The Combined Basic LP and Affine IP Relaxation for Promise VCSPs on Infinite Domains

Viola, Caterina; Živný, Stanislav

doi:10.1145/3458041

Cited by 9 publications

(17 citation statements)

References 49 publications

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“…We call (A, B) a PVCSP template if the sets of Yes and No instances are disjoint. We show in Proposition 3 that this least restrictive meaningful requirement on a PVCSP template coincides with the choice taken in [25].…”

Section: Pvcspmentioning

confidence: 70%

“…Recall that the equivalence of (iv) and (v) still holds for PVCSPs [25]; we give a streamlined presentation of the proof using Proposition 3. We also mention, in Example 2, some (P)(V)CSPs that satisfy these conditions, and thus also satisfy the equivalent statements in the main result.…”

Section: Contributionsmentioning

confidence: 98%

“…Second, the approach via generalized polymorphisms, so successful in the above special cases, is still available [9] (the work is not yet published). The only published work on PVCSP that we are aware of is [25] where the authors, among other results, generalize (iv) ⇐⇒ (v) in Theorem 1 to the PVCSP setting and even consider the more general infinite-domain case.…”

Section: Promise Valued Cspmentioning

confidence: 99%

“…For two valued σ-structures A and B, the Promise Valued CSP over (A, B) [9,25], denoted PVCSP(A, B), is defined as follows: given a pair (I, τ ), where I is a non-negative finite-valued σstructure and τ ∈ Q is a threshold, output Yes if Opt(I, A) ≤ τ , and output No if Opt(I, B) > τ . We call (A, B) a PVCSP template if the sets of Yes and No instances are disjoint.…”

Section: Pvcspmentioning

confidence: 99%

“…An n-ary fractional polymorphism [25] of two valued σ-structures (A, B) is a probability distribution ω on the set B A n := {f : A n → B} such that for every R ∈ σ and every list of n tuples a 1 , . .…”

Section: Polymorphismsmentioning

confidence: 99%

See 4 more Smart Citations

Weisfeiler-Leman Invariant Promise Valued CSPs

Barto¹,

Butti²

2022

Preprint

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In a recent line of work, Butti and Dalmau have shown that a fixed-template Constraint Satisfaction Problem is solvable by a certain natural linear programming relaxation (equivalent to the basic linear programming relaxation) if and only if it is solvable on a certain distributed network, and this happens if and only if its set of Yes instances is closed under Weisfeiler-Leman equivalence. We generalize this result to the much broader framework of fixed-template Promise Valued Constraint Satisfaction Problems. Moreover, we show that two commonly used linear programming relaxations are no longer equivalent in this broader framework.

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Section: Pvcspmentioning

confidence: 70%

Section: Contributionsmentioning

confidence: 98%

Section: Promise Valued Cspmentioning

confidence: 99%

Section: Pvcspmentioning

confidence: 99%

Section: Polymorphismsmentioning

confidence: 99%

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Weisfeiler-Leman Invariant Promise Valued CSPs

Barto¹,

Butti²

2022

Preprint

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Algebraic Theory of Promise Constraint Satisfaction Problems, First Steps

Barto

2019

Fundamentals of Computation Theory

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What makes a computational problem easy (e.g., in P, that is, solvable in polynomial time) or hard (e.g., NP-hard)? This fundamental question now has a satisfactory answer for a quite broad class of computational problems, so called fixed-template constraint satisfaction problems (CSPs) -it has turned out that their complexity is captured by a certain specific form of symmetry. This paper explains an extension of this theory to a much broader class of computational problems, the promise CSPs, which includes relaxed versions of CSPs such as the problem of finding a 137coloring of a 3-colorable graph.

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The Quantum Cyclic Rotation Gate

Pavone,

Viola

2024

SN COMPUT. SCI.

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A circular shift operator (or cyclic rotation gate) $${{\,\mathrm{\texttt {ROT}}\,}}_k$$ ROT k applies a rightward (or leftward) shift to an input register of n qubits o by as many positions as encoded by an additional input $$k \in \mathbb N$$ k ∈ N . Specifically, the qubit at position x is moved to position $$(x+k) \mod n$$ ( x + k ) mod n . While it is known that there exists a quantum rotation operator that can be implemented in $${{\,\mathrm{\mathcal {O}}\,}}(\log (n))$$ O ( log ( n ) ) -time, through the repeated parallel application of the elementary $${{\,\mathrm{\texttt {Swap}}\,}}$$ Swap operators, there is no systematic procedure that concretely constructs the quantum operator $${{\,\mathrm{\texttt {ROT}}\,}}$$ ROT for variable size n of the quantum register and a variable parameter k. We fill the gap, providing a systematic implementation of the cyclic rotation operator (denoted $${{\,\mathrm{\texttt {ROT}}\,}}$$ ROT ) in a quantum circuit model of computation whose depth is $${{\,\mathrm{\mathcal {O}}\,}}(\log (n))$$ O ( log ( n ) ) . We show how the circular shift operator can be utilized in quantum approaches to text processing, focusing on the problem of getting all possible cyclic rotations of a string in $${{\,\mathrm{\mathcal {O}}\,}}(\log ^2(n))$$ O ( log 2 ( n ) ) depth.

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The Combined Basic LP and Affine IP Relaxation for Promise VCSPs on Infinite Domains

Cited by 9 publications

References 49 publications

Weisfeiler-Leman Invariant Promise Valued CSPs

Weisfeiler-Leman Invariant Promise Valued CSPs

Algebraic Theory of Promise Constraint Satisfaction Problems, First Steps

The Quantum Cyclic Rotation Gate

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