Algorithms & LPs for k-Edge Connected Spanning Subgraphs

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Algorithms & LPs for k-EdgeConnected Spanning Subgraphs. Dave Pritchard University of Waterloo CMU Theory Lunch, Dec 2 09. k-Edge Connected Graph. k edge-disjoint paths between every u, v at least k edges leave S, for all S V (k-1) edge failures still leaves G connected. - PowerPoint PPT Presentation

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  • Algorithms & LPs fork-EdgeConnectedSpanning Subgraphs Dave Pritchard University of Waterloo

    CMU Theory Lunch, Dec 2 09

  • k-Edge Connected Graphk edge-disjoint paths between every u, vat least k edges leave S, for all S V(k-1) edge failures still leaves G connectedS|(S)| k

  • k-ECSS & k-ECSM Optimization Problemsk-edge connected spanning subgraph problem: given an initial graph (possibly with edge costs), find k-edge connected subgraph including all vertices, w/ |E| (or cost) minimalk-ecs multisubgraph problem (k-ECSM): can buy as many copies as you like of any edge3-edge-connected multisubgraph of G, |E|=9G

  • Overview of TalkAlgorithms/ComplexityLinear ProgramsApproximationalgorithmsHardness constructionsParsimonious PropertyAlg. designVertex connectivitySubset k-ECSMIntricate extreme point solutionsTSP

  • Motivating QuestionsWhat is the best possible approximation ratio (assuming PNP) for these problems?What qualities of these various problems make them computationally easy or hard?Can we learn some new useful broad techniques from the study of these problems?

  • Approximation: State of the Art(Worst-case ratio from optimal)1+O(1/k)?1+ [P.]

    Unit CostsArbitrary CostsLower boundUpper boundLower boundUpper boundk-ECSS1+/k [GGTW]~1+0.5/k [CT, GG]1+/k [GGTW]2 [KV, J]k-ECSM? 1+, k=2~1+1.9/k [GGTW,GG]? 1+, k=2~3/2 [GB]

  • An Initial ObservationFor the k-ESCM (multisubgraph) problem, we may assume edge costs are metric, i.e. cost(uv) cost(uw) + cost(wv)since replacing uv with uw, wv maintains k-ECuSvw

  • Whats Hard About Hardness?A 2-VCSS is a 2-ECSS is a 2-ECSM.For metric costs, can split-off conversely, e.g.

    All APX-hard, i.e. no 1+ approx [BBHKPSU]2-ECSM2-ECSS2-VCSS

  • Whats Hard About Hardness?1+ hardness for 2-VCSS implies 1+ hardness for k-VCSS, for all k 2

    But this approach fails for k-ECSS, k-ECSMG, a hard instance for 2-VCSS Instance for 3-VCSS with same hardnessGzero-cost edges to V(G)

  • k-ECSS is APX-hard (1/2)We reduce MinTreeCoverByPaths to k-ECSSInput: a tree T, collection X of paths in TA subcollection Y of X is a cover if the union of {E(p) | p in Y} equals E(T)Goal: min-size subcollection of X that is a coversize-2 cover

  • k-ECSS is APX-hard (2/2)Replace each edge e of T by k-1 zero-cost parallel edges; replace each path p in X by a unit-cost edge connecting endpoints of p

    k-ECSS problem = min |X| to cover T.

    0 x (k-1)0 x (k-1)0 x (k-1)0 x (k-1)0 x (k-1)0 x (k-1)1111

  • Part 2:Complexity Linear Programs

  • From Hardness to ApproximabilityConjecture [P.] For some constant C, there is a (1+C/k)-approximation algorithm for k-ECSM.Holds for C=1, k 2. Next: definition of LP-relative;similar theorems known to be true;motivating consequence.an LP-relative

  • LP-Relative (1/2)Term LP-relative hides a specific reference to a particular undirected linear programming relaxation of the k-ECSM problem:Introduce variables xe 0 for all edges e of G.Min xecost(e) s.t. x((S)) k for all S VSe in (S) xe k0.41.21.4

  • LP-Relative (2/2)k-ECSM corresponds to integral LP solutions, but LP also has fractional solutionsSo LP-OPT OPT (of k-ECSM)-approx algorithm: ALG k-OPTDefinition: an algorithm is LP-relative -apx if ALG LP-OPT+ALGOPTLP-OPT(integrality gap)

  • Similar True TheoremsWidth W of an integer linear program is the max ratio of RHS entry to LHS coefficient in the same row. (In case of k-ECSM IP it is k)Conj: 1+O(1/W) LP-rel approx for k-ECSM1+O(1/W) LP-rel holds, and is tight, forsparse integer programs multicommodity flow/covering in trees

    LP structure for k-ECSM multiflow in tree

  • Background:(Per-vertex) Network DesignIn input, each vertex v has requirement rv ZObjective: find a min-cost subgraph s.t. for all vertices u, v, there are at least min{ru, rv} edge-disjoint paths connecting u and vHas a similar undirected LP relaxation: x((S)) min{ru, rv} if S separates u from v[GB] showed LP has parsimonious property: without loss of generality, x(({v})) = rv for all v0.50.51.5

  • Consequence of ConjectureSubset k-ECSM: rv {0,k} for all vvertices are required (rv= k) or optional (rv= 0) By parsimonious property, Subset k-ECSM has the same LP as k-ECSM on required subsetConsequence of parsimony: LP-relative -approx algorithm for k-ECSM implies a same quality approx for Subset k-ECSMconj.

  • A Combinatorial Approach?Is the following true for some constant C?For every A, B > 0, every (A + B + C)-edge-connected graph contains a disjoint A-ECSM and B-ECSM?

  • Part 3:LPs & Extreme Point Structure

  • LPs & Extreme Point Properties (Part 3)Compare k-ECSM LP and Held-Karp TSP LPIntroduce standard structural propertiesShow how this gives the elegant algorithm of [GGTW] for k-ECSSWe undertake goal of finding an object as unstructured as possible:[P.] extreme points on n vertices with maximum degree n/2 and minimum value 1/Fibonacci(n/2)

  • k-ECSM LP by any other namek-ECSM (using parsimony): xe 0, x((S)) k, x(({v})) = kHeld-Karp relaxation of TSP (outer form): xe 0, x((S)) 2, x(({v})) = 2Therefore these LPs (for all k) are the same up to uniform scalingi.e., x feasible for first iff 2x/k feasible for second

  • Structural Property [CFN]Held-Karp LP is large (2|V|-1 constraints, |V|2 variables) but:every extreme point / basic / vertex solution x has at most 2|V|-3 nonzero coordinatesonly 2|V|-3 constraints are needed to uniquely define this x, and we can pick a well-structured such set (laminar family)Note: some optimal solution is basic

  • 1+O(1/k) Algorithm forUnit-Cost k-ECSM [GGTW]Solve LP to get a basic optimal solution x*Round every value in x* up to the next highest integer and return the corresponding multigraph(k=4)

  • AnalysisOptimal k-ECSM has degree k or more at each vertex, hence at least k|V|/2 edgesThe fractional LP solution x* has value (fractional edge count) k|V|/2There are at most 2|V|-3 nonzero coordinatesRounding up increases cost by at most 2|V|-3ALG/OPT (k|V|/2 + 2|V|-3)/(k|V|/2) < 1 + 4/k

  • What Is Known about HK?[BP]: minimum nonzero value of x* can be ~1/|V|

    [C]: max degree can be ~|V|1/2

  • What Is New? Edge values of the form Fibi/Fib|V|/2 and 1 - Fibi/Fib|V|/2 Maximum degree |V|/2

  • How Was It Found?Computational methods plus some cleverness can enumerate all extreme points on a small number of verticesWe got up to 10; Boyd & coauthors have data available online up to 12Look for most complex extreme points:Big maximum degree, big denominatorTry to find a pattern & prove it

  • Small Extreme Examplesn=6, denom=2n=7, =4n=8, denom=3n=9, =5n=9, denom=4n=10, denom==5

  • Laminar Set-FamilyAny S, T in have ST, TS, or S,T disjointMaximal: cannot add any new sets and retain laminarity

  • Proof that this is indeed a family of extreme points Need to show x* is feasible, extremeFirst, show x*((S))=2 holds for a maximal laminar system L*Argue x* is unique such solution (long part)Suppose x*((S))
  • Could It Get Worse?Determinant bound shows denominator of extreme point is at most ~|V||V|Size of laminar family can be used to show max degree is at most n-3This construction does not attain maximal denominator on 12 vertices

  • ReviewWe found a hardness construction for k-edge-connected spanning subgraphNo good hardness known for k-edge-connected spanning multisubgraphLP-relative 1+O(1/k) algorithm for k-ECSM would give one for subset k-ECSMExtremely extreme extreme points

  • Thesis PlugInvestigated hypergraphic LP relaxations of Steiner tree problem

    Showed equivalences, structure, gap bounds[joint with D. Chakrabarty & J. Knemann]

  • Thanks for Attending!

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