Improved Approximation for Orienting Mixed Graphs Iftah Gamzu CS Division, The Open Univ., and CS School, Tel-Aviv University Moti Medina EE 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. – E D - is the set of directed edges. – E U - 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 E U. 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.

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]

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! Those are my principles, and if you don't like them... well, I have others. Local Requests: v 1 v 2 v 3 v 2 v 1 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. s 1 t 1, s 2 t 2. – Hence there is e in E U 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. d d 2, d d 1. A contradiction!

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

Algorithm Outline 1 st phase: – While there is a request in conflict with other requests: Orient it, and reject the conflicting requests. 2 nd phase: – Pick a heavy vertex. – Orient its local requests Local-to-Global. Budget: a way of going broke methodically.

Main Result - Proof Proof outline: – We show that in each phase: – 1 st phase: This holds by design of the alg. – 2 nd phase: Pigeon-Hole Principle. Local-to-global.

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.

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