19 research outputs found
The Power of the Combined Basic LP and Affine Relaxation for Promise CSPs
In the field of constraint satisfaction problems (CSP), promise CSPs are an
exciting new direction of study. In a promise CSP, each constraint comes in two
forms: "strict" and "weak," and in the associated decision problem one must
distinguish between being able to satisfy all the strict constraints versus not
being able to satisfy all the weak constraints. The most commonly cited example
of a promise CSP is the approximate graph coloring problem--which has recently
seen exciting progress [BKO19, WZ20] benefiting from a systematic algebraic
approach to promise CSPs based on "polymorphisms," operations that map tuples
in the strict form of each constraint to tuples in the corresponding weak form.
In this work, we present a simple algorithm which in polynomial time solves
the decision problem for all promise CSPs that admit infinitely many symmetric
polymorphisms, which are invariant under arbitrary coordinate permutations.
This generalizes previous work of the first two authors [BG19]. We also extend
this algorithm to a more general class of block-symmetric polymorphisms. As a
corollary, this single algorithm solves all polynomial-time tractable Boolean
CSPs simultaneously. These results give a new perspective on Schaefer's classic
dichotomy theorem and shed further light on how symmetries of polymorphisms
enable algorithms. Finally, we show that block symmetric polymorphisms are not
only sufficient but also necessary for this algorithm to work, thus
establishing its precise powerComment: 17 pages, to appear in SICOM
Algebraic Theory of Promise Constraint Satisfaction Problems, First Steps
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 137-coloring of a 3-colorable graph
Promises Make Finite (Constraint Satisfaction) Problems Infinitary
The fixed template Promise Constraint Satisfaction Problem (PCSP) is a
recently proposed significant generalization of the fixed template CSP, which
includes approximation variants of satisfiability and graph coloring problems.
All the currently known tractable (i.e., solvable in polynomial time) PCSPs
over finite templates can be reduced, in a certain natural way, to tractable
CSPs. However, such CSPs are often over infinite domains. We show that the
infinity is in fact necessary by proving that a specific finite-domain PCSP,
namely (1-in-3-SAT, Not-All-Equal-3-SAT), cannot be naturally reduced to a
tractable finite-domain CSP, unless P=NP
The combined basic LP and affine IP relaxation for promise VCSPs on infinite domains
Convex relaxations have been instrumental in solvability of constraint
satisfaction problems (CSPs), as well as in the three different generalisations
of CSPs: valued CSPs, infinite-domain CSPs, and most recently promise CSPs. In
this work, we extend an existing tractability result to the three
generalisations of CSPs combined: We give a sufficient condition for the
combined basic linear programming and affine integer programming relaxation for
exact solvability of promise valued CSPs over infinite-domains. This extends a
result of Brakensiek and Guruswami [SODA'20] for promise (non-valued) CSPs (on
finite domains).Comment: Full version of an MFCS'20 pape
Approximate Graph Colouring and the Hollow Shadow
We show that approximate graph colouring is not solved by constantly many
levels of the lift-and-project hierarchy for the combined basic linear
programming and affine integer programming relaxation. The proof involves a
construction of tensors whose fixed-dimensional projections are equal up to
reflection and satisfy a sparsity condition, which may be of independent
interest.Comment: Generalises and subsumes results from Section 6 in arXiv:2203.02478;
builds on and generalises results in arXiv:2210.0829
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