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For any n ≥ 3 and q ≥ 3, we prove that the Equality function (= n ) on n variables over a domain of size q cannot be realized by matchgates under holographic transformations. This is a consequence of our theorem on the structure of blockwise symmetric matchgate signatures. This has the implication that the standard holographic algorithms based on matchgates, a methodology known to be universal for #CSP over the Boolean domain, cannot produce P-time algorithms for planar #CSP over any higher domain q ≥ 3.

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“…odd). It was proved in [4] that matchgate signatures are characterized by the following two sets of conditions.…”

Section: Preliminariesmentioning

confidence: 99%

“…(alternating sum by flipping in sequence the bits p i and the bits in P \ {p i }), where α + β denotes the XOR of α and β. Actually in [4] it is shown that MGI implies the Parity Condition. But in practice, it is easier to apply the Parity Condition first.…”

Section: Preliminariesmentioning

confidence: 99%

On blockwise symmetric matchgate signatures and higher domain #CSP

Yang

Yin

2019

Information and Computation

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“…even) weight are zero. The matchgate signatures are characterized by the following two sets of conditions (see [5] for a self-contained proof). (1) The Parity Condition: either all even entries are 0 or all odd entries are 0.…”

Section: Matchgate Signaturesmentioning

confidence: 99%

“…Actually in [5] it is shown that MGI implies the Parity Condition. But in practice, it is easier to apply the Parity Condition first.…”

Section: Matchgate Signaturesmentioning

confidence: 99%

Holographic algorithm with matchgates is universal for planar #CSP over boolean domain

Cai

2017

Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing

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We prove a complexity classification theorem that classifies all counting constraint satisfaction problems (#CSP) over Boolean variables into exactly three categories: (1) Polynomial-time tractable;(2) #P-hard for general instances, but solvable in polynomial-time over planar graphs; and (3) #P-hard over planar graphs. The classification applies to all sets of local, not necessarily symmetric, constraint functions on Boolean variables that take complex values. It is shown that Valiant's holographic algorithm with matchgates is a universal strategy for all problems in category (2).Proof. One direction is trivial, since (= 4 ) ∈ EQ 2 . For the other direction we prove by induction. For k = 1, we have (= 2 ) ∈ EQ. For k = 2, we have (= 4 ) given. Assume that we have (= 2(k−1) ). Then connecting (= 2(k−1) ) and (= 4 ) by one edge we get (= 2k ).By (2.

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“…Define an n × n skew-symmetric matrix B where its (i, j) entry, for 1 ≤ i < j ≤ n, is 11...1⊕e i ⊕e j , the signature value of on the bit pattern that has two 1's at the i-th and j-th bit positions and 0 elsewhere. The theory of matchgate signatures ( [3], see also [1,2]) implies that for any…”

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confidence: 99%