Single source unsplittable flows with arc-wise lower and upper - - PowerPoint PPT Presentation

Single source unsplittable flows with arc-wise lower and upper bounds

Sarah Morell (TU Berlin) Joint work with Martin Skutella (TU Berlin) Aussois, January 6, 2020

Introduction

Single source unsplittable flow problem [Kleinberg 1996]

Given Digraph D ✏ ♣V , Aq, k commodities with common source s P V , sinks t1, . . . , tk P V and demands d1, . . . , dk → 0. Task Route each flow unsplittably, i.e. find flow y with path decomposition ♣yPi q1↕i↕k s.th. each path Pi is an s-ti-path and yPi ✏ di. 4 2 4 5 1 3 2 4 4 s 1 2 In the following, ✆ each node v P V lies on an s-ti-path for some commodity i, ✆ each flow has to satisfy the demands d1, . . . , dk.

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Motivation

Theorem [Dinitz, Garg, Goemans 1999] For any fractional flow x, there is an unsplittable flow y s.th. ya ↕ xa dmax for all a P A. Given any fractional flow x, ✆ does there exist an unsplittable flow satisfying arc-wise lower bounds w.r.t. flow x? ✆ does there exist an unsplittable flow satisfying both arc-wise lower and upper bounds w.r.t. flow x?

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Outlook of this talk

Theorem For any fractional flow x, there is an unsplittable flow y s.th. ya ➙ xa ✁ dmax for all a P A. ✆ Proof via lower bound preserving augmentation steps Theorem If all demands are integer multiples of each other: For any fractional flow x, there is an unsplittable flow y s.th. xa ✁ dmax ↕ ya ↕ xa dmax for all a P A. For arbitrary demands, there is an unsplittable flow y s.th. xa 2 ✁ dmax ↕ ya ↕ 2xa dmax for all a P A. ✆ Proof via an iterative rounding procedure

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Flows with arc-wise lower bounds

Definition of reachability

Let x be a fractional and y be an unsplittable flow. Definition Node v P V is reachable for commodity i w.r.t. y if v P Pi and ya ➙ xa for all arcs a on the s-v-subpath of Pi. s s 1 1 2 2

1 2 5 2

1

3 2

1 1 2 2 2

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Lower bound preserving augmentation steps

Let v P V be a node that is reachable for commodity i w.r.t. flow y. Definition An LBP augmentation step reroutes demand di from the s-v-subpath of Pi along an arbitrary s-v-path Q. ti v s Pi ≥ xa Ñ ti v s ˜ Pi ≥ xa − di

× × ×

A finite sequence of augmentation steps from y to ˜ y is denoted by y ˜ y.

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Key Lemma

Key Lemma For any unsplittable flow y and any v P V , there is a flow ˜ y with y ˜ y s.th. v is reachable w.r.t. ˜ y. Proof Assume: there is a flow y s.th. set Xy of eventually reachable nodes is not V . If y ˜ y, then X˜

y ❸ Xy.

Choose y with set Xy inclusion-wise minimal. Then, X˜

y ✏ Xy ✏: X for all ˜

y with y ˜ y.

s

∃˜ y : ˜ y ≥ x ∃y′ : y′ ≥ x

ti tj Sarah Morell: Single source unsplittable flows with arc-wise lower and upper bounds 6/16

Key Lemma

s

∃˜ y : ˜ y ≥ x ∃y′ : y′ ≥ x

The following properties are satisfied: (P1) y♣δin♣Xqq cannot decrease. Choose y with y♣δin♣Xqq maximal. (P2) δin♣Xq ✏ ❍. (P3) There is an arc a P δout♣Xq s.th. ya → 0 and ya ➙ xa. (P4) Flow on arcs in δout♣Xq cannot be added or deleted.

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Key Lemma

s

∃˜ y : ˜ y ≥ x ti v

Let a ✏ ♣v, wq P Pi: v is not eventually reachable w.r.t. i, so for every ˜ y there is an arc b P ˜ Pi⑤v with ˜ yb ➔ xb. Let by denote such an arc that is closest to v. Choose y s.th. the following two criteria are met in the given order: (i) the number of arcs succeeding by on Py

i ⑤v is maximal,

(ii) value yby is maximal.

s

∃˜ y : ˜ y ≥ x ∃yq : yq ≥ x ti v u tj

Node v is reachable w.r.t. some ˜ y with y ✏ y0 Ñ . . . Ñ yp ✏ ˜ y. Let q be the smallest index s.th. some node u succeeding by on Py

i is reachable for some

j w.r.t. yq.

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Key Lemma

s

∃˜ y : ˜ y ≥ x ∃yq : yq ≥ x ti v u tj

By choice of index q, ✆ flow has not been decreased on any arc of Py

i succeeding by,

✆ paths Py

i and P yq i

contain by ✏ byq and have identical subpaths from by to node v. Rerouting commodity j along Py

i ⑤u increases flow on arc by, a contradiction. Sarah Morell: Single source unsplittable flows with arc-wise lower and upper bounds 9/16

Proving the Theorem via Key Lemma

Corollary For any unsplittable flow y and any arc a P A, there is a flow ˜ y with y ˜ y s.th. ˜ ya ➙ xa. Proof Let y have maximal flow on a ✏ ♣v, wq P A with ya ➔ xa. By KL, node w is reachable for some i w.r.t. ˜ y with y ˜

• y. Rerouting di increases flow on a.

Theorem For any fractional flow x, there is an unsplittable flow y s.th. ya ➙ xa ✁ dmax for all a P A.

ti v s Pi ≥ xa

Ñ

ti v s ˜ Pi ≥ xa − di

× × ×

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Further results

Theorem [Dinitz, Garg, Goemans 1999] For any fractional flow x, there is an unsplittable flow y s.th. ya ↕ xa dmax for all a P A. ✆ Simpler proof via upper bound preserving augmentation steps

ti v s Pi ≤ x

Ñ

ti v s ˜ Pi ≤ x + di

× × ×

Conjecture [Goemans 2000] Given arbitrary cost c on the arcs and any fractional flow x, there is an unsplittable flow y s.th. c♣yq ↕ c♣xq and ya ↕ xa dmax for all a P A.

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Combining lower and upper bounds

Flows with arc-wise lower and upper bounds

Theorem If all demands are integer multiples of each other: For any fractional flow x, there is an unsplittable flow y s.th. xa ✁ dmax ↕ ya ↕ xa dmax for all a P A. For arbitrary demands, there is an unsplittable flow y s.th. xa 2 ✁ dmax ↕ ya ↕ 2xa dmax for all a P A. ✆ Proof via iterative rounding procedure, see [Kolliopoulos Stein 2002] and [Skutella 2002].

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Solving the special case

Consider a flow x satisfying demands d1 ⑤ d2 ⑤ . . . ⑤ dk. Theorem If all demands d1, . . . , dk are integral, then any flow x can be turned into an integral flow ˜ x s.th. txa✉ ↕ ˜ xa ↕ rxas for all a P A. While x ✘ 0: ✆ Turn x into a dmin-integral flow whose flow values differ by at most dmin. ✆ Route all commodities i of value di ✏ dmin unsplittably by iteratively choosing a flow-carrying s-ti-path Pi. Decrease flow along Pi and delete commodity i. Set y :✏ ♣yPi q1↕i↕k: its flow values differ from x by at most dmax ✁ dmin.

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Solving the general case

Consider a flow x satisfying arbitrary demands d1, . . . , dk. ✆ Round down all demands di to ¯ di so that ¯ d1 ⑤ ¯ d2 ⑤ . . . ⑤ ¯ dk. ✆ Compute a flow ¯ x satisfying ¯ d1, . . . , ¯ dk s.th. ¯ xa ➙ xa

2 for all a P A.

✆ Apply previous rounding procedure to ¯ x, yielding flow ¯ y ✏ ♣¯ yPi q1↕i↕k. ✆ Increase flow ¯ y by di ✁ ¯ di along Pi in order to get flow y. The final unsplittable flow y satisfies xa 2 ✁ dmax ↕ ya ↕ 2xa dmax for all a P A.

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Concluding remarks

Summary

Theorem For any fractional flow x, there is an unsplittable flow y s.th. ya ➙ xa ✁ dmax for all a P A. ✆ Proof via lower bound preserving augmentation steps Theorem If all demands are integer multiples of each other: For any fractional flow x, there is an unsplittable flow y s.th. xa ✁ dmax ↕ ya ↕ xa dmax for all a P A. For arbitrary demands, there is an unsplittable flow y s.th. xa 2 ✁ dmax ↕ ya ↕ 2xa dmax for all a P A. ✆ Proof via an iterative rounding procedure

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Open questions

Regarding flows with arc-wise lower bounds ✆ Are there any efficient strategies for applying the augmentation steps? ✆ Can we bound the length of these sequences polynomially? Regarding flows with arc-wise lower and upper bounds ✆ Can we find a proof via augmentation steps? ✆ Do the tighter bounds hold in the case of arbitrary demands? Conjecture (strengthening of [Goemans 2000]) Given arbitrary cost c on the arcs and any fractional flow x, there is an unsplittable flow y s.th. c♣yq ↕ c♣xq and xa ✁ dmax ↕ ya ↕ xa dmax for all a P A.

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