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 , A q , k commodities with common source s P V , sinks t 1 , . . . , t k P V and demands d 1 , . . . , d k → 0. Task Route each flow unsplittably, i.e. find flow y with path decomposition ♣ y P i q 1 ↕ i ↕ k s.th. each path P i is an s - t i -path and y P i ✏ d i . 4 4 In the following, 5 1 0 ✆ each node v P V lies on an 3 s - t i -path for some commodity i , s 2 1 4 4 2 ✆ each flow has to satisfy the demands d 1 , . . . , d k . 0 2 Sarah Morell: Single source unsplittable flows with arc-wise lower and upper bounds 1/16
Motivation Theorem [Dinitz, Garg, Goemans 1999] For any fractional flow x , there is an unsplittable flow y s.th. y a ↕ x a � d max 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 ? Sarah Morell: Single source unsplittable flows with arc-wise lower and upper bounds 2/16
Outlook of this talk Theorem For any fractional flow x , there is an unsplittable flow y s.th. y a ➙ x a ✁ d max 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. x a ✁ d max ↕ y a ↕ x a � d max for all a P A . For arbitrary demands, there is an unsplittable flow y s.th. x a 2 ✁ d max ↕ y a ↕ 2 x a � d max for all a P A . ✆ Proof via an iterative rounding procedure Sarah Morell: Single source unsplittable flows with arc-wise lower and upper bounds 3/16
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 P i and y a ➙ x a for all arcs a on the s - v -subpath of P i . 5 2 2 1 0 3 2 2 2 s 2 s 1 1 1 2 2 1 1 Sarah Morell: Single source unsplittable flows with arc-wise lower and upper bounds 4/16
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 d i from the s - v -subpath of P i along an arbitrary s - v -path Q . t i t i ˜ P i P i ≥ x a ≥ x a − d i v v × Ñ × × s s A finite sequence of augmentation steps from y to ˜ y is denoted by y � ˜ y . Sarah Morell: Single source unsplittable flows with arc-wise lower and upper bounds 5/16
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 X y of t i eventually reachable nodes is not V . ∃ ˜ y ≥ x y : ˜ s If y � ˜ y , then X ˜ y ❸ X y . Choose y ∃ y ′ : y ′ ≥ x with set X y inclusion-wise minimal. Then, t j y ✏ X y ✏ : X for all ˜ y with y � ˜ y . X ˜ Sarah Morell: Single source unsplittable flows with arc-wise lower and upper bounds 6/16
Key Lemma ∃ ˜ y : ˜ y ≥ x s ∃ y ′ : y ′ ≥ x The following properties are satisfied: y ♣ δ in ♣ X qq cannot decrease. (P1) Choose y with y ♣ δ in ♣ X qq maximal. δ in ♣ X q ✏ ❍ . (P2) There is an arc a P δ out ♣ X q s.th. y a → 0 and y a ➙ x a . (P3) Flow on arcs in δ out ♣ X q cannot be added or deleted. (P4) Sarah Morell: Single source unsplittable flows with arc-wise lower and upper bounds 7/16
Key Lemma t i Let a ✏ ♣ v , w q P P i : v is not eventually ∃ ˜ y : ˜ y ≥ x reachable w.r.t. i , so for every ˜ y there is an v arc b P ˜ s P i ⑤ v with ˜ y b ➔ x b . Let b y 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 b y on P y i ⑤ v is maximal, (ii) value y b y is maximal. Node v is reachable w.r.t. some ˜ y with t i y ✏ y 0 Ñ . . . Ñ y p ✏ ˜ y . ∃ ˜ y : ˜ y ≥ x t j v s u Let q be the smallest index s.th. some node u succeeding b y on P y i is reachable for some ∃ y q : y q ≥ x j w.r.t. y q . Sarah Morell: Single source unsplittable flows with arc-wise lower and upper bounds 8/16
Key Lemma t i ∃ ˜ y : ˜ y ≥ x t j v s u ∃ y q : y q ≥ x By choice of index q , ✆ flow has not been decreased on any arc of P y i succeeding b y , contain b y ✏ b y q and have identical subpaths from b y y q ✆ paths P y i and P i to node v . Rerouting commodity j along P y i ⑤ u increases flow on arc b y , 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. ˜ y a ➙ x a . Proof Let y have maximal flow on a ✏ ♣ v , w q P A with y a ➔ x a . By KL, node w is reachable for some i w.r.t. ˜ y with y � ˜ y . Rerouting d i increases flow on a . Theorem For any fractional flow x , there is an unsplittable flow y s.th. y a ➙ x a ✁ d max for all a P A . t i t i ˜ P i ≥ x a P i ≥ x a − d i v v Ñ × × × s s Sarah Morell: Single source unsplittable flows with arc-wise lower and upper bounds 10/16
Further results Theorem [Dinitz, Garg, Goemans 1999] For any fractional flow x , there is an unsplittable flow y s.th. y a ↕ x a � d max for all a P A . ✆ Simpler proof via upper bound preserving augmentation steps t i t i ˜ P i P i v v × × Ñ × s s ≤ x ≤ x + d i Conjecture [Goemans 2000] Given arbitrary cost c on the arcs and any fractional flow x , there is an unsplittable flow y s.th. c ♣ y q ↕ c ♣ x q and y a ↕ x a � d max for all a P A . Sarah Morell: Single source unsplittable flows with arc-wise lower and upper bounds 11/16
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. x a ✁ d max ↕ y a ↕ x a � d max for all a P A . For arbitrary demands, there is an unsplittable flow y s.th. x a 2 ✁ d max ↕ y a ↕ 2 x a � d max for all a P A . ✆ Proof via iterative rounding procedure , see [Kolliopoulos Stein 2002] and [Skutella 2002]. Sarah Morell: Single source unsplittable flows with arc-wise lower and upper bounds 12/16
Solving the special case Consider a flow x satisfying demands d 1 ⑤ d 2 ⑤ . . . ⑤ d k . Theorem If all demands d 1 , . . . , d k are integral, then any flow x can be turned into an integral flow ˜ x s.th. t x a ✉ ↕ ˜ x a ↕ r x a s for all a P A . While x ✘ 0: ✆ Turn x into a d min -integral flow whose flow values differ by at most d min . ✆ Route all commodities i of value d i ✏ d min unsplittably by iteratively choosing a flow-carrying s - t i -path P i . Decrease flow along P i and delete commodity i . Set y : ✏ ♣ y P i q 1 ↕ i ↕ k : its flow values differ from x by at most d max ✁ d min . Sarah Morell: Single source unsplittable flows with arc-wise lower and upper bounds 13/16
Solving the general case Consider a flow x satisfying arbitrary demands d 1 , . . . , d k . ✆ Round down all demands d i to ¯ d i so that ¯ d 1 ⑤ ¯ d 2 ⑤ . . . ⑤ ¯ d k . x satisfying ¯ d 1 , . . . , ¯ x a ➙ x a ✆ Compute a flow ¯ d k s.th. ¯ 2 for all a P A . ✆ Apply previous rounding procedure to ¯ x , yielding flow ¯ y ✏ ♣ ¯ y P i q 1 ↕ i ↕ k . y by d i ✁ ¯ ✆ Increase flow ¯ d i along P i in order to get flow y . The final unsplittable flow y satisfies x a 2 ✁ d max ↕ y a ↕ 2 x a � d max for all a P A . Sarah Morell: Single source unsplittable flows with arc-wise lower and upper bounds 14/16
Concluding remarks
Summary Theorem For any fractional flow x , there is an unsplittable flow y s.th. y a ➙ x a ✁ d max 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. x a ✁ d max ↕ y a ↕ x a � d max for all a P A . For arbitrary demands, there is an unsplittable flow y s.th. x a 2 ✁ d max ↕ y a ↕ 2 x a � d max for all a P A . ✆ Proof via an iterative rounding procedure Sarah Morell: Single source unsplittable flows with arc-wise lower and upper bounds 15/16
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 ♣ y q ↕ c ♣ x q and x a ✁ d max ↕ y a ↕ x a � d max for all a P A . Sarah Morell: Single source unsplittable flows with arc-wise lower and upper bounds 16/16
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