G.P. Kogan and J.A. Makowsky. Computing the permanent over fields of characteristic 3: Where and why it becomes difficult. Technical 34 report, Technion--Israel Institute of Technology, February 1996, submitted for publication to Computational Complexity, http://cs.technion.ac.il/admlogic/TR/readme.html.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... M over Zof linear rank r (1) the permanent of M can be computed using g per (r) Delta O(n b ) arithmetic operations for some constant b = O(r) and (2) the hamiltonian of M can be computed using g ham (r) Delta O(n c ) arithmetic operations for some constant c = O(r 2 ) Kogan in [52] looked at the linear rank of MM T Gamma 1 rather than of M and showed that in fields of characteristic 3 and for matrices M with rk(MM T Gamma 1) 1 the permanent per(M) can be computed in polynomial time. However, he also shows that for rk(MM T Gamma 1) 2 the function is as difficult ....

....metafinite structures can now be defined as depending only on the underlying finite structure. We conjecture that theorems 31 and 32 have their suitable analogues. 30 Acknowledgements The second author is indebted to Gregory Kogan for introducing him to the world of permanents as documented in [52,55,53,54]. He would also like to thank Peter Burgisser, who made an early version of [17] available, to M. Grohe, who pointed out the bound on the treewidth of the incidence graph, and to R. Bar Yehuda, who reminded us of SAT . The final version of this papers was prepared during a visit of the second ....

G.P. Kogan. Computing the permanent over fields of characteristic 3: Where and why it becomes difficult. In FOCS'96, pages 108--114. IEEE, 1996.

....p Gamma 1 times, p 0. Then adding a multiple of this column to any other column doesn t change perrank(A) 2 This lemma suggests that we can use elementary operations to study the permanent. This idea has also been discovered independently by Kogan and Makowsky; see section 3. 5 of [8]. The following result (true for p = 3 only) is an application of this idea. Theorem 1.5 If p = 3, then perrank(A) perrank(A Gamma1 ) for any nonsingular A. Proof. Let T 1 ; T 2 ; T 3 ; Delta Delta Delta be a sequence of elementary column operations that change A n Thetan to I. Consider ....

G. Kogan and J.A. Makowsky, Computing Permanents over Fields of Characteristic 3: Where and Why it Becomes Difficult, to appear in Computational Complexity.

.... matrices M over Zof linear rank r (i) the permanent of M can be computed using g per (r) Delta O(n b ) arithmetic operations for some constant b = O(r) and (ii) the Hamiltonian of M can be computed using g ham (r) Delta O(n c ) arithmetic operations for some constant c = O(r 2 ) Kogan in [Kog96] looked at the linear rank of MM T Gamma 1 rather than of M and showed that in fields of characteristc 3 and for matrices M with rk(MM T Gamma 1) 1 the permanent per(M ) can be computed in polynomial time. However, he also shows that for rk(MM T Gamma 1) 2 the problem is again ]P hard. ....

....of a metafinite structures can now be defined as depending only on the underlying finite structure. We conjecture that theorems 5 and 4 have their suitable analogues. Acknowledgements The second author is indebted to Gregory Kogan for introducing him to the world of permanents as documented in [Kog96,KM96,KM97,KM99]. He would also like to thank Peter Burgisser, who made an early version of [Bur98] available, to M. Grohe, who pointed out the bound on the treewidth of the incidence graph, and to R. Bar Yehuda, who reminded us of SAT . ....

G.P. Kogan. Computing the permanent over fields of characteristic 3: Where and why it becomes difficult. In FOCS'96, pages 108--114. IEEE, 1996.

.... matrices M over Zof linear rank r (i) the permanent of M can be computed using g per (r) Delta O(n b ) arithmetic operations for some constant b = O(r) and (ii) the Hamiltonian of M can be computed using g ham (r) Delta O(n c ) arithmetic operations for some constant c = O(r 2 ) Kogan in [Kog96] looked at the linear rank of MM T Gamma 1 rather than of M and showed that in fields of characteristc 3 and for matrices M with rk(MM T Gamma 1) 1 the permanent per(M ) can be computed in polynomial time. However, he also shows that for rk(MM T Gamma 1) 2 the problem is again ]P hard. ....

....of a metafinite structures can now be defined as depending only on the underlying finite structure. We conjecture that theorems 5 and 6 have their suitable analogues. Acknowledgements The second author is indebted to Gregory Kogan for introducing him to the world of permanents as documented in [Kog96, KM96, KM97, KM99]. He would also like to thank Peter Burgisser, who made an early version of [Bur98] available, and to M. Grohe, who improved the bound on the treewidth of the incidence graph. ....

G.P. Kogan. Computing the permanent over fields of characteristic 3: Where and why it becomes difficult. In FOCS'96, pages 108--114. IEEE, 1996.

....metafinite structures can now be defined as depending only on the underlying finite structure. We conjecture that theorems 31 and 32 have their suitable analogues. 30 Acknowledgements The second author is indebted to Gregory Kogan for introducing him to the world of permanents as documented in [52,55,53,54]. He would also like to thank Peter Burgisser, who made an early version of [17] available, to M. Grohe, who pointed out the bound on the treewidth of the incidence graph, and to R. Bar Yehuda, who reminded us of SAT . The final version of this papers was prepared during a visit of the second ....

G.P. Kogan and J.A. Makowsky. Computing the permanent over fields of characteristic 3: Where and why it becomes difficult. Technical 34 report, Technion--Israel Institute of Technology, February 1996, submitted for publication to Computational Complexity, http://cs.technion.ac.il/admlogic/TR/readme.html.

....of a metafinite structures can now be defined as depending only on the underlying finite structure. We conjecture that theorems 5 and 4 have their suitable analogues. Acknowledgements The second author is indebted to Gregory Kogan for introducing him to the world of permanents as documented in [Kog96,KM96,KM97,KM99]. He would also like to thank Peter Burgisser, who made an early version of [Bur98] available, to M. Grohe, who pointed out the bound on the treewidth of the incidence graph, and to R. Bar Yehuda, who reminded us of SAT . ....

G.P. Kogan and J.A. Makowsky. Computing the permanent over fields of characteristic 3: Where and why it becomes difficult. Technion--Israel Institute of Technology, February 1996, submitted, 1996.

....presented these results first in September 1995 at the Technion, both orally and leaving a manuscript in Russian. The current version of the paper was prepared by the second author on the basis of notes taken during these lectures. An expanded version of section 2 was essentially published as [Kog96]. We are indebted to M. Kaminski, who was present when the first author lectured and helped throughout in making these lectures comprehensible. We would like to thank M. Kaminski for allowing us to use his notes of some of the material in sections 3.5, 3.6 and 6. We would like to thank I. ....

G.P. Kogan. Computing the permanent over fields of characteristic 3: Where and why it becomes difficult. In FOCS'96, pages 108--114. IEEE, 1996.

....two kinds of reductions, by projections, as defined by Valiant, and by subroutines, introduced here in section 3. Valiant in [Val79] cf. BCS97, theorem 21.17] proved Theorem 1.1 (Valiant) The family HC is VNP(k) complete via projections. The same is true for PER provided char(k) 6= 2. In [Kog96, KMb] we studied the complexity of PER restricted to unitary and semi unitary matrices in fields k with char(k) 3. A matrix U is unitary if U Delta U T Gamma 1n = 0n . A matrix V is k semi unitary if the rank rg(V Delta V T Gamma 1n ) k. Our main results in [Kog96, KMb] can be summarized as ....

....char(k) 6= 2. In [Kog96, KMb] we studied the complexity of PER restricted to unitary and semi unitary matrices in fields k with char(k) 3. A matrix U is unitary if U Delta U T Gamma 1n = 0n . A matrix V is k semi unitary if the rank rg(V Delta V T Gamma 1n ) k. Our main results in [Kog96, KMb] can be summarized as follows: Theorem 1.2 (Kogan) Let k be a field with char(k) 3. i) The family SU 1 PER, PER restricted to 1 semi unitary matrices, is in VP(k) ii) The family SU 2 PER, PER restricted to 2 semi unitary matrices, is VNP(k) complete by subroutine reductions. Now, let ....

G.P. Kogan. Computing the permanent over fields of characteristic 3: Where and why it becomes difficult. In FOCS'96, pages 108--114. IEEE, 1996.

....two kinds of reductions, by projections, as defined by Valiant, and by subroutines, introduced here in section 3. Valiant in [Val79] cf. BCS97, theorem 21.17] proved Theorem 1.1 (Valiant) The family HC is VNP(k) complete via projections. The same is true for PER provided char(k) 6= 2. In [Kog96, KMb] we studied the complexity of PER restricted to unitary and semi unitary matrices in fields k with char(k) 3. A matrix U is unitary if U Delta U T Gamma 1n = 0n . A matrix V is k semi unitary if the rank rg(V Delta V T Gamma 1n ) k. Our main results in [Kog96, KMb] can be summarized as ....

....char(k) 6= 2. In [Kog96, KMb] we studied the complexity of PER restricted to unitary and semi unitary matrices in fields k with char(k) 3. A matrix U is unitary if U Delta U T Gamma 1n = 0n . A matrix V is k semi unitary if the rank rg(V Delta V T Gamma 1n ) k. Our main results in [Kog96, KMb] can be summarized as follows: Theorem 1.2 (Kogan) Let k be a field with char(k) 3. i) The family SU 1 PER, PER restricted to 1 semi unitary matrices, is in VP(k) ii) The family SU 2 PER, PER restricted to 2 semi unitary matrices, is VNP(k) complete by subroutine reductions. Now, let ....

G.P. Kogan and J.A. Makowsky. Computing the permanent over fields of characteristic 3: Where and why it becomes difficult. Technion-- Israel Institute of Technology, February 1996, Journal version in preparation.

....of a metafinite structures can now be defined as depending only on the underlying finite structure. We conjecture that theorems 5 and 6 have their suitable analogues. Acknowledgements The second author is indebted to Gregory Kogan for introducing him to the world of permanents as documented in [Kog96, KM96, KM97, KM99]. He would also like to thank Peter Burgisser, who made an early version of [Bur98] available, and to M. Grohe, who improved the bound on the treewidth of the incidence graph. ....

G.P. Kogan and J.A. Makowsky. Computing the permanent over fields of characteristic 3: Where and why it becomes difficult. Technion--Israel Institute of Technology, February 1996, submitted, 1996.

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