2 1 2 2 2 + ε 2 2 2 2 + ε 2 2 + ε Broadca model Broadcast model Po number Port numbering Unique identif Unique identifiers

Czygrinow et al. 2008 Lenzen & Wattenhofer 2008

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Deterministic distributed algorithms for vertex cover: approximation ratios

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Time lower upper lower upper lower upper O(n) f(Δ) + polylog(n) f(Δ) + O(log* n) f(Δ) 2 2 2 1 2 2 2 2 2 2 2 + ε 2 2 2 2 + ε 2 2 + ε Broadca model Broadcast model Po number Port numbering Unique identif Unique identifiers

Trivial (cycles)

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Deterministic distributed algorithms for vertex cover: approximation ratios

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Time lower upper lower upper lower upper O(n) f(Δ) + polylog(n) f(Δ) + O(log* n) f(Δ) 2 2 2 1 2 2 2 2 2 2 2 + ε 2 2 2 2 + ε 2 2 + ε Broadca model Broadcast model Po number Port numbering Unique identif Unique identifiers

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Deterministic distributed algorithms for vertex cover: approximation ratios

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Time lower upper lower upper lower upper O(n) f(Δ) + polylog(n) f(Δ) + O(log* n) f(Δ) 2 ? 2 2 1 2 ? 2 2 2 2 ? 2 2 + ε 2 2 ? 2 2 + ε 2 2 + ε Broadca model Broadcast model Po number Port numbering Unique identif Unique identifiers

Could we have 2? Anything here?

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Deterministic distributed algorithms for vertex cover: approximation ratios

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Time lower upper lower upper lower upper O(n) f(Δ) + polylog(n) f(Δ) + O(log* n) f(Δ) 2 ? 2 2 1 2 ? 2 2 2 2 ? 2 2 2 2 ? 2 2 2 2 Broadca model Broadcast model Po number Port numbering Unique identif Unique identifiers

DISC 2009

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Deterministic distributed algorithms for vertex cover: approximation ratios

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Time lower upper lower upper lower upper O(n) f(Δ) + polylog(n) f(Δ) + O(log* n) f(Δ) 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Broadca model Broadcast model Po number Port numbering Unique identif Unique identifiers + faster and more general solution here

Latest results

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Deterministic distributed algorithms for vertex cover: approximation ratios

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Time lower upper lower upper lower upper O(n) f(Δ) + polylog(n) f(Δ) + O(log* n) f(Δ) 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Broadca model Broadcast model Po number Port numbering Unique identif Unique identifiers Let’s study this case first...

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Vertex cover in the port-numbering model

Convenient to study a more general problem:

minimum-weight vertex cover

More general problems

are sometimes easier to solve?

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Notation: w(v) = weight of v 5 1 3 9 6 6 9 6

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Edge packings and vertex covers

Edge packing: weight y(e) ≥ 0 for each edge e
Packing constraint: y[v] ≤ w(v) for each node v,

where y[v] = total weight of edges incident to v

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5 1 3 9 6 6 9 6 2 3 4 6 edge packing ≈ fractional matching (LP relaxation)

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Edge packings and vertex covers

Node v is saturated if y[v] = w(v)
Total weight of edges incident to v is equal to w(v),

i.e., the packing constraint holds with equality

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5 1 3 9 6 6 9 6 2 3 4 6 y[v] = w(v) y[v] < w(v)

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Edge packings and vertex covers

Edge e is saturated if

at least one endpoint of e is saturated

Equivalently: edge weight y(e) can’t be increased

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5 1 3 9 6 6 9 6 2 3 4 6 2 + ε would violate a packing constraint

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Edge packings and vertex covers

Maximal edge packing: all edges saturated

⇐⇒ none of the edge weights y(e) can be increased ⇐⇒ saturated nodes form a vertex cover

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5 1 3 9 6 6 9 6 2 3 4 6

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Edge packings and vertex covers

Maximal edge packing: all edges saturated

⇐⇒ saturated nodes form a vertex cover

... and saturated nodes are 2-approximation of

minimum-weight vertex cover (Bar-Yehuda & Even 1981)

How to find a maximal edge packing...?
Phase I: “greedy but safe”, cf. Khuller et al. (1994),

Papadimitriou & Yannakakis (1993)

Phase II: if phase I fails to saturate an edge e = {u,v},

we can break symmetry between u and v; exploit it!

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Finding a maximal edge packing: phase I

y[v] = total weight of edges incident to node v
Residual capacity of node v: r(v) = w(v) − y[v]
Saturated node:

r(v) = 0

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5 1 3 9 6 6 9 6 2 3 4 6 1 6 4 3 1 w(v): r(v):

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Finding a maximal edge packing: phase I

Start with a trivial edge packing y(e) = 0

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5 1 3 9 6 6 6 1 6 3 6 6 5 9 9 9 w(v): r(v):

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Finding a maximal edge packing: phase I

Each node v offers r(v)/deg(v) units to each incident edge

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5 1 3 9 6 6 6 1 6 3 6 6 5 9 9 9 w(v): r(v): 9

ffer:

1 5/2 3/2 6 3/2 3/2 6

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Finding a maximal edge packing: phase I

Each edge accepts the smallest of the 2 offers it received Increase y(e) by this amount

Safe, can’t violate

packing constraints

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5 1 3 9 6 6 6 1 3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2 1 6 3 6 6 5 9 9 9 9 1 5/2 3/2 6 3/2 3/2 6

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Finding a maximal edge packing: phase I

Update residuals...

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5 1 3 9 6 6 6 1 3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2 9/2 3/2 2 1/2 9 15/2

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Finding a maximal edge packing: phase I

Update residuals, discard saturated nodes and edges...

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5 1 3 9 6 6 6 1 3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2 9/2 3/2 2 1/2 9 15/2

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Finding a maximal edge packing: phase I

Update residuals, discard saturated nodes and edges, repeat... Offers...

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5 1 3 9 6 6 6 1 3/2 3/2 3/2 3/2 3/2 3/2 3/2 3/2 9/2 3/2 2 1/2 9 15/2 15/2 2 1/6 9/2 –

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Finding a maximal edge packing: phase I

Update residuals, discard saturated nodes and edges, repeat... Offers... Increase weights...

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5 1 3 9 6 6 6 1 5/3 5/3 3/2 3/2 3/2 3/2 3/2 9/2 3/2 2 1/2 9 15/2 15/2 2 1/6 9/2 – 5/3

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Finding a maximal edge packing: phase I

Update residuals, discard saturated nodes and edges, repeat... Offers... Increase weights... Update residuals...

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5 1 3 9 6 6 6 1 3/2 3/2 3/2 3/2 3/2 13/3 3/2 10/3 9 22/3 5/3 5/3 5/3

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Finding a maximal edge packing: phase I

Update residuals, discard saturated nodes and edges, repeat... Offers... Increase weights... Update residuals and graph, etc.

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5 1 3 9 6 6 6 1 3/2 3/2 3/2 3/2 3/2 13/3 3/2 10/3 9 22/3 5/3 5/3 5/3

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Finding a maximal edge packing: phase I

We are making some progress towards finding a maximal edge packing...

But this is too slow! How to make it faster?

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1 2 4 8 16 32 64 128 1 2 4 8 16 32 64 128 1 1 2 4 8 16 32 64 1 1 2 4 8 16 32 64 1 2 4 8 16 32 1 1 2 4 8 16 1 1 2 4 8 16 1 2 4 8 1 1 2 4 1 1 2 4 1 2

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Finding a maximal edge packing: colouring trick

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4 3 2 2 5

Offer is a local minimum:
Node will be saturated
And all edges incident to it

will be saturated as well 2 2 2 2 Residual capacity was 8, will be 0

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Finding a maximal edge packing: colouring trick

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4 3 2 2 5

Offer is a local minimum:
Node will be saturated
Otherwise there is a neighbour

with a different offer:

Interpret the offer

sequences as colours

Nodes u and v have

different colours: {u, v} is multicoloured 1 2 2 2 2

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Finding a maximal edge packing: colouring trick

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4 3 2 2 5

Progress guaranteed:
On each iteration, for each node,

at least one incident edge becomes saturated or multicoloured

Such edges are be discarded

in phase I; maximum degree ∆ decreases by at least one

Hence in ∆ rounds all edges

are saturated or multicoloured 1 2 2 2 2

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Finding a maximal edge packing: colouring trick

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Colours are sequences of

∆ offers (rational numbers)

Assume that node weights

are integers 1, 2, ..., W

Then offers are rationals
f the form q/(∆!)∆ with

q ∈ {1, 2, ..., W(∆!)∆} (2, 2/3, 1/6, 1/24) (2, 2/3, 1/6, 1/12)

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Finding a maximal edge packing: colouring trick

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Colours are sequences of

∆ offers (rational numbers)

Assume that node weights

are integers 1, 2, ..., W

Then offers are rationals
f the form q/(∆!)∆ with

q ∈ {1, 2, ..., W(∆!)∆}

k = (W(∆!)∆)∆ possible

colours, replace with integers 1, 2, ..., k 2789 1378

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Finding a maximal edge packing: phase II

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Proper k-colouring of the unsaturated subgraph
Orient from lower to higher colour
Partition in ∆ forests
Use Cole–Vishkin (1986) style

colour reduction algorithm

Use colour classes

to saturate edges

O(∆ + log* W) rounds

2789 1378 3 2

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Finding a maximal edge packing: summary

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Maximal edge packing and

2-approximation of vertex cover in time O(∆ + log* W)

W = maximum node weight
Unweighted graphs:

running time simply O(∆), independent of n

Everything can be implemented

in the port-numbering model

2789 1378 3 2

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Vertex cover and set cover in anonymous networks: summary

2-approximation of vertex cover in time O(∆)

in the port-numbering model

Idea: consider a more general problem,

minimum-weight vertex cover

2-approximation of vertex cover in time poly(∆)

in the broadcast model?

Idea: consider a more general problem,

minimum-weight set cover!

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Take-home messages

Algorithms that we saw today are strictly local
Running time independent of the number of nodes
Output of a node depends only on its local neighbourhood
Very efficient, can be used in arbitrarily large networks
Deterministic, highly fault-tolerant
There are non-trivial graph problems that

can be solved with strictly local algorithms!

More: www.cs.helsinki.fi/jukka.suomela/local-survey

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