A. Bley, Aussois, 12-Jan-2006 1 Overview Practical background: Internet = shortest path routing Properties of unsplittable shortest path systems Integer programming models Implementation and computational results Design of capacitated networks with unsplittable shortest path routing Andreas Bley Zuse Institute Berlin (ZIB) Practical Background: Internet Routing Internet: Shortest Path Routing (OSPF, BGP, IS-IS, RIP, …) Assign routing weights to links Send data packets via shortest paths Administrative Routing Control Only by changing the routing weights Variants Link-state vs. Distance-vector Unsplittable vs. Multi-path (ECMP) 3 4 5 4 1 1 1 2 1 1 1 3 3 1 1 1 1 HH H L K F Ka M N B S
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Design of capacitated networks with unsplittable shortest path … · 2016-01-13 · A. Bley, Aussois, 12-Jan-2006 2 Unsplittable shortest path routing Setting •Digraph • Commodities
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A. Bley, Aussois, 12-Jan-2006 1
Overview
Practical background: Internet = shortest path routing
Properties of unsplittable shortest path systems
Integer programming models
Implementation and computational results
Design of capacitated networks withunsplittable shortest path routing
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.
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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: P1 and P2 have the B-property ifP1[u,v]=P2[u,v] for all u,v with P1[u,v] ≠ ∅ and P2[u,v] ≠∅. Otherwise P1 and P2 conflict.
Obs: Paths of an USPR have B-property.
for all pairs of conflicting paths P1 and P2
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Bellman property
Obs [BenAmeur00]: In undirected cycles and hat-cycles, any path setwith 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)
Other non-combinatorial properties [Brostroem and Holmberg ’05]
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Obs: Shortest Path Systems form an indepenendence system . (But not a matroid!)
Representation: weakly stable sets in conflict hypergraph
Shortest Path Systems
Maximal SPS = bases in indep. system = maximal weakly stable setsMinimal Non-SPS = circuits in indep. system = conflict hyperedgesConflicting paths = rank 1 circuits = simple conflict edges
Def: Path set is Shortest Path System (SPS) if compatible lengths exists (each is unique shortest path).
Theorem: One can decide polynomially whether or not.
Shortest Path Systems
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).
Inequalities (1) polynomially separable via 2-shortest path algorithm.
ISP is equivalent to solving the linear system:
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 factorof min( |V|/2, maxP∈ Q|P| ).
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Shortest Path Systems
Corollary: Given a non-SPS , one can find in polynomialtime 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.
Corollary: Computing the rank of an arbitrary path set is NP-hard.
Thm [B’04]: Finding the maximum cardinality or maximum weight SPSfor some is NP-hard.
Obs: Rank-quotient of may become arbitrarily large.
Reduction from Maximum-3-SAT yields inapproximability within 8/7-ε.
Path-ILP
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Path-ILP
Circuit inequalities for suffice for ILP description.
Model with exponentially many variables and constraints!
Algorithmic properties of Path-ILP
Obs: There are instances, where the optimal LP solution has exponentially many non-zero path variables xP.
Thm: Separation problem for inequalities (*) is NP-hard for x∈ [0,1]P.
Equivalent to finding a minimum weight non-SPS.
Thm: Separation problem for inequalities (*) is polynomial for x∈ 0,1P.
Equivalent to finding some irreducible non-SPS
We can at least cut-off infeasible binary vectors x∈ 0,1P efficiently in a branch-and-cut algorithm.
Thm: Pricing problem for xP is NP-hard.
But: Single pricing iteration can be solved polynomially in the size of the currentrestricted formulation (via k-shortest path algorithm).
LP Models that solve small instances to optimality and produce good solutions and bounds for medium size instances.
Not well-suited for large instances (yet)Problem: Interdependencies among commodities routings, whichare handled explicitly in these models.(Lagrangean relaxation and heuristics are better for large instances.)