Improved Approximation for Orienting Mixed Graphs

Iftah GamzuCS Division, The Open Univ., and

CS School, Tel-Aviv University

Moti MedinaEE School, Tel-Aviv University

Interactions!

– Biological networks, communication networks…and more

Problem Definition: Maximum Mixed Graph Orientation

Input:• Mixed graph– V - is the set vertices.

• |V|=n.

– ED - is the set of directed edges.

– EU - is the set of undirected edges.

• Set of source-target requests .

Output:• An orientation of G

– A directed graph .– - single direction for each edge

in EU.

VVP

),( UD EEVG ),( UD EEVG

UE

A request (s,t) is satisfied if there is a directed path

s t ⇝ in . G

Goal:Maximize the number of satisfied requests.

Before I speak, I have something important to say.

An Example

• Four requests:– – – –

• We satisfied ¾ of the requests.

All the edges are now oriented

This edge is directed

Previous Work• NP-completeness proof [Arkin and Hassin 2002].• [Elberfeld et al. 2011] – NP-hardness to approximate within a factor of 7/8.– Several Polylog approximation algs for tree-like mixed graphs.

– General Setting: An - approximation greedy alg, where .

• Experimental work– Polynomial-size integer linear program formulation [Silverbush, Elberfeld,

Sharan 2011]

nM log

17071.0

||,max PnM

Our Results• Local-to-Global property.• Deterministic approximation algorithm for

maximizing the number of satisfied requests.– - approximation.– Greedy.– Applying the Local-to-Global property.

• More results:– “Shaving” log factors for tree like inputs.– Other variants of the problem…

Who are you going to believe, me or your own eyes?

From Local to Global Orientation• Orientation of a “local” neighborhood ⇒ orientation of a

“Global” neighborhood• Some definitions:

– Local neighborhood of .– Request ↦ shortest path in G.– shortest path in G ↦ Local Request (and hence a local path).– The local graph orientation problem.

Think Global!

Orient Local!

Vv

Those are my principles, and if you don't like them... well, I have others.

Local Requests:• v1 →v2

• v3 → v2

• v1 → v

From Local to Global Orientation, cont.• Lemma:– Given a local orientation that satisfies a set of

local paths, then– there is a global orientation that satisfies the set

of corresponding global paths.• Proof:– Proof by contradiction: assume that two global

paths are in conflict.• s1 → t1, s2 → t2 .

– Hence there is e in EU that gets “different” directions.

e

From Local to Global Orientation, cont.– Two main cases.

1. Edge e appears after v in both paths.2. Edge e appears after v in the first path and before v in

the second.

• Conclusion– A constant fraction of the local requests can be

oriented globally.

No man goes before his time - unless the boss leaves early.

d1 + 1 ≤ d2,d2 + 1 ≤ d1.A contradiction!

2/3 of the talk..

Improved Approximation for the General Case• Techniques– Greedy approach.– Local-to-global orientation property.

• Main result

Algorithm Outline• 1st phase:–While there is a request in conflict with other requests:• Orient it, and reject the conflicting requests.

• 2nd phase:–Pick a “heavy” vertex.–Orient its local requests• Local-to-Global.

Budget: a way of going broke methodically.

Maximal number of requests cross it

Main Result - Proof

• Proof outline:–We show that in each phase:

–1st phase: • This holds by design of the alg.

–2nd phase:• Pigeon-Hole Principle.• Local-to-global.

REJECTSAT

SAT

##

#

REJECTSAT ##

3/2

3/4

n

P

PPn 3/1

PPnn

P 3/13/1

3/21

Open Problems• Improve the approximation ratio.– O(1) vs. .

• Study variants of the problem– Orientation with fixed paths• NP hard to approximate within a factor of 1/|P|.• Designing such an algorithm is trivial.

– Orientation in grid networks • Better “lower bounds”.• The undirected case is easy.

Time flies like an arrow. Fruit flies like a banana.

THANK YOU!

You haven’t stopped talking since we got here! You must have been vaccinated with a phonograph needle!