Overview Overview Probabilistic existence of regular combinatorial objects

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Overview Overview Probabilistic existence of regular combinatorial objects

Concordia College, Selma, US has reference to this Academic Journal, Probabilistic existence of regular combinatorial objectsShachar Lovett (UCSD)Joint alongside Greg Kuperberg (UC Davis), Ron Peled (Tel-Aviv university)OverviewRegular combinatorial objectsProbabilistic modelMain Theorem: random walks on latticesProof: Fourier analysis in addition to codes Summary in addition to open problemsOverviewRegular combinatorial objectsProbabilistic modelMain Theorem: random walks on latticesProof: Fourier analysis in addition to codes Summary in addition to open problems

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Regular objects?highly symmetric? objectsRegular graphsRegular hyper-graphs (aka designs)K-wise permutationsOrthogonal arraysq-analogs?Constructions known in some special casesThis work: First existence proofs in consideration of (nearly) all underlying parameters by a probabilistic argumentRegular graphs(n,d) regular graph ? n vertices, all of degree dEasy so that constructRegular hyper-graphsAlso known as designst-(n,k,?) design: k-uniform hyper-graph on n vertices; any t vertices belong so that exactly ? edges1-(n,2,d) design: regular graph

Regular hyper-graphst-(n,k,?) design: k-uniform hyper-graph on n vertices; any t vertices belong so that exactly ? edgesConstructions:Small values based on group symmetries[Teirlinck?87] first asymptotic construction of t-(n,t+1,?) designs in consideration of infinitely many n, ?Few other asymptotic constructionsRegular hyper-graphst-(n,k,?) design: k-uniform hyper-graph on n vertices; any t vertices belong so that exactly ? edgesExistence unknown in consideration of most parameters:Steiner systems: t-(n,k,1) designs, open in consideration of t>5Hadamard matrices: 2-(4m-1,2m-1,m-1) designsIn general, constructions (and existence) unknown in consideration of almost all parametersK-wise permutationsFamily of permutations acting uniformly on k elementsA set F?Sn is k-wise if in consideration of any k distinct elements i1,?,ik in addition to j1,?,jk125346251643361254

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K-wise permutationsFamily of permutations acting uniformly on k elementsConstructions:Subgroups of Sn: k=1,2,3 (e.g. in consideration of k=2 in addition to n prime, subgroup of affine maps {x->ax+b})Fail in consideration of k>3 (only Sn or An are 4-wise in consideration of n>24)[Finucane-Peled-Yaari?12] Combinatorial constructions in consideration of small k of exponential size 125346251643361254Other examplesOrthogonal arrays: subsets of [c]n where any k coordinates get all values equally often (aka k-wise independent functions [n]->[c])q-analogs: Family of k-dimensional subspaces of Fqn which cover uniformly all the t-dimensional subspaces (eg designs in consideration of the Grassmanian)Spherical designs: family of points on Sn which allow so that integrate low degree polynomials by summing over the pointsOur approachProbabilistic constructionGeneral technique so that prove existence of regular objectsProve existence of designs, k-wise permutations, orthogonal arrays, in consideration of (nearly) all underlying parameters; of optimal size up so that polynomial overhead

OverviewRegular combinatorial objectsProbabilistic modelMain Theorem: random walks on latticesProof: Fourier analysis in addition to codes Summary in addition to open problemsProbabilistic modelRunning example: t-(n,k,?) designsk-uniform hyper-graph on n vertices; any t vertices belong so that exactly ? edgesRandom construction: Sample N=N(n,k,t,?) edges uniformlyAnalyze probability that any t vertices covered by exactly ? edgesVery unlikely eventProbabilistic modelRandom constructions unlikely so that workBut is probability zero or tiny but positive?How can we analyze ?rare events? ?Standard tools fail, e.g.Limited dependence (e.g. Lovasz Local lemma) doesn?t holdSpencer?s method not relevant

Probabilistic modelAnother viewpoint: sum of matrix rows010010100110010110000000101110?0010110100Edges: k-subsets of [n]t-subsets of [n]Sample few rowsAnalyze probability that sum is (?,?,?)Pr[sum=expected sum]Incidence matrixProbabilistic modelYet another viewpoint: short random walk on a latticeLattice spanned by rowsSteps: rowsProbability that a short random walk will end in a specific pointCan we guarantee fast convergence?010010100110010110000000101110?0010110100OverviewRegular combinatorial objectsProbabilistic modelMain Theorem: random walks on latticesProof: Fourier analysis in addition to codes Summary in addition to open problems

General setupMatrix Sample N rowsPr[sum of rows= expected sum of rows]When can we guarantee it is positive?010010100110010110000000101110?0010110100General setup[Alon-Vu?97] example of regular hyper-graphs on n vertices, ~nn/2 edges, alongside no regular sub-hypergraphsPr[sum of rows= expected sum of rows]=0Conclusion: need so that assume some structure010010100110010110000000101110?0010110100Main TheoremMain theorem: set of conditions that guarantee thatN is polynomial in underlying parametersIn our applications we get optimal N (up so that polynomial factors)Can approximate probability up so that 1+o(1)010010100110010110000000101110?0010110100Pr[sum of N rows= expected sum of rows]>0

Main TheoremSome notationA ? set of columnsB ? set of rows (|B| >> |A|)V ? linear space spanned by columnsrow(b) ? row in index b?BWe want S?B of size |S|=N such that010010100110010110000000101110?0010110100ABMain TheoremPre-condition: divisibilityWe want |S|=N in consideration of whichLet c1 be minimal integer such thatN must be a multiple of c1010010100110010110000000101110?0010110100ABMain TheoremPre-condition: divisibilityExample: t-(n,k,?) designs[Wilson?73, Graver-Jurkat?73] analyze divisibility of incidence matrices N multiple of 010010100110010110000000101110?0010110100AB

Main TheoremMain condition: column spanV = space spanned by columnsNeed:(a) V has transitive symmetry group(b) V spanned by short integer vectors in l?(c) V? spanned by short integer vectors in l1 (in coding terms, V is an LDPC)(d) V contains the constant vectors010010100110010110000000101110?0010110100ABMain TheoremV = space spanned by columnsExample: t-(n,k,?) designs(a) V has transitive symmetry groupSn acts as permutations on k-subsets (rows) in addition to t-subsets (columns)Acts transitively on rows (e.g. B)010010100110010110000000101110?0010110100ABMain TheoremV = space spanned by columnsExample: t-(n,k,?) designs(b) V spanned by short integer vectors in l?Immediate since matrix has small elements, so columns are such a basis in consideration of V010010100110010110000000101110?0010110100AB

Main TheoremV = space spanned by columnsExample: t-(n,k,?) designs(c) V? spanned by short integer vectors in l1Usually the hardest condition so that verify; in consideration of designs, requires some combinatorial calculations010010100110010110000000101110?0010110100ABMain TheoremV = space spanned by columnsExample: t-(n,k,?) designs(d) V contains the constant vectorSum of columns is constant010010100110010110000000101110?0010110100ABMain TheoremB x A matrix, V=span(columns)Assumec1 divisibility constantV spanned by integer vectors alongside l? bound c2V? spanned by integer vectors alongside l1 bound c3V has transitive symmetry groupV contains the constant vectors Then in consideration of N=poly(|A|,c1,c2,c3), Pr[sum of N rows= expected sum]>0 In fact, we approximate the probability up so that 1+o(1)010010100110010110000000101110?0010110100AB

SummaryProof technique: Fourier analysisMake new connections between coding theory in addition to Fourier analysis in order so that bound Fourier coefficientsOpen problemsApplicationsWork in progress (with Kuperberg in addition to Peled): spherical designsWork in progress (with Vardy): q-analogsOpen problemsAlgorithmsCurrent proof is purely existentialDon?t know how so that find objects efficiently, even using randomnessOther probabilistic techniques in consideration of rare events were made algorithmic, so there is hopeLovasz local lemma: Moser, Moser-TardosSpencer?s theorem: Bansal, L-Meka

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Journal Ratings by Concordia College, Selma

This Particular Journal got reviewed and rated by Main TheoremMain condition: column spanV = space spanned by columnsNeed:(a) V has transitive symmetry group(b) V spanned by short integer vectors in l?(c) V? spanned by short integer vectors in l1 (in coding terms, V is an LDPC)(d) V contains the constant vectors010010100110010110000000101110?0010110100ABMain TheoremV = space spanned by columnsExample: t-(n,k,?) designs(a) V has transitive symmetry groupSn acts as permutations on k-subsets (rows) in addition to t-subsets (columns)Acts transitively on rows (e.g. B)010010100110010110000000101110?0010110100ABMain TheoremV = space spanned by columnsExample: t-(n,k,?) designs(b) V spanned by short integer vectors in l?Immediate since matrix has small elements, so columns are such a basis in consideration of V010010100110010110000000101110?0010110100AB and short form of this particular Institution is US and gave this Journal an Excellent Rating.