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1 Approximation Algorithms for Min-Max Generalization Problems Piotr Berman and Sofya Raskhodnikova Pennsylvania State University TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A

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Input: data items with weights, a lower bound w lb Goal: partition items into groups of weight w lb while minimizing the maximum weight of a group Rules of legal partitioning for MIN-MAX generalization problems MIN-MAX GRAPH PARTITION a graph into connected subgraphs MIN-MAX RECTANGLE TILING 2-dim array into non-overlapping contiguous rectangles MIN-MAX BIN COVERING unstructured data into arbitrary groups Min-Max Generalization [Du Eppstein Goodrich Lucker 09] 2 5 8 3 1 2 4 6 1 11 2 3 2 1 1 3 2 1 9 image source for BIN COVERING: http://www.developerfusion.com/article/5540/bin-packing/4 and Michael Goodrichs slides on [DEGL 09] 3 4 5 6 3 1 1 1 1 3 2 3 1 1 2 1 2 3 4 1 3 4 5 6 3

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Motivation [Du Eppstein Goodrich Lucker 09] Generalization is used in privacy preserving data mining to achieve k-anonymity [Samarati Sweeney 98] replace each value with a more general one so that no category has fewer than k members (weight lower bound w lb = k) unstructured data MIN-MAX BIN COVERING GPS coordinates MIN-MAX RECTANGLE TILING ZIP codes MIN-MAX GRAPH PARTITION 3 Image source: -Diversity: Privacy Beyond k-Anonymity Ashwin Machanavajjhala Johannes Gehrke Daniel Kifer Muthuramakrishnan Venkitasubramaniam Department of Computer Science, Cornell University Disclaimer: Known issues with k-anonymity Use differential privacy [Dwork McSherry Nissim Smith 06] when possible Image source: http://www.knowldentitlow.com/PrivacyPolicy.php

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Geographic Information Systems –Partition geographic information into pages that can be transmitted to a mobile device / retrieved from secondary storage –Graph of road network MIN-MAX GRAPH PARTITION –Coordinate data MIN-MAX RECTANGLE TILING Scheduling –Assign jobs to workers to minimize makespan –Union rules: each worker must be hired for w lb hours MIN-MAX BIN COVERING Additional Motivation 4 image sources: http://gisagents.blogspot.com/2009/08/new-publication-random-planar-graphs.html, Grandfather Frost –Distribute presents of different values to kids –Each kid must get a bundle of value at least w lb –Minimize the value of max bundle, to avoid jealousy MIN-MAX BIN COVERING –Maximize the number of kids who get presents classical BIN COVERING http://www.travelforlife.ru/articles/troyka.jpg.att.jpg, http://thedailymind.com/wp-content/uploads/2008/04/bad-job-china.jpg

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Results on Min-Max Generalization Problems 5 Min-Max ProblemHardness [DEGL 09] Ratio in [DEGL 09] Our Ratio GRAPH PARTITION -- 3 on 3-connected planar graphs 1.33 4 on 4-connected planar graphs -- 3 2.5 BIN COVERING 22 + ε in time exp in ε -1 2 RECTANGLE TILING 1.335 4 with 0-1 entries -- 3 Main Result Completely resolved 3 4 5 6 3 1 1 1 1 3 2 3 1 1 2 1 2 3 4 1 3 4 5 6 3

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Other Related Work Classical BIN COVERING [Assmann Johnson Kleitman Leung 84, Csirik Johnson Kleitman 01, Jansen Solis-Oba 03, …] Rectangle tiling problems with other optimization criteria [Manne 93, Khanna Muthukrishnan Paterson 98, Sharp 99, Smith Suri 99, Muthukrishnan Poosala Suel 99, Berman DasGupta Muthukrishnan Ramaswami 01, Berman DasGupta Muthukrishnan 02, …] 6

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Our 3-Approximation for MIN-MAX GRAPH PARTITION Input: undirected graph with vertex weights, a lower bound w lb Goal: partition the graph into connected subgraphs of weight w lb while minimizing the maximum weight of a part Phases of the Algorithm 1.Construct a preliminary 2-tier partition 2.Improve the initial 2-tier partition 3.Run a 2-approximation algorithm of [Lenstra Shmoys Tardos 90] for Scheduling on Unrelated Parallel Machines (SUPM) 4.Repair parts of insufficient weight SUPM Input: m parallel machines, n jobs, processing times p ji for job j on machine i Goal: schedule each job on some machine while minimizing the makespan The reduction to SUPM is gray-box: black-box in the algorithm, but we look inside the box for the analysis. 7 5 8 3 1 2 4 6 1 11 2 3 2 1 1 3 2 1 9

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MIN-MAX GRAPH PARTITION : Definitions/Assumptions A node (or a subgraph) is lean if its weight is < w lb ; fat otherwise. W.l.o.g. assume the input graph is connected and w lb =1. In this talk: assume all nodes are lean. 8

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2-Tier Partition 1.Partition nodes into lean groups 2.Partition groups into fat supergroups Nodes in a group (supergroup) must induce a connected subgraph Intuition: supergroups parts in a legal partition groups nearly indivisible subgraphs Recall: want a 3-approximation if all supergroups have 3 groups, done. 9 wt 1 wt<1

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Types of Supergroups 10 Group-pair: consists of 2 groups Triangle: consists of 3 groups pairwise connected by an edge Star: consists of 3 groups When the central group is removed, remaining groups are connected components central group Example: stargroup-pairtriangle central group

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1.Form groups greedily: a.Make each node a group. b.While there are adjacent groups G, H such that G H is lean, merge G and H. 2.Form supergroups greedily: a.Form one supergroup that includes all groups. b.While there is a supergroup with 4 groups that is not star, split it. Claim: In the initial 2-tier partition 1.G H is fat for all adjacent groups G,H. 2.Each supergroup is a group-pair, a triangle or a star. Phase 1: Obtaining Initial 2-Tier Partition 11

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Mobile Groups A group is mobile if it is not in a group-pair and it is not a central group. I.e., if it is removed from its supergroup, the supergroup is still of one of the allowed types. 12 m m m mmm

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Goal of Phase 2 of the Algorithm 13 m m m m m m m Structured supergroups stars with central nodes Other supergroups supergroups with 3 groups Mobile groups of structured supergroups are adjacent only to central nodes of structured supergroups. They will be repartitioned among using a reduction to SUPM. m m m m m

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Transformations: Perform the first that applies 1.Form a group-pair from 2 adjacent mobile groups if they belong to different supergroups or to a supergroup with 4 groups. 2.Split central group if it has nodes u,v to which mobile groups connect. 14 mm uv mm uv mm Result: No adjacent mobile groups, except for groups in the same triangle. Result: Each central group has a unique central node to which mobiles connect.

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What Remains to Achieve Our Goal 15 m m m m m m m Structured supergroups stars with central nodes Other supergroups supergroups with 3 groups Mobile groups of structured supergroups are adjacent only to central nodes of structured supergroups. They will be repartitioned among using a reduction to SUPM. m m m m m

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3.If a mobile from a star with 4 groups chain connects to a group-pair, chain reconnect. Last Transformation 16 Result: Goal achieved. m mm m mm

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Phase 3 of the Algorithm 17 1.Other" supergroups become parts in the final partition. 2.Each central node of a structured supergroup gets its own part. 3.Mobiles of structured supergroups are repartitioned among central nodes. 4.SUPM instance: are machines, and are jobs, weights are processing times. 5. Other nodes from central groups join parts of their central nodes. m m m m m m m Structured supergroups stars with central nodes Other supergroups supergroups with 3 groups m m m m m m Analysis highlights 1.SUPM algorithm gives output of value OPT f +1, where OPT f is a fractional optimum. 2.OPT f our OPT 3.Our guarantee: OPT f +1 + wt(nodes in a central group) OPT+2 Mobile groups of structured supergroups are adjacent only to central nodes of structured supergroups.

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Results on Min-Max Generalization Problems 18 Min-Max ProblemHardness [DEGL 09] Ratio in [DEGL 09] Our Ratio GRAPH PARTITION -- 3 on 3-connected planar graphs 1.33 4 on 4-connected planar graphs -- 3 2.5 BIN COVERING 22 + ε in time exp in ε -1 2 RECTANGLE TILING 1.335 4 with 0-1 entries -- 3 Main Result Completely resolved

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Transformation 1 Transformations: Perform the first that applies 1.Combine adjacent groups G, H if G H is lean and H is mobile. 19 m G H G H

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Transformations: Perform the first that applies 1.Form a group-pair from 2 adjacent mobile groups if they belong to different supergroups or to a supergroup with 4 groups. 2.Split central group G if it has nodes u,v adjacent to mobile groups H u,H v, resp. Split supergroup S containing G into S u,S v by attaching each mobile to G u or G v. S u,S v form supergroups if they are fat, groups in a group-pair if they are lean, get repackaged if one is fat, another is lean. 20 mm HuHu HvHv uv m m HuHu HvHv uv m m G GvGv GuGu Result: No adjacent mobile groups, except for groups in the same triangle. Result: Each central group has a unique central node to which mobiles connect.

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Obtaining Initial 2-Tier Partition 21 1.Form groups greedily: a.Make each node a group b.While there are adjacent groups G, H such that G H is lean, merge G and H 2.Form group-pairs greedily: While there are adjacent groups G,H not in a supergroup, form a supergroup G H. 3.Insert remaining groups into supergroups: For each group G still not in a supergroup, pick an adjacent group H. (H must be in some group-pair created in Step 2.) Insert G into Hs supergroup. 4.Break down supergroups with 4 groups that are not stars

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