Online Algorithms for Mul2-commodity Network Design
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1 Online Algorithms for Mul2-commodity Network Design Debmalya Panigrahi Joint work with Deeparnab Chakrabarty, Alina Ene, and Ravishankar Krishnaswamy Thanks Alina for sharing slides!
2 Online Network Design: a quick review (Un)directed graph G = (V, E) given offline Every subgraph has an associated cost (fixed edge cost, fixed node cost, buy at bulk, two cost) Terminals/terminal pairs arrive online Algorithm has to connect: Terminal to a root (single sink) Terminal pair (mulc-commodity) CompeCCve RaCo = Online Alg / (Offline) Opt 16/04/29 Online Algorithms for Node-weighted Network Design 2
3 Online Network Design: a bit of history Edge-weighted Steiner tree (Imase-Waxman, 91) Greedy algorithm Charge cost of algorithm to log n layers of duals Layer i has ball of radius 2 i-2 centered at v if cost of conneccng v is (approximately) [2 i-1, 2 i ] 2 i-2 2 i-2 2 i-1 Compe22ve ra2o: O(log n) 16/04/29 Online Algorithms for Node-weighted Network Design 3
4 Online Network Design: a bit of history Edge-weighted Steiner forest (Awerbuch-Azar-Bartal, 94) Greedy algorithm Charge cost of algorithm to log n layers of duals Layer i has ball of radius 2 i /log n centered at s i or t i if cost of conneccng this pair is (approximately) [2 i-1, 2 i ] 2 i-1 accoun2ng edge 2 i /log n Compe22ve ra2o: O(log 2 n) Modified greedy [Berman-Coulston, 97]: O(log n) 16/04/29 Online Algorithms for Node-weighted Network Design 4
5 Online Network Design: more recent history Node-weighted Steiner tree (Naor-P.-Singh, 11) Greedy algorithm fails because node-weighted Steiner tree captures set cover Main idea: minimal cost sharing conneccng path = greedy path + one shared vertex use online set cover for shared vertex, greedy for the rest 2 i-2 2 i-2 2 i-1 Compe22ve ra2o: O(poly log n) 16/04/29 Online Algorithms for Node-weighted Network Design 5
6 Online Network Design: more recent history Node-weighted Steiner forest (Hajiaghayi-Liaghat-P., 13) Node-weighted Steiner tree approach fails sharing on log n vercces required: label cover hard Charge cost of algorithm to log n layers of duals set cover instances OpCmum of each set cover instance lower bounds the primal If new terminals far from exiscng vercces, opcmum set cover cost changes; if not, use accouncng edge as earlier 2 i-1 2 i /log n accoun2ng edge Compe22ve ra2o: O(poly log n) 16/04/29 Online Algorithms for Node-weighted Network Design 6
7 Online Network Design: moral of the story MulC-commodity problems have historically required new techniques Not any more J Single-sink algorithm This talk MulC-commodity algorithm 16/04/29 Online Algorithms for Node-weighted Network Design 7
8 What about offline? Junc2on Trees [Chekuri, Kortsarz, Hajiaghayi, Salava2pour 06] Single commodity instance Mul2-commodity instance Single commodity instance Single commodity instance Route a subset of the pairs via a junc2on vertex; overall cost is sum of individual junc2on tree costs
9 What about offline? Junc2on Trees [Chekuri, Kortsarz, Hajiaghayi, Salava2pour 06] Single commodity instance Mul2-commodity instance Single commodity instance Single commodity instance Route a subset of the pairs via a junc2on vertex; overall cost is sum of individual junc2on tree costs Existence of a near-op2mal junc2on tree solu2on easy for Steiner forest, more involved for buy at bulk cost func2ons Par22on the terminal pairs via a greedy covering algorithm, where each greedy step runs a min-density single sink algorithm
10 Online junc2on trees Arguably even more useful! The log set cover loss is not limi2ng Focus on obtaining a frac2onal solu2on first Frac2onal assignment Single Sink Instance Inner LP New (s,t) pair Outer LP Single Sink Instance Inner LP Single Sink Instance Inner LP
11 The Composite (Outer + Inner) LP for buy at bulk
12 Solve the Composite LP Use the online primal-dual framework of [Buchbinder, Naor, and others] Some new ideas, does not fall in previous packing/ covering/mixed templates Outer LP runs mul2plica2ve weight updates, as usual Flow-type constraints in inner LP, uses a min cost (max) flow gadget
13 Rounding Pains Outer LP: easy, randomized rounding Inner LP: no idea! (how to round Steiner tree LP online?) Do not round! J Use bounded integrality gap (from offline rounding known) to claim that each inner LP has integer solu2on of ``small value Run a separate online Steiner tree instance for each inner LP in parallel
14 Results First non-trivial online algorithms for mul2- commodity buy at bulk network design with edge costs, node costs, and in directed graphs Simpler alterna2ve proof for online Steiner forest Easily extends to other cost frameworks such as prize-collec2ng problems
15 Open Problems Node-weighted algorithms are quasi-polynomial 2me using this template, but polynomial 2me using specific algorithms Can we avoid losing a log D factor for non-uniform demands, where D is max-to-min demand ra2o? How do you round (even) the Steiner tree LP online?
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