dagstuhl 2010 university of puerto rico computer science department the power of group algebras for...

Dagstuhl 2010 University of Puerto Rico Computer Science Department

The power of group algebras for constrained multilinear

monomial detection

Yiannis KoutisComputer Science Department

(CMU)University of Puerto Rico


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]


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]


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].



problem

[this talk]In time O*(2k)


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.


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


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



• On top of the usual group operator, we introduce an addition (mod 2) operator to define

ABLERAG = Z2 [Z2 k]

• 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.



• Theorem: ABLERAG is nearly interesting [K08]

1. For each xi pick a random vector vi

2. 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



• 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.



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].



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 Z2 k



problem• Input: A graph G, with blue and red nodes.

Upper bound kb on the number of blue variables on a valid k-multilinear monomial. Upper bound kr on the number of blue (red) variables on a valid k-multilinear monomial.

• The algorithm:

1. Pick a random kb -dimensional subspace B of Z2k (same with R)

2. Assign to each blue variable a random vector from B

3. Evaluate the circuit as in the non-constrained case.



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 pb pr (a constant loss comparing to non-constrained)



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


The weighted version of monomial detection

i.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.




• 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]




• 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 ) deteroriation in time and space comparing to the decision version




• 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 uses 1 at 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.


open questions

• Is there an O*(2k) algorithm for the general constrained problem ?

• Extend applications to weighted versions.


Thank you !

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