115 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
When do homomorphism counts help in query algorithms?
A query algorithm based on homomorphism counts is a procedure for determining
whether a given instance satisfies a property by counting homomorphisms between
the given instance and finitely many predetermined instances. In a left query
algorithm, we count homomorphisms from the predetermined instances to the given
instance, while in a right query algorithm we count homomorphisms from the
given instance to the predetermined instances. Homomorphisms are usually
counted over the semiring N of non-negative integers; it is also meaningful,
however, to count homomorphisms over the Boolean semiring B, in which case the
homomorphism count indicates whether or not a homomorphism exists. We first
characterize the properties that admit a left query algorithm over B by showing
that these are precisely the properties that are both first-order definable and
closed under homomorphic equivalence. After this, we turn attention to a
comparison between left query algorithms over B and left query algorithms over
N. In general, there are properties that admit a left query algorithm over N
but not over B. The main result of this paper asserts that if a property is
closed under homomorphic equivalence, then that property admits a left query
algorithm over B if and only if it admits a left query algorithm over N. In
other words and rather surprisingly, homomorphism counts over N do not help as
regards properties that are closed under homomorphic equivalence. Finally, we
characterize the properties that admit both a left query algorithm over B and a
right query algorithm over B.Comment: 24 page
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
On the Algebraic Proof Complexity of Tensor Isomorphism
The Tensor Isomorphism problem (TI) has recently emerged as having connections to multiple areas of research within complexity and beyond, but the current best upper bound is essentially the brute force algorithm. Being an algebraic problem, TI (or rather, proving that two tensors are non-isomorphic) lends itself very naturally to algebraic and semi-algebraic proof systems, such as the Polynomial Calculus (PC) and Sum of Squares (SoS). For its combinatorial cousin Graph Isomorphism, essentially optimal lower bounds are known for approaches based on PC and SoS (Berkholz & Grohe, SODA \u2717). Our main results are an ?(n) lower bound on PC degree or SoS degree for Tensor Isomorphism, and a nontrivial upper bound for testing isomorphism of tensors of bounded rank.
We also show that PC cannot perform basic linear algebra in sub-linear degree, such as comparing the rank of two matrices (which is essentially the same as 2-TI), or deriving BA = I from AB = I. As linear algebra is a key tool for understanding tensors, we introduce a strictly stronger proof system, PC+Inv, which allows as derivation rules all substitution instances of the implication AB = I ? BA = I. We conjecture that even PC+Inv cannot solve TI in polynomial time either, but leave open getting lower bounds on PC+Inv for any system of equations, let alone those for TI. We also highlight many other open questions about proof complexity approaches to TI
An Analysis of Core-Guided Maximum Satisfiability Solvers Using Linear Programming
Many current complete MaxSAT algorithms fall into two categories: core-guided or implicit hitting set. The two kinds of algorithms seem to have complementary strengths in practice, so that each kind of solver is better able to handle different families of instances. This suggests that a hybrid might match and outperform either, but the techniques used seem incompatible. In this paper, we focus on PMRES and OLL, two core-guided algorithms based on max resolution and soft cardinality constraints, respectively. We show that these algorithms implicitly discover cores of the original formula, as has been previously shown for PM1. Moreover, we show that in some cases, including unweighted instances, they compute the optimum hitting set of these cores at each iteration. We also give compact integer linear programs for each which encode this hitting set problem. Importantly, their continuous relaxation has an optimum that matches the bound computed by the respective algorithms. This goes some way towards resolving the incompatibility of implicit hitting set and core-guided algorithms, since solvers based on the implicit hitting set algorithm typically solve the problem by encoding it as a linear program
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
Local consistency as a reduction between constraint satisfaction problems
We study the use of local consistency methods as reductions between
constraint satisfaction problems (CSPs), and promise version thereof, with the
aim to classify these reductions in a similar way as the algebraic approach
classifies gadget reductions between CSPs. This research is motivated by the
requirement of more expressive reductions in the scope of promise CSPs. While
gadget reductions are enough to provide all necessary hardness in the scope of
(finite domain) non-promise CSP, in promise CSPs a wider class of reductions
needs to be used.
We provide a general framework of reductions, which we call consistency
reductions, that covers most (if not all) reductions recently used for proving
NP-hardness of promise CSPs. We prove some basic properties of these
reductions, and provide the first steps towards understanding the power of
consistency reductions by characterizing a fragment associated to
arc-consistency in terms of polymorphisms of the template. In addition to
showing hardness, consistency reductions can also be used to provide feasible
algorithms by reducing to a fixed tractable (promise) CSP, for example, to
solving systems of affine equations. In this direction, among other results, we
describe the well-known Sherali-Adams hierarchy for CSP in terms of a
consistency reduction to linear programming
Pliability and approximating max-CSPs
We identify a sufficient condition, treewidth-pliability, that gives a polynomial-time
algorithm for an arbitrarily good approximation of the optimal value in a large class of
Max-2-CSPs parameterised by the class of allowed constraint graphs (with arbitrary constraints on an unbounded alphabet). Our result applies more generally to the maximum
homomorphism problem between two rational-valued structures.
The condition unifies the two main approaches for designing a polynomial-time approximation scheme. One is Baker’s layering technique, which applies to sparse graphs
such as planar or excluded-minor graphs. The other is based on Szemer´edi’s regularity
lemma and applies to dense graphs. We extend the applicability of both techniques to
new classes of Max-CSPs. On the other hand, we prove that the condition cannot be used
to find solutions (as opposed to approximating the optimal value) in general.
Treewidth-pliability turns out to be a robust notion that can be defined in several
equivalent ways, including characterisations via size, treedepth, or the Hadwiger number.
We show connections to the notions of fractional-treewidth-fragility from structural graph
theory, hyperfiniteness from the area of property testing, and regularity partitions from
the theory of dense graph limits. These may be of independent interest. In particular
we show that a monotone class of graphs is hyperfinite if and only if it is fractionallytreewidth-fragile and has bounded degree
- …