Improved Approximation for Orienting Mixed Graphs

1. Improved Approximation for Orienting Mixed Graphs Moti Medina EE School, Tel-Aviv University Iftah Gamzu CS Division, The Open Univ., and CS School, Tel-Aviv University

2. Interactions! • Biological networks, communication networks…and more

3. 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. A request (s,t) is satisfied if there is a directed path s ⇝ t in . Goal: Maximize the number of satisfied requests. Before I speak, I have something important to say.

4. An Example This edge is directed All the edges are now oriented • Four requests: • We satisfied ¾ of the requests.

5. 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]

6. 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?

7. Think Global! Orient Local! 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. • Local Requests: • v1 →v2 • v3 → v2 • v1 → v Those are my principles, and if you don't like them... well, I have others.

8. From Local to Global Orientation, cont. • Lemma: • Given a localorientation 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

9. From Local to Global Orientation, cont. • Two main cases. • Edge e appears after v in both paths. • 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. d1 + 1 ≤ d2, d2 + 1 ≤ d1. A contradiction! No man goes before his time - unless the boss leaves early.

10. 2/3 of the talk..

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

12. 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. Maximal number of requests cross it Budget: a way of going broke methodically.

13. 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. 

14. 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.

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