Download presentation

Presentation is loading. Please wait.

Published byEmilia Armer Modified over 3 years ago

1
Network Design with Degree Constraints Guy Kortsarz Joint work with Rohit Khandekar and Zeev Nutov

2
Problems we consider Minimum cost 2-vertex-connected spanning subgraph with degree constraints Minimum-degree arborescence/tree spanning at least k vertices Minimum-degree diameter-bounded tree spanning at least k vertices Prize-collecting Steiner network design with degree constraints

3
Our results (1) Minimum cost 2-vertex-connected spanning subgraph with degree constraints – First paper to deal with vertex-connectivity – (6b(v) + 6)-degree violation, 4-approximation – Algorithm: Compute an MST T with degree constraints Augment T to a 2-vertex-connected spanning subgraph without violating the degree constraints by too much

4
Our results (2) Minimum-degree arborescence/tree spanning at least k vertices – Much harder than minimum-degree arborescence – Large integrality gap for natural LP: k ½ = n ¼ – No o(log n)-approximation unless NP=Quasi(P) – Combinatorial ((k log k)/OPT) ½ -approximation Minimum-degree diameter-bounded (undirected) tree spanning at least k vertices – No o(log n)-approximation unless NP=Quasi(P)

5
Our results (3) Prize-collecting Steiner network design with degree constraints – (α b(v) + β)-degree violation ρ-approximation algorithm for SNDP with degree constraints implies (α (1+1/ρ) b(v) + β)-degree violation (ρ+1)-approximation algorithm for prize-collecting SNDP with degree constraints – For example, ρ = 2, α = 1, β = 6r+3 where r = maximum connectivity requirement

6
Outline 1.Minimum cost 2-vertex-connected spanning subgraph with degree constraints 2.Minimum-degree arborescence/tree spanning at least k vertices

7
Vertex connectivity with degree constraints Minimum cost 2-vertex-connected spanning subgraph with degree constraints Algorithm: Compute an MST T with degree constraints Augment T to a 2-vertex-connected spanning subgraph without violating the degree constraints by too much

8
Augmenting vertex connectivity with degree constraints Tree T S Set S is violated if S has only one neighbor Γ(S) in T S + Γ(S) V Γ(S)

9
Vertex connectivity with degree constraints Tree T S Mengers theorem: T + F is 2-vertex-connected if and only if each violated S has an edge outside S + Γ(S). Γ(S)

10
Natural LP with degree bounds Minimize cost of the picked edges such that: – For each violated set S, there is a picked edge from S to V \ (S + Γ(S)) – Degree of any vertex v is at most b(v) Main theorem: One can iteratively round this LP to get a solution such that: – The cost is at most 3 times LP value – degree(v) 2 δ T (v) + 3 b(v) + 3 Since δ T (v) b(v) + 1, the final degree of v is at most 6 b(v) + 6.

11
Outline 1.Minimum cost 2-vertex-connected spanning subgraph with degree constraints 2.Minimum-degree arborescence/tree spanning at least k vertices

12
k ½ integrality gap … Degree = k LP sets x e = 1/k and has maximum degree = 1 Any integral solution has maximum degree k ½

13
Minimum-degree k-arborescence: Our approach Consider the optimum k-arborescence with max- degree OPT

14
Minimum-degree k-arborescence: Our approach There exist (k OPT) ½ subtrees each containing at most (k OPT) ½ terminals Portal

15
Minimum-degree k-arborescence: Our approach Algorithmic goal: find at most (k OPT) ½ portals, each sending at most (k OPT) ½ flow, such that at least k terminals receive a unit flow each Portal Constraint: each vertex can support at most OPT flow (the degree constraint)

16
Minimum-degree k-arborescence: Our approach This is a submodular cover problem We can get O(log k) approximation Thus overall ((k OPT) ½ log k)-degree k-arborescence Portal

17
Open questions How hard is it to approximate the minimum- degree arborescence spanning at least k vertices? – Can we get polylog k approximation? – Can we show k ε hardness of approximation? Is the problem easier in undirected graphs?

Similar presentations

OK

Minimum Spanning Tree Algorithms. What is A Spanning Tree? u v b a c d e f Given a connected, undirected graph G=(V,E), a spanning tree of that graph.

Minimum Spanning Tree Algorithms. What is A Spanning Tree? u v b a c d e f Given a connected, undirected graph G=(V,E), a spanning tree of that graph.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on series and parallel circuits video Ppt on accounting standard 10 fixed assets Download ppt on foundry technology Ppt on economic and labour force in india Ppt on teachers day wishes Ppt on preservation of public property auctions Ppt on asteroids and comets Download free ppt on srinivasa ramanujan Ppt on kinetic energy for class 9 Ppt on latest technology in electronics and communication free download