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The power of group algebras for constrained multilinear monomial detection

The power of group algebras for constrained multilinear monomial detection. Yiannis Koutis Computer Science Department (CMU)  University of Puerto Rico. the k-path problem.

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The power of group algebras for constrained multilinear monomial detection

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  1. The power of group algebras for constrained multilinear monomial detection Yiannis Koutis Computer Science Department (CMU)University of Puerto Rico

  2. the k-path problem • How fast can we decide the following: Given a graph G, does it contain a k-vertex path ? • In O*((2e)k ) time [Alon, Yuster, Zwick 94] • In O*(4k) time [Chen et.al. 07] • In O*(2k) time [K, Williams 08] • In O*(1.66k) time [Bjoerklund et. al. 10]

  3. the k-multilinear monomial detection problem • How fast can we decide the following: Given an n-variate polynomial P presented as an arithmetic circuit does the sum-product expansion of P contain a square-free monomial of total degree k? • In O*((2e)k ) time [Alon, Yuster, Zwick 94] • In O*(2k) time [K, Williams 08]

  4. the constrained k-multilinear monomial detection problem • How fast can we decide the following: Given a graph G with vertices colored red and blue, does it contain a k-path that uses at most 3k/4 red nodes and at most 3k/4 blue nodes? • A special case of the problem, MULTISET MAX MOTIF appears to be useful in computational biology. • Previous algorithms use color-coding. • In time O*(4k) using [Guillemot, Sikora MFCS10].

  5. the constrained k-multilinear monomial detection problem [this talk] In time O*(2k)

  6. the color-coding approach • Observation 1: k-multilinear monomials in k-variate polynomials can be detected in time O*(2k). • Observation 2:Pick a random mapping of the n variables X to a set of k variables Y. Then with probability 1/ek the k-multilinear monomial stays multilinear. Then obs1 can be applied.

  7. the algebraic approach • Try to decide the instance by only evaluating the circuit over an interesting commutative algebra • What is the definition of interesting ? • Squares of variables annihilate, i.e. evaluate to 0 • Sum of multilinear terms evaluate to something non-zero • The complexity of the algebra must be small

  8. the group algebra approach • Fix the group to be Z2 k (k-dimensional 0-1 vectors) • Commutative group multiplication (xor-ing vectors v and w) • Group identity, the zero vector

  9. the group algebra approach • On top of the usual group operator, we introduce an addition (mod 2) operator to define ABLERAG = Z2 [Z2k] • The elements of the algebra are sums of vectors • For example: v+u+w is an element • The only possible simplification: v+v = 0 • Multiplication of algebra elements apply the usual rules, e.g.

  10. the group algebra approach • Theorem:ABLERAG is nearly interesting [K08] • For each xi pick a random vector vi • Assign to xi the value v0 + vi and evaluate • Squares annihilate: • Any given multilinear term is non-zero if and only if the corresponding k vectors are linearly independent. This happens with prob>0.25 • Circuit can be evaluated in O*(2k) time (and pspace) • If polynomial contains an odd number of square-free terms then their sum evaluates to non-zero with prob>0.25

  11. the group algebra approach • Theorem:ABLERAG is interesting [nearly-officemate-08] • Multiply the edges of the circuit by random univariate polynomials of degree O(k). This essentially hashes the terms of P: • At least one Pi contains an odd number of square-free monomials if P contains square-free monomials.

  12. the constrained k-multilinear monomial detection problem • How fast can we decide the following: Given a graph G with vertices colored red and blue, does it contain a k-path that uses at most 3k/4 red nodes and at most 3k/4 blue nodes? • In time O*(4k) using ABLERAG[Guillemot, Sikora MFCS10]. • In time O*(2k) [this talk].

  13. the constrained k-multilinear monomial detection problem • We will still assign to the variables elements of ABLERAG that look exactly like the ones we used three slides ago. • But NOT SO RANDOMLY! • Any given t-multilinear term is non-zero if and only if the corresponding t vectors are linearly independent. This happens with prob>0.25 • Force any (3k/4+1)blue vectors to be linearly dependent (same for red) • Pick the blue vectors from a (3k/4)-dimensional subspace of Z2k

  14. the constrained k-multilinear monomial detection problem • Input:A graph G, with blue and red nodes. Upper bound kbon the number of blue variables on a valid k-multilinear monomial. Upper bound kron the number of blue (red) variables on a valid k-multilinear monomial. • The algorithm: • Pick a random kb -dimensional subspace B of Z2k(same with R) • Assign to each blue variable a random vector from B • Evaluate the circuit as in the non-constrained case.

  15. the constrained k-multilinear monomial detection problem • Analysis: • Every invalid multilinear monomial evaluates to 0. • For a given valid multilinear monomial, the blue vectors are linearly independent with constant probability pb>0.25. Similarly define pr. Let’s call these the ok-subspace probabilities. • The subspace spanned by the blue vectors does not contain a red vector with probability g>1/2. Let’s call this the glue probability. • The three events are independent, so probability of success is greater than g pbpr (a constant loss comparing to non-constrained)

  16. the constrained k-multilinear monomial detection problem • Arbitrarily many color classes? • In time O*(4k) using ABLERAG[Guillemot, Sikora MFCS10] • In time O*(6k/2) using the obvious generalization of the two-colored algo. • We pick up a constant loss in probability of sucess for every class • But the ok-subspace probabilities become progressively better • The glue probability is always at least 0.25 • Worst case is when the solution contains k/2 color classes • Carefully working out the probabilities gives the bound

  17. The weighted version of monomial detectioni.e. the TSP-like analogue • Each variable xi is associated with a weight wi. • The weight of a multilinear monomial is defined as the sum of the weights of the variables in it.

  18. The weighted version of monomial detectioni.e. the TSP-like analogue • How fast can we decide the following: Given a polynomial P, what is the minimum weight k-multilinear monomial in P ? • In O*((2e)k ) time but exp(k) space [color coding] • In O*(2k *k*max_weight) and pspace [ABLERAG+ Lokshtanov&Nederlof 10]

  19. The weighted version of monomial detectioni.e. the TSP-like analogue • Consider for example the case where variables have weight 0 or 1 • We introduce a new indeterminate z, and in P(X) we multiply each variable of weight 1 by z. This essentially hashes the terms of P: • Evaluate with ABLERAG and find the smallest i, for which the coefficient of zi is non-zero. • An O*(k ) deteroriationin time and spacecomparing to the decision version

  20. The weighted version of monomial detectioni.e. the TSP-like analogue • We can view weights 0 and 1, as color classes • We can answer in time essentially the same as the decision problem the following question: Is there a k-multilinear monomial that uses1at most t times? • We can then find the minimum number of ones with binary search • An O*(log k ) slowdown with respect to the decision version • No blowup in space requirements.

  21. open questions • Is there an O*(2k) algorithm for the general constrained problem ? • Extend applications to weighted versions.

  22. Thank you !

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