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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!
Improved Approximation for the General Case• Techniques– Greedy approach.– Local-to-global orientation property.
• Main result
I think you’ve got something there, but I’ll wait outside until you clean it up.
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!