Design of capacitated networks with unsplittable shortest path - PDF document

Design of capacitated networks with unsplittable shortest path routing Andreas Bley Zuse Institute Berlin (ZIB) Overview Practical background: Internet = shortest path routing Properties of unsplittable shortest path systems Integer programming models Implementation and computational results Practical Background: Internet Routing Internet: Shortest Path Routing (OSPF, BGP, IS-IS, RIP, …) HH Assign routing weights to links � 4 3 � Send data packets via shortest paths 4 5 H B Administrative Routing Control 2 1 1 1 L Only by changing the routing weights � K 1 1 F Variants 3 1 3 � Link-state vs. Distance-vector N Ka � Unsplittable vs. Multi-path (ECMP) 1 1 1 1 S M A. Bley, Aussois, 12-Jan-2006 1

Unsplittable shortest path routing Setting • Digraph • Commodities with demands Def: Def: Lengths Lengths define an unsplittable shortest path routing if define an unsplittable shortest path routing if there is a unique shortest there is a unique shortest -path -path for each for each Induced arc flows Relation to other routing schemes Multicommodity flow Fractional flow for each All commodities independent . Unsplittable flow Single path for each All commodities independent . Unsplittable shortest path routing Unique shortest path for each Interdependencies among all Shortest Multi-Path routing All shortest paths for each Interdependencies among A. Bley, Aussois, 12-Jan-2006 2

Relation to other routing schemes Multicommodity flow Fractional flow for each All commodities independent . Unsplittable flow Single path for each All commodities independent . Unsplittable shortest path routing Unique shortest path for each Interdependencies among all Shortest Multi-Path routing All shortest paths for each Interdependencies among Relation to other routing schemes Multicommodity flow Fractional flow for each All commodities independent . Unsplittable flow Single path for each All commodities independent . Unsplittable shortest path routing Unique shortest path for each Interdependencies among all Shortest Multi-Path routing All shortest paths for each Interdependencies among A. Bley, Aussois, 12-Jan-2006 3

Relation to other routing schemes Multicommodity flow Fractional flow for each All commodities independent . Unsplittable flow Single path for each All commodities independent . Unsplittable shortest path routing Unique shortest path for each Interdependencies among all Shortest Multi-Path routing All shortest paths for each Interdependencies among Relation to other routing schemes Multicommodity flow Fractional flow for each All commodities independent . Unsplittable flow Single path for each All commodities independent . Unsplittable shortest path routing Unique shortest path for each Interdependencies among all Shortest Multi-Path routing All shortest paths for each Interdependencies among A. Bley, Aussois, 12-Jan-2006 4

Relation to other routing schemes Multicommodity flow Fractional flow for each All commodities independent . Unsplittable flow Single path for each All commodities independent . Unsplittable shortest path routing Unique shortest path for each Interdependencies among all Shortest Multi-Path routing All shortest paths for each Interdependencies among L USPR ≥ Ω (V 2 ) { L MCF , L UFP , L ECMP } Obs: where L := min max a f a /c a Capacitated network design with USPR Instance Digraph Available arc capacities with costs Commodities with demands Solution Arc lengths that define an USPR for , Capacity installation such that Objective Practice additional hop & delay constraints symmetric bidirected capacities node hardware, technical compatibility, budgets symmetric routing A. Bley, Aussois, 12-Jan-2006 5

Previous and related work Structure of path sets of undirected USPR [BenAmeurGourdin00,BrostroemHolmberg05,Farago+98] Necessary conditions for paths to form an USPR LP for finding lengths that induce given paths Weight-based approaches Modify lengths ⇒ Evaluate effects on routing • Local Search, Genetic Algorithms, ... [BleyGrötschelWessäly98, FaragoSzentesiSzvitatovszki98, FortzThorup00, EricssonResendePardalos01, BuriolResendeRibeiroThorup03, ...] • Lagrangian Approaches [LinWang93, Bley03, ...] Flow-based approaches Optimize end-to-end flows ⇔ Find compatible weights • Integer linear programming [BleyKoch02, HolmbergYuan01, Prytz02, ...] General approach Idea of our flow based models: 1. Start with an ILP formulation for unsplittable flow version. 2. Add constraints ensuring that paths form an USPR . 3. Optimize over this model. 4. Find compatible arc lengths afterwards . A. Bley, Aussois, 12-Jan-2006 6

Path-ILP model Version 1: binary variables for end-to-end routing paths Path-ILP: Question: What is USPR (inequalities) ? Bellman property (or subpath property) Def: P 1 and P 2 have the B-property if P 1 [u,v]= P 2 [u,v] for all u,v with P 1 [u,v] ≠ ∅ and P 2 [u,v] ≠ ∅ . Otherwise P 1 and P 2 conflict. Obs: Paths of an USPR have B-property. for all pairs of conflicting paths P 1 and P 2 A. Bley, Aussois, 12-Jan-2006 7

Bellman property Obs [BenAmeur00]: In undirected cycles and hat-cycles, any path set with the B-property is an USPR. • with parallel edges • sufficient for blocks Bellman property Obs: There are path sets with (gen.) B-property that are not USPRs. Further USPR properties: (all necessary but insufficient ) Generalized B-property = B-property w.r.t. subgraphs subsumes cyclic & generalized cyclic comparability of [BenAmeurGourdin’00] Other non-combinatorial properties [Brostroem and Holmberg ’05] A. Bley, Aussois, 12-Jan-2006 8

Shortest Path Systems Def: Path set is Shortest Path System (SPS) if compatible lengths exists (each is unique shortest path). Obs: Shortest Path Systems form an indepenendence system . (But not a matroid!) Representation: weakly stable sets in conflict hypergraph Maximal SPS = bases in indep. system = maximal weakly stable sets Minimal Non-SPS = circuits in indep. system = conflict hyperedges Conflicting paths = rank 1 circuits = simple conflict edges Shortest Path Systems Theorem: One can decide polynomially whether or not. Inverse Shortest Paths (ISP) problem (with uniqueness) Given: Digraph D=(V,A) and path set Q. Task: Find compatible lengths for Q (or prove that none exist). ISP is equivalent to solving the linear system: Inequalities (1) polynomially separable via 2-shortest path algorithm. Remark: Only small integer lengths admissible in practice . Thm [B‘04]: Finding min integer λ is APX-hard. Thm [BenAmeurGourdin00]: Finding min integer λ is approximable within a factor of min( |V|/2, max P ∈ Q |P| ). A. Bley, Aussois, 12-Jan-2006 9

Shortest Path Systems Corollary: Given a non-SPS , one can find in polynomial time an irreducible non-SPS with . Algorithm: Greedily remove paths from Q and check if remainder is SPS. Thm [B’04]: Finding the minimum cardinality or minimum weight irreducible non-SPS for is NP-hard. Reduction from Minimum Vertex Cover yields inapproximability within 7/6- ε. Thm [B’04]: Finding the maximum cardinality or maximum weight SPS for some is NP-hard. Reduction from Maximum-3-SAT yields inapproximability within 8/7- ε. Corollary: Computing the rank of an arbitrary path set is NP-hard. Obs: Rank-quotient of may become arbitrarily large. Path-ILP A. Bley, Aussois, 12-Jan-2006 10

Path-ILP Circuit inequalities for suffice for ILP description. Model with exponentially many variables and constraints ! Algorithmic properties of Path-ILP Separation problem for inequalities (*) is NP-hard for x ∈ [0,1] P . Thm: Equivalent to finding a minimum weight non-SPS. Separation problem for inequalities (*) is polynomial for x ∈ {0,1} P . Thm: Equivalent to finding some irreducible non-SPS We can at least cut-off infeasible binary vectors x ∈ {0,1} P efficiently in a branch- and-cut algorithm. Thm: Pricing problem for x P is NP-hard. Obs: There are instances, where the optimal LP solution has exponentially many non-zero path variables x P . But: Single pricing iteration can be solved polynomially in the size of the current restricted formulation (via k-shortest path algorithm). A. Bley, Aussois, 12-Jan-2006 11

Valid inequalities for Path-ILP Path-ILP is intersection of and • superadditive metric inequalities [BelaidouniBenAmeur’03] • rank inequalities for Thm: Separation is NP-hard. (Even computing rhs for given Q!) • induced cover inequalities for arc-capacity knapsacks Induced cover inequalities Capacity constraints define knapsacks with precedence constraints P 5 P 4 P 7 P 3 P 2 P 6 P 1 Paths in network Precedences from subpath relation A. Bley, Aussois, 12-Jan-2006 12

Induced cover inequalities Capacity constraints define knapsacks with precedence constraints Induced cover: such that Induced cover inequality [Boyd‘93, vdLeensel+‘97,ParkPark’95]: Previous slide‘s example: y a = 6, all demands = 1 Induced cover inequality Cover inequality Induced cover inequalities Capacity constraints define knapsacks with precedence constraints Induced cover: such that Induced cover inequality [Boyd‘93, vdLeensel+‘97,ParkPark’95]: Network has bounded degree → precedence graph has bounded tree width → separable in pseudopolynomially via dynamic programming [JohnsonNiemi’83] A. Bley, Aussois, 12-Jan-2006 13