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Finding the smallest H -subgraph in real weighted graphs and related problems

Finding the smallest H -subgraph in real weighted graphs and related problems. Raphael Yuster. University of Haifa. Joint work with: Virginia Vassilevska, Ryan Williams CMU. The min H -subgraph problem. H = K 3 real vertex weights. 1/2. 2/3. 1/3. 1/2. 3/4.

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Finding the smallest H -subgraph in real weighted graphs and related problems

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  1. Finding the smallest H-subgraph in real weighted graphs and related problems Raphael Yuster University of Haifa Joint work with:Virginia Vassilevska, Ryan Williams CMU

  2. The min H-subgraph problem H = K3 real vertex weights 1/2 2/3 1/3 1/2 3/4 Input: A vertex-weighted graph G=(V,E), where |E|=m and |V|=n. Output: An (induced) H-subgraph with minimum total weight, if exists. Similarly define the edge-weighted version.

  3. The min H-subgraph problem • We work in the addition-comparison model. • All results apply to directed graphs as well. • The naïve algorithm solves themin H-subgraph problem in O(nh) time. • The goal: reduce the exponent (problem is conjectured to be not fixed parameter tractable already in the unweighted case). • Case h=3 (triangles) studied in [VW – STOC06] who gave the first sub-cubic algorithm.

  4. Results – the vertex weighted case For fixed H with h vertices, a minimum weight (induced) H-subgraph can be found in O(nt(ω,h)) time. ω < 2.376[Coppersmith-Winograd ’90] is the exponent of fast square matrix multiplication(possibly ω = 2+on(1) someday). The exponentt(ω,h)in our theorem is obtained by solving a small integer program.

  5. t(ω,h) : definition t(ω,h) → 3h/(6-ω)≤ 0.828h

  6. Improves (ω+3)/2 VW-STOC06

  7. Proof – the vertex weighted case • Assume H=Kh • G=(V,E) V={1,2,…,n} w: V → R • For a positive integer t let Stbe the set of allt-subsets of V sorted by their total weight.|St| < nt and can be sorted in O(ntlog n) time. • Suppose h=a+b+c where a,b,c are positive integers. • Create two 0-1 matricesM1=M1(G,a,b) M2=M2(G,b,c)

  8. Proof – the vertex weighted case The matrix M1 Sb (sorted) Y 1: XUY = Ka+b 0:otherwise Sa X

  9. Proof – the vertex weighted case The matrix M2 Sc Z 1: YUZ = Kb+c 0:otherwise Sb (sorted) Y

  10. Proof – the vertex weighted case • Compute M3=M1M2 • Suppose (i) M3(X,Z) > 0 (ii) X U Z induces a Ka+c YSb so that M1(X,Y)=1 M2(Y,Z)=1 hence X U Y U Z is a Kh • Y is called a witness for the pair (X,Z).If Yis the minimal witness in the sorted Sb thenw(X)+w(Y)+w(Z) is the weight of the smallest Kh containing X U Z.

  11. Proof – the vertex weighted case • Let W be the minimal witness matrix of the product of two 0-1 matrices C=AB. 0 : C[i,j]=0k : A[i,k]=B[k,j]=1, A[i.k’]B[k’,j]=0 k’<k W[i,j]= • Computing W for dimensions n1,n2,n3 denotedf(n1,n2,n3) • Need to prove:mina+b+c=hf(na,nb,nc) = O(nt(ω,h))

  12. Computing minimal witnesses • Computing a matrix of witnesses can be done in essentially the same time needed to perform the product ([Seidel ’95], [Alon+Naor ’96]). • Problem: Gives no clue on the minimum witness. • [Kowaluk and Lingas ’05] f(n,n,n) = O(n2+1/(4-ω))They considered a different problem:computing all pairs lca in a dag • This already gives the case t(ω,3)=2+1/(4-ω) • We need to extend their result to other dimensions and optimize upon the choices of a,b,c.

  13. a+b+c=h 0< μ≤b μ split nb nc B1 nμ … na A1 Ap . . . . . . nb … nμ nμ Ci = AiBii=1,…pC=AB= ∑Ci Bp nμ

  14. Computing W[i,j] • Let r be the smallest index for which Cr[i,j]>0 • If r does not exist then W[i,j]=0. • Otherwise let k be the smallest index forwhich Ar[i,k]=Br[k,j]=1. • W[i,j] = (r-1)nμ+k.

  15. Computing W Computing a single witness: O(p+nμ) Computing all witnesses:O(na+c(p+nμ)) Computing a single Ci:O(na-μnc-μnω μ) Computing all Ci:O(p(na-μnc-μnω μ)) Overall running time O(na+c(p+nμ) + p(na+c+(ω-2)μ)) =O(nh-(3- ω)μ+nh-b+μ)

  16. Computing W • For h=3 take a=1 b=1 c=1μ=1/(4-ω) • For h=4 take a=1 b=2 c=1μ=1 • For h=5 take a=2 b=2 c=1μ=1 • For h=7 takea=2 b=3 c=2 μ=3/(4-ω) • For h=9 take a=3 b=4 c=2μ=2

  17. Additional features • Number of comparisons is pretty small:h=3 n log n + mh=4 n2 log n + mh=5 n2 log n + mnh=6 n2 log n + m2 • Can solve the decision problem:Given a real interval I R is there an (induced)H-subgraph whose weight is within I ?Use binary search within the matrices C1, …,CpRunning time increases only by log n factor.

  18. Additional features • Can find smallest K2,k in O(n2+1/(4-ω)) < O(n2.616)time. • Result is meaningful also when the weights are large integers: β(G,H) the H edge-covering numbermaximum number of edges incident with H-subgraph 4 2 1 β(G,K3)=8 4 3 1 3

  19. Sparse graphs A triangle can be found in O(m2ω/(ω+1)) < O(m1.41) time[Alon+Y+Zwick ’97] Finding the smallest weighted triangle • Let Δ be a parameter. • Vertices in Y having degree > Δ. |Y| < 2m/Δ • Triangles with vertex in V-Y examined in O(mΔ) time. • Smallest triangle inside Y found in O(|Y|2+1/(4-ω)) time. • Overall: O(mΔ+(m/Δ)2+1/(4-ω)) • Optimizing on Δ: O(m(18-4ω)/(13-3ω)) < O(m1.45)

  20. Edge weighted graphs The color coding method [Alon+Y+Zwick ’95] • H has k vertices. • Color V(G) with k colors, randomly. • Colorful H sometimes easier to find. • Probability of being colorful: k!/kk > e-k • Can be derandomized.

  21. Edge weighted graphs Distance products • A and B two compatible matrices in RU ∞ • The distance product D=AB defined byD[i,j] = minkA[i,k]+B[k,j] • If A and B are square matrices of ordern, then D can be computed in O(n3) time. • [Chan ’05] Only O(n3/log n) time needed! • Consequence: Min H-subgraph in O(nk/log n) time.

  22. Edge weighted graphs A minimum weight k-cycle can be found, w.h.p., in2O(k)n3/log n time.For k=o(log log n) this is sub-cubic. • For each pair u,v find the shortest colorful path of length k-1 connecting them in2O(k)n3/log n time. • Assume k is a power of 2 (we use recursion). • Let C1 be a set of k/2 colors. C2 = C-C1. • Vi– the vertices colored with a color from Ci. • Gi– the subgraph induced by Vi.

  23. Edge weighted graphs • Recursively find, for each pair in Vi, a colorful path in Gi of length k/2-1. Record the results in matrices A1 and A2. • B – the matrix with rows indexed by V1 and columns indexed by V2. B[u,v]=w(u,v). • The distance product DC1,C2=A1B A2 gives for each pair of vertices, the shortest colorful path of length k-1 where the first k/2 vertices are colored from C1 and the last k/2 vertices are colored from C2.

  24. Edge weighted graphs • By considering all choices for (C1,C2) (less than 2k choices) we obtain an nn matrix D where D[u,v] is the shortest colorful path of length k-1 between u and v. • The number of distance products using this approach satisfies t(k) ≤ 2kt(k/2).Thus, t(k) = 2O(k). • Each product costs O(n3/log n).

  25. Chromatic subgraphs • G=(V,E) an edge-colored graph (number of colors is arbitrary). • We consider • The monochromatic H-subgraphproblem • Therainbow H-subgraphproblem • Both problems are not easier than the (uncolored) H-subgraph existence problem. • Are they also not harder?

  26. Rainbow subgraphs • We use color reduction. • Suppose H has t edges. • f : C → {1,2,…,t}. We now have only t colors. • A rainbow H now is also originally rainbow. • f random – an original rainbow H will remain rainbow after reduction with probability t!/tt. • Do this 2O(t) times and it will happen w.h.p. at least once. • Can be derandomized. Cost is only O(log n) factor.

  27. Rainbow subgraphs • When we have only t colors, same algorithm for the uncolored version can be used to detect a monochromatic H. Example K6

  28. Monochromatic subgraphs For any connected fixed graph H with 3k+j vertices (j=0,1,2), a monochromatic H can be detected in O(nωk+j), except of H=K3. • Running time identical to (uncolored) detection. • Triangles can be found only in O(n(3+ω)/2) < o(n2.688). Example K6

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