1 / 23

Braess’s Paradox, Fibonacci Numbers, and Exponential Inapproximability

Braess’s Paradox, Fibonacci Numbers, and Exponential Inapproximability. Henry Lin * Tim Roughgarden ** Éva Tardos † Asher Walkover †† * UC Berkeley ** Stanford University † Cornell University †† Google. Overview. Selfish routing model and Braess’s Paradox

torin
Download Presentation

Braess’s Paradox, Fibonacci Numbers, and Exponential Inapproximability

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Braess’s Paradox, Fibonacci Numbers, and Exponential Inapproximability Henry Lin* Tim Roughgarden** Éva Tardos† Asher Walkover†† *UC Berkeley **Stanford University †Cornell University ††Google

  2. Overview • Selfish routing model and Braess’s Paradox • New lower and upper bounds on Braess’s Paradox in multicommodity networks • Connections to the price of anarchy with respect to the maximum latency objective • Open questions

  3. Routing in congested networks • a directed graph: G = (V,E) • for each edge e, a latency function: ℓe(•) • nonnegative, nondecreasing, and continuous • one or more commodities: (s1, t1, r1) … (sk, tk, rk) • for i=1 to k, a rate ri of traffic to route from si to ti Single Commodity Example (k=1): r1=1 v ℓ(x)=1 ℓ(x)=x Flow = ½ s1 t1 ℓ(x)=x Flow = ½ ℓ(x)=1 u

  4. Selfish Routing and Nash Flows How do we model selfish behavior in networks? Def: A flow is at Nash equilibrium (or is a Nash flow) if all flow is routed on min-latency paths [at current edge congestion] • Note: at Nash Eq., all flow must have same s to t latency • Always exist & are unique [Wardrop, Beckmann et al 50s] An example Nash flow: v k=1, r1=1 ℓ(x)=1 ℓ(x)=x Flow = ½ s1 t1 ℓ(x)=x Flow = ½ ℓ(x)=1 u

  5. Braess’s Paradox • Common latency is 1.5 • Adding edge increased latency to 2! • Replacing x with xd yields more severe example where latency increases from 1 to 2 v ½ 1 ½ 1 x 0 s t ½ 1 ½ x 1 u

  6. Previous results on Braess’s Paradox In single-commodity networks: • Thm: [R 01]Adding 1 edge to a graph can increase common latency by at least a factor of 2 • Thm: [LRT 04]Adding 1 edge to a graph can increase common latency by at most a factor of 2 What about multicommodity networks?

  7. New results for BPin multicommodity networks In a network with k ≥ 2 commodities, n nodes, m edges: • Thm: Adding 1 edge to a graph can increase common latency by at least a factor of 2Ω(n) or 2Ω(m), even if k = 2 • Thm: Adding 1 edge to a graph can increase common latency at most a factor of 2O(m·logn)or 2O(kn),whichever is smaller

  8. Braess’s Paradox in MC networks t2 r1 = r2 = 1 • All unlabelled edges have 0 latency (at current flow) • Only edge leaving s1 has latency 1 • Latency between s1 and t1 is 1 • Latency between s2 and t2 is 0 1 s1 t1 s2

  9. Braess’s Paradox in MC networks t2 r1 = r2 = 1 • All unlabelled edges have 0 latency (at current flow) 1 s1 t1 1 -½ flow +½ flow s2

  10. Braess’s Paradox in MC networks t2 r1 = r2 = 1 • All unlabelled edges have 0 latency (at current flow) 1 s1 t1 1 1 -¼ flow +¼ flow s2

  11. Braess’s Paradox in MC networks t2 r1 = r2 = 1 • All unlabelled edges have 0 latency (at current flow) 1 1 s1 t1 1 1 -⅛ flow +⅛ flow s2

  12. Braess’s Paradox in MC networks t2 r1 = r2 = 1 • All unlabelled edges have 0 latency (at current flow) 1 1 s1 t1 2 1 1 -1/16 flow +1/16 flow s2

  13. Braess’s Paradox in MC networks t2 r1 = r2 = 1 • All unlabelled edges have 0 latency (at current flow) 3 -1/32 flow +1/32 flow 1 1 s1 t1 2 1 1 s2

  14. Braess’s Paradox in MC networks t2 -1/64 flow +1/64 flow 3 1 1 s1 t1 2 5 1 1 s2 • All unlabelled edges have 0 latency (at current flow)

  15. Braess’s Paradox in MC networks t2 8 -1/128 flow +1/128 flow 3 1 1 s1 t1 2 5 1 1 s2 • All unlabelled edges have 0 latency (at current flow)

  16. Braess’s Paradox in MC networks t2 • Latency between s1 and t1 increased from 1 to 9 • Latency between s2 and t2 increased from 0 to 13 8 3 1 1 s1 t1 2 5 1 1 s2 • All unlabelled edges have 0 latency (at current flow)

  17. t2 8 3 1 1 s1 t1 2 5 1 1 s2 Braess’s Paradox in MC networks • In a general network with O(p) nodes: • Latency between s1 and t1 can increase from 1 to Fp-1+1 • Latency between s2 and t2 can increased from 0 to Fp • (where Fp is the pth fibonacci number) • In fact, adding 1 edge is enough to cause this bad example

  18. Proving Upper Bounds To prove 2O(m·logn)bound, let: f be the flow before edges were added g be the flow after edges were added Main Lemma: For any edge e: ℓe(ge) ≤ 2O(m·logn)·maxe’єE(ℓe’(fe’))

  19. Proving Main Lemma Main Lemma: For any edge e: ℓe(ge) ≤ 2O(m·logn)·maxe’єE(ℓe’(fe’)) Proof (sketch):Let f, g, and ℓe(fe) be fixed. Resulting latencies ℓe(ge) must be: • nonnegative • nondecreasing • at Nash equilibrium Requirements can be formulated as a set of linear constraints on ℓe(ge)

  20. Proving Main Lemma Main Lemma: For any edge e: ℓe(ge) ≤ 2O(m·logn)·maxe’єE(ℓe’(fe’)) Proof (sketch):Let f, g, and ℓe(fe) be fixed. In fact, finding maximum ℓe(ge) can be formulated as a linear program • can show maximum occurs at extreme point • can bound extreme point solution with Cramer’s rule and a bound on the determinant

  21. Price of Anarchy with respect to Maximum Latency Objective In the Braess’s Paradox example: • The maximum si-ti latency at Nash Eq. is 2Ω(n) • An optimal flow avoiding the extra edges can have maximum si-ti latency equal to 1 New Thm:The price of anarchy wrt to the maximum latency is at least 2Ω(n). Disproves conjecture that PoA for multicommodity networks is no worse than for single-commodity networks

  22. Price of Anarchy with respect to Maximum Latency Objective • Linear programming technique not specific to Braess’s Paradox • Provides same bound for price of anarchy wrt maximum latency New Thm:The price of anarchy wrt to the maximum latency is at most 2O(m·logn) or 2O(kn), whichever is smaller

  23. Open Questions • Can the upper bounds be improved to 2O(n)or 2O(m)? • Can the lower bounds be improved to 2Ω(m·logn) or 2Ω(kn)? • What are upper and lower bounds on Braess’s Paradox and price of anarchy for atomic splittable instances?

More Related