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Polynomial Identity Testing Problems Schwartz-Zippel Lemma Bipartite Matching References

Polynomial Identity Testing

Anil Maheshwari

anil@scs.carleton.ca School of Computer Science Carleton University Canada

Polynomial Identity Testing Problems Schwartz-Zippel Lemma Bipartite Matching References

String Equality Testing

Alice-Bob String Testing Problem

Assume Alice has a binary string A = a1a2 . . . an and Bob has a binary string B = b1b2 . . . bn. What is minimum amount of communication required to test whether A = B?

Polynomial Identity Testing Problems Schwartz-Zippel Lemma Bipartite Matching References

Randomized Algorithm

Define A(x) =

n

• i=1

aixi and B(x) =

n

• i=1

bixi, where x ∈ F. F is a Field defined with modular arithmetic for a large prime number p. Algorithm:

1

Pick a random element α ∈ F

2

Alice computes A(α) and sends (α, A(α)) to Bob

3

Bob computes B(α)

4

Bob communicates to Alice True if B(α) = A(α), else False

Polynomial Identity Testing Problems Schwartz-Zippel Lemma Bipartite Matching References

Analysis of Randomized Algorithm

Case I: A = B: Algorithms reports TRUE as A(α) = B(α) no matter what is the value α ∈ F Case II: Assume A = B. Can algorithm make an error? Yes, if α is the root of the polynomial (A − B)x = 0. Pr(a random element of F is root of (A − B)x) ≤ n/|F| Question: How to increase the success probability? Communication Complexity: O(log |F|) bits.

Polynomial Identity Testing Problems Schwartz-Zippel Lemma Bipartite Matching References

Multivariate Polynomials

Determine if the multivariate polynomial Q(x1, x2, . . . , xn) ≡ 0? Example I: Q(x1, x2, x3, x4) = (x3

1 − x2 2)(−x2 1 − x4 3)(x3 4 − 2x1x2) ≡ 0

Example II: Det x1 − x2

2

x3 − x1 x2

4

x4 − x1 −x4

2

x2 − x4 2x3 − 7x1 x2

2 − x2 3

x3

1

x2 − x1 x4 − x3 x3

2

x3

2

x4 − 2x2 x1 − x2

3

x3

1 − x3 2

≡ 0

Polynomial Identity Testing Problems Schwartz-Zippel Lemma Bipartite Matching References

Schwartz–Zippel Lemma

Schwartz–Zippel Lemma

Let Q(x1, x2, . . . , xn) ≡ 0 be a multivariate polynomial of total degree d, where each xi takes value from a finite field F. Fix any finite set S ⊆ F and let r1, . . . , rn be chosen uniformly at random from S. Then Pr(Q(r1, . . . , rn) = 0) ≤

d |S|

Polynomial Identity Testing Problems Schwartz-Zippel Lemma Bipartite Matching References

Testing Determinants

Is Det x1 − x2

2

x3 − x1 x2

4

x4 − x1 −x4

2

x2 − x4 2x3 − 7x1 x2

2 − x2 3

x3

1

x2 − x1 x4 − x3 x3

2

x3

2

x4 − 2x2 x1 − x2

3

x3

1 − x3 2

≡ 0? Choose a large enough prime number p, and choose random values for x1, x2, x3, x4 from {0, . . . , p − 1}. Evaluate the determinant. Probability of one sided error ≤ d

p,

where d is the degree of the polynomial.

Polynomial Identity Testing Problems Schwartz-Zippel Lemma Bipartite Matching References

Bipartite Matching

Let G = (U ∪ V, E) be a bipartite graph, where |U| = |V | = n. M ⊆ E is a perfect matching if

1

|M| = n

2

Edges in M are independent, i.e. vertex disjoint.

Polynomial Identity Testing Problems Schwartz-Zippel Lemma Bipartite Matching References

Adjacency Matrix

Define n × n matrix A where, Aij =

• xij, if uivj ∈ E

0, otherwise x11 x12 x21 x22 x31 x32 x33 x11 x12 x13 x22 x32

Polynomial Identity Testing Problems Schwartz-Zippel Lemma Bipartite Matching References

Edmonds Theorem

Edmonds

A bipartite graph G has a perfect matching if and only if det(A) = 0.

Polynomial Identity Testing Problems Schwartz-Zippel Lemma Bipartite Matching References

Decision Problem

Input: Given a bipartite graph G = (U ∪ V, E), where |U| = |V | Output: TRUE if G has perfect matching, otherwise FALSE Randomized Algorithm:

1

Choose a large enough prime number p.

2

For each edge uivj, set xij to be a random value in {0, . . . , p − 1} uniformly at random.

3

Compute det(A)

4

Return TRUE iff det(A) = 0.

Polynomial Identity Testing Problems Schwartz-Zippel Lemma Bipartite Matching References

Analysis

1

Choose a large enough prime number.

2

For each edge uivj, set xij to be a random value in {0, . . . , p − 1} uniformly at random.

3

Compute det(A)

4

Return TRUE iff det(A) = 0. Case 1: If G has no perfect matching = ⇒ det(A) = 0 Case 2: If G has perfect matching = ⇒ det(A) = 0 (Edmonds) Degree of determinant polynomial is ≤ n = |U| Pr(det(A)=0 given that G has a perfect matching) ≤ n/p (Schwartz-Zippel) Choose p ≈ 1000n, Probability of success ≥ 1 − 1/1000

Polynomial Identity Testing Problems Schwartz-Zippel Lemma Bipartite Matching References

How to find a perfect matching

Isolation Lemma (MVV87)

Assume we have set system S on a ground set of n

• elements. Assign weights to each element uniformly and

at random from {1, 2, . . . , 2n}. The probability that there is a unique minimum weight set in S is ≥ 1

2

Polynomial Identity Testing Problems Schwartz-Zippel Lemma Bipartite Matching References

Finding a Perfect Matching

• Let G = (U ∪ V, E) and |E| = m.
• Assume G has a perfect matching.
• For each edge e ∈ E, assign a weight in {1, . . . , 2m}

uniformly at random.

• Let M= Set system consisting of all perfect matchings
• Isolation Lemma: ∃M ∈ M of unique minimum weight

with probability ≥ 1/2.

New Problem

Find (unique) minimum weight perfect matching M in G

Polynomial Identity Testing Problems Schwartz-Zippel Lemma Bipartite Matching References

Unique MWPM

Let unique MWPM has a total weight W ≤ 2m2. For each edge e = (uivj) ∈ E with weight w(e), set xij = 2w(e) in det(A). Consider the non-zero terms in the expansion of det(A). Observation: Only one term is 2W and all other terms are ≥ 2W+1 = 2 ∗ 2W .

Polynomial Identity Testing Problems Schwartz-Zippel Lemma Bipartite Matching References

Unique MWPM (contd.)

Note: det(A) 2k =     

• dd, if k = W

even, if k < W fractional, if k > W Algorithm:

1

Find k: Guess k and check parity of det(A)

2k

2

For each edge e = (uv), it is in unique MWPM if and

• nly if MWPM in G \ {u, v} has weight W − w(e).

Note: Computation of det(A), Guess & Check k, and Testing ∀e ∈ E is part of unique MWPM are parallelizable.

Polynomial Identity Testing Problems Schwartz-Zippel Lemma Bipartite Matching References

Matching in General Graphs

Let G = (V, E) be a general graph. Define Aij =      +xij, if vivj ∈ E and i<j −xij, if vivj ∈ E and i>j 0, otherwise

Tutte

G has a perfect matching if and only if det(A) = 0.

Polynomial Identity Testing Problems Schwartz-Zippel Lemma Bipartite Matching References

References

1

Mitzenmacher and Upfal, Probability and Computing, Cambridge.

2

Motwani and Raghavan, Randomized Algorithm, Cambridge.

3

Mulmuley, Vazirani and Vazirani, Matching is as easy as matrix inversion, Combinatorica 7(1):105-113, 1987.

4

DeMillo and Lipton, A probabilistic remark on algebraic program testing, Information Processing Letters 7(4):193-195, 1978.