Design of capacitated networks with unsplittable shortest path - - PDF document

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Design of capacitated networks with unsplittable shortest path - - PDF document

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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

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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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Unsplittable shortest path routing Setting

  • Digraph
  • Commodities

with demands Induced arc flows Def: Lengths define an unsplittable shortest path routing if there is a unique shortest

  • path

for each Def: Lengths define an unsplittable shortest path routing if there is a unique shortest

  • path

for each

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

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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

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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

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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

Obs:

LUSPR≥Ω(V2) { LMCF, LUFP, LECMP }

where L := min maxa fa/ca

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

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Previous and related work

Flow-based approaches

Optimize end-to-end flows ⇔ Find compatible weights

  • Integer linear programming [BleyKoch02, HolmbergYuan01, Prytz02, ...]

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, ...]

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.

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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 if P1[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 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]

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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 sets Minimal Non-SPS = circuits in indep. system = conflict hyperedges Conflicting 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 factor

  • f min( |V|/2, maxP∈ Q|P| ).
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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.

Corollary: Computing the rank of an arbitrary path set is NP-hard. Thm [B’04]: Finding the maximum cardinality or maximum weight SPS for 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∈ {0,1}P.

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 xP is NP-hard.

But: Single pricing iteration can be solved polynomially in the size of the current restricted formulation (via k-shortest path algorithm).

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  • superadditive metric inequalities [BelaidouniBenAmeur’03]
  • rank inequalities for
  • induced cover inequalities for arc-capacity knapsacks

Valid inequalities for Path-ILP

Thm: Separation is NP-hard. (Even computing rhs for given Q!) Path-ILP is intersection of and

Induced cover inequalities

Capacity constraints define knapsacks with precedence constraints

Paths in network Precedences from subpath relation P5 P4 P3 P2 P1 P6 P7

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Induced cover: such that Induced cover inequality [Boyd‘93, vdLeensel+‘97,ParkPark’95]:

Induced cover inequalities

Capacity constraints define knapsacks with precedence constraints Previous slide‘s example: ya = 6, all demands = 1 Induced cover inequality Cover inequality Induced cover: such that Induced cover inequality [Boyd‘93, vdLeensel+‘97,ParkPark’95]:

Induced cover inequalities

Capacity constraints define knapsacks with precedence constraints Network has bounded degree → precedence graph has bounded tree width → separable in pseudopolynomially via dynamic programming [JohnsonNiemi’83]

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Induced cover: such that Induced cover inequality [Boyd‘93, vdLeensel+‘97,ParkPark’95]:

Induced cover inequalities

Capacity constraints define knapsacks with precedence constraints Obs: If network is `sufficiently‘ connected, any induced cover facet of precedence constrained arc-capacity knapsack is facet of Path-ILP. Separation 1. Strengthened metric inequalities (heuristically find tight network-cuts) 2. Induced cover inequalities (Greedily grow covers) 3. Clique inequalities in simple conflict graph

(Greedily find disjoint maximal covers)

4. CSPS inequalities (Greedy + IIS of infeasible ISP-LP) Pricing

  • k-shortest path to find best paths not in restricted formulation

Branching

  • few capacity levels: interchange between xP and za

t

  • many capacity levels: only on xP
  • biggest demands / incremental capacities first

Heuristics

  • λ = duals of capacity constraints
  • λ = solution ISP-LP for integer and near-integer routing paths

Implementation

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Results Path-ILP

Problem V E K Initial LP Root LB UB Gap Time (sec) BB-Nodes B-WiN 10 45 90 1795 7417 7417 7417 0.0 78 14 G-WiN 11 47 110 1643 2205 2988 2988 0.0 3021 25633 X-WiN 42 63 250 9641 11567 12087 12356 2.2 10000 75752 Atlanta 15 22 210 4765 40994 95700 95700 0.0 41 101 DiYuan 11 42 22 3161 7189 8124 8124 0.0 15 133 PDH 11 34 24 4594 11680 13037 13037 0.0 12 178 Polska 12 18 66 5866 7114 8555 8555 0.0 23 291 Nobel-EU 28 41 378 8172 8401 8454 9374 10.9 10000 1382 Nobel-GER 17 26 121 1475 1565 1621 1789 10.3 10000 5472 Nobel-US 14 21 91 1212 10266 11866 13334 12.4 10000 34793 France 25 45 300 1888 1990 1994 2340 17.4 10000 297 Norway 27 51 702 8536 63731 63767 63767 0.0 1299 1 TA1 24 55 391 2345 10776 10776 10776 0.0 56 1 Newyork 16 49 240 3610 16527 16527 35124 112.5 10000 3

Major problems:

  • inefficient pricing
  • huge conflict graphs even with very tight length restrictions

France: 2233 paths, 538.940 simple conflicts Norway: 4260 paths, 2.645.995 simple conflicts Sun: 5106 paths, 6.250.921 simple conflicts

Path length restrictions: shortest path length + 1

Arc-Flow ILP model

Alternative: Arc-flow based formulation. (No pricing necessary)

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Shortest Forwarding Arc Systems Analog to Path Formulation:

  • such that each

is contained in the unique shortest path r-arborescence for some compatible lengths form an independence system

  • is polynomially solvable.
  • Max-weight-independent-set and min-weight-circuit in are APX-hard.

Apply same machinery to solve Arc-Flow ILP 1. Strengthened metric inequalities 2. Induced cover inequalities

(Weeker than for Path-ILP, but stronger than plain cover inequalities)

3. Special rank inequalities

  • forcing paths to form arborescences and anti-arborescences
  • ensuring B-property for integer solutions

4. Circuit-rank inequalities (Greedy + IIS of infeasible ISP-LP)

Results Arc-Flow-ILP

Problem V E K Initial LP Root LB UB Gap Time (sec) BB-Nodes B-WiN 10 45 90 1795 5698 7417 7417 0.0 183 100 G-WiN 11 47 110 1643 2165 2944 2988 1.1 10000 32610 X-WiN 42 63 250 9641 11830 12268 12268 0.0 718 6089 Atlanta 15 22 210 4765 65156 95700 95700 0.0 24 36 DiYuan 11 42 22 3161 7129 8124 8124 0.0 18 124 PDH 11 34 24 4594 11377 13037 13037 0.0 12 280 Polska 12 18 66 5866 7329 8555 8555 0.0 28 250 Nobel-EU 28 41 378 8172 8395 8461 9267 9.5 10000 1816 Nobel-GER 17 26 121 1475 1576 1645 1789 8.8 10000 14406 Nobel-US 14 21 91 1212 10387 11835 13334 12.7 10000 24131 France 25 45 300 1888 1991 1997 2320 16.2 10000 2388 Norway 27 51 702 8536 62763 63767 63767 0.0 88 6 TA1 24 55 391 2345 10515 10776 10776 0.0 30 8 Newyork 16 49 240 3610 15030 18152 18152 0.0 206 266

Results comparable with path-flow model on most instances. No pricing deadlock.

Path length restrictions: shortest path length + 1

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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, which are handled explicitly in these models.

(Lagrangean relaxation and heuristics are better for large instances.)

Generalizes straightforward to survivable USPR.

Conclusions

Enjoy Dinner!