40 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
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\u2720] for promise (non-valued) CSPs (on finite domains)
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
Beyond PCSP(1-in-3, NAE)
The promise constraint satisfaction problem (PCSP) is a recently introduced vast generalisation of the constraint satisfaction problem (CSP) that captures approximability of satisfiable instances. A PCSP instance comes with two forms of each constraint: a strict one and a weak one. Given the promise that a solution exists using the strict constraints, the task is to find a solution using the weak constraints. While there are by now several dichotomy results for fragments of PCSPs, they all consider (in some way) symmetric PCSPs.
1-in-3-SAT and Not-All-Equal-3-SAT are classic examples of Boolean symmetric (non-promise) CSPs. While both problems are NP-hard, Brakensiek and Guruswami showed [SODA\u2718] that given a satisfiable instance of 1-in-3-SAT one can find a solution to the corresponding instance of (weaker) Not-All-Equal-3-SAT. In other words, the PCSP template (?-in-?,NAE) is tractable.
We focus on non-symmetric PCSPs. In particular, we study PCSP templates obtained from the Boolean template (?-in-?, NAE) by either adding tuples to ?-in-? or removing tuples from NAE. For the former, we classify all templates as either tractable or not solvable by the currently strongest known algorithm for PCSPs, the combined basic LP and affine IP relaxation of Brakensiek and Guruswami [SODA\u2720]. For the latter, we classify all templates as either tractable or NP-hard
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
SDPs and Robust Satisfiability of Promise CSP
For a constraint satisfaction problem (CSP), a robust satisfaction algorithm
is one that outputs an assignment satisfying most of the constraints on
instances that are near-satisfiable. It is known that the CSPs that admit
efficient robust satisfaction algorithms are precisely those of bounded width,
i.e., CSPs whose satisfiability can be checked by a simple local consistency
algorithm (eg., 2-SAT or Horn-SAT in the Boolean case). While the exact
satisfiability of a bounded width CSP can be checked by combinatorial
algorithms, the robust algorithm is based on rounding a canonical Semidefinite
programming(SDP) relaxation.
In this work, we initiate the study of robust satisfaction algorithms for
promise CSPs, which are a vast generalization of CSPs that have received much
attention recently. The motivation is to extend the theory beyond CSPs, as well
as to better understand the power of SDPs. We present robust SDP rounding
algorithms under some general conditions, namely the existence of particular
high-dimensional Boolean symmetries known as majority or alternating threshold
polymorphisms. On the hardness front, we prove that the lack of such
polymorphisms makes the PCSP hard for all pairs of symmetric Boolean
predicates. Our method involves a novel method to argue SDP gaps via the
absence of certain colorings of the sphere, with connections to sphere Ramsey
theory.
We conjecture that PCSPs with robust satisfaction algorithms are precisely
those for which the feasibility of the canonical SDP implies (exact)
satisfiability. We also give a precise algebraic condition, known as a minion
characterization, of which PCSPs have the latter property.Comment: 62 pages, to appear in STOC 202
Valued Constraint Satisfaction Problems over Infinite Domains
The object of the thesis is the computational complexity of certain combinatorial optimisation problems called \emph{valued constraint satisfaction problems}, or \emph{VCSPs} for short. The requirements and optimisation criteria of these problems are expressed by sums of \emph{(valued) constraints} (also called \emph{cost functions}). More precisely, the input of a VCSP consists of a finite set of variables, a finite set of cost functions that depend on these variables, and a cost ; the task is to find values for the variables such that the sum of the cost functions is at most .
By restricting the set of possible cost functions in the input, a great variety of computational optimisation problems can be modelled as VCSPs. Recently, the computational complexity of all VCSPs for finite sets of cost functions over a finite domain has been classified. Many natural optimisation problems, however, cannot be formulated as VCSPs over a finite domain.
We initiate the systematic investigation of infinite-domain VCSPs by studying the complexity of VCSPs for piecewise linear (PL) and piecewise linear homogeneous (PLH) cost functions.
The VCSP for a finite set of PLH cost functions can be solved in polynomial time if the cost functions are improved by fully symmetric fractional operations of all arities. We
show this by (polynomial-time many-one) reducing the problem to a finite-domain VCSP which can be solved using a linear programming relaxation. We apply this result to show the polynomial-time tractability of VCSPs for {\it submodular} PLH cost functions, for {\it convex} PLH cost functions, and for {\it componentwise increasing} PLH cost functions; in fact, we show that submodular PLH functions and componentwise increasing PLH functions form maximally tractable classes of PLH cost functions.
We define the notion of {\it expressive power} for sets of cost functions over arbitrary domains, and discuss the relation between the expressive power and the set of fractional operations improving the same set of cost functions over an arbitrary countable domain.
Finally, we provide a polynomial-time algorithm solving the restriction of the VCSP for {\it all} PL cost functions to a fixed number of variables
Fractional Homomorphism, Weisfeiler-Leman Invariance, and the Sherali-Adams Hierarchy for the Constraint Satisfaction Problem
Given a pair of graphs ? and ?, the problems of deciding whether there exists either a homomorphism or an isomorphism from ? to ? have received a lot of attention. While graph homomorphism is known to be NP-complete, the complexity of the graph isomorphism problem is not fully understood. A well-known combinatorial heuristic for graph isomorphism is the Weisfeiler-Leman test together with its higher order variants. On the other hand, both problems can be reformulated as integer programs and various LP methods can be applied to obtain high-quality relaxations that can still be solved efficiently. We study so-called fractional relaxations of these programs in the more general context where ? and ? are not graphs but arbitrary relational structures. We give a combinatorial characterization of the Sherali-Adams hierarchy applied to the homomorphism problem in terms of fractional isomorphism. Collaterally, we also extend a number of known results from graph theory to give a characterization of the notion of fractional isomorphism for relational structures in terms of the Weisfeiler-Leman test, equitable partitions, and counting homomorphisms from trees. As a result, we obtain a description of the families of CSPs that are closed under Weisfeiler-Leman invariance in terms of their polymorphisms as well as decidability by the first level of the Sherali-Adams hierarchy
Fractional homomorphism, Weisfeiler-Leman invariance, and the Sherali-Adams hierarchy for the Constraint Satisfaction Problem
Given a pair of graphs and , the problems of
deciding whether there exists either a homomorphism or an isomorphism from
to have received a lot of attention. While graph
homomorphism is known to be NP-complete, the complexity of the graph
isomorphism problem is not fully understood. A well-known combinatorial
heuristic for graph isomorphism is the Weisfeiler-Leman test together with its
higher order variants. On the other hand, both problems can be reformulated as
integer programs and various LP methods can be applied to obtain high-quality
relaxations that can still be solved efficiently. We study so-called fractional
relaxations of these programs in the more general context where
and are not graphs but arbitrary relational structures. We give a
combinatorial characterization of the Sherali-Adams hierarchy applied to the
homomorphism problem in terms of fractional isomorphism. Collaterally, we also
extend a number of known results from graph theory to give a characterization
of the notion of fractional isomorphism for relational structures in terms of
the Weisfeiler-Leman test, equitable partitions, and counting homomorphisms
from trees. As a result, we obtain a description of the families of CSPs that
are closed under Weisfeiler-Leman invariance in terms of their polymorphisms as
well as decidability by the first level of the Sherali-Adams hierarchy.Comment: Full version of a MFCS'21 pape