44 research outputs found
Robust algorithms with polynomial loss for near-unanimity CSPs
An instance of the Constraint Satisfaction Problem (CSP) is given by a family of constraints on overlapping sets of variables, and the goal is to assign values from a xed domain to the variables so that all constraints are satised. In the optimization version, the goal is to maximize the number of satised constraints. An approximation algorithm for CSP is called robust if it outputs an assignment satisfying an (1????g("))-fraction of constraints on any (1????")-satisable instance, where the loss function g is such that g(") ! 0 as " ! 0. We study how the robust approximability of CSPs depends on the set of constraint relations allowed in instances, the so-called constraint language. All constraint languages admitting a robust polynomial-time algorithm (with some g) have been characterised by Barto and Kozik, with the general bound on the loss g being doubly exponential, specically g(") = O((log log(1="))= log(1=")). It is natural to ask when a better loss can be achieved: in particular, polynomial loss g(") = O("1=k) for some constant k. In this paper, we consider CSPs with a constraint language having a nearunanimity polymorphism. This general condition almost matches a known necessary condition for having a robust algorithm with polynomial loss. We give two randomized robust algorithms with polynomial loss for such CSPs: one works for any near-unanimity polymorphism and the parameter k in the loss depends on the size of the domain and the arity of the relations in ????, while the other works for a special ternary near-unanimity operation called dual discriminator with k = 2 for any domain size. In the latter case, the CSP is a common generalisation of Unique Games with a xed domain and 2-Sat. In the former case, we use the algebraic approach to the CSP. Both cases use the standard semidenite programming relaxation for CSP
Homomorphism Tensors and Linear Equations
Lov\'asz (1967) showed that two graphs and are isomorphic if and only
if they are homomorphism indistinguishable over the class of all graphs, i.e.
for every graph , the number of homomorphisms from to equals the
number of homomorphisms from to . Recently, homomorphism
indistinguishability over restricted classes of graphs such as bounded
treewidth, bounded treedepth and planar graphs, has emerged as a surprisingly
powerful framework for capturing diverse equivalence relations on graphs
arising from logical equivalence and algebraic equation systems.
In this paper, we provide a unified algebraic framework for such results by
examining the linear-algebraic and representation-theoretic structure of
tensors counting homomorphisms from labelled graphs. The existence of certain
linear transformations between such homomorphism tensor subspaces can be
interpreted both as homomorphism indistinguishability over a graph class and as
feasibility of an equational system. Following this framework, we obtain
characterisations of homomorphism indistinguishability over two natural graph
classes, namely trees of bounded degree and graphs of bounded pathwidth,
answering a question of Dell et al. (2018).Comment: 33 pages, accepted for ICALP 202
Non-Homogenizable Classes of Finite Structures
Homogenization is a powerful way of taming a class of finite structures with several interesting applications in different areas, from Ramsey theory in combinatorics to constraint satisfaction problems (CSPs) in computer science, through (finite) model theory. A few sufficient conditions for a class of finite structures to allow homogenization are known, and here we provide a necessary condition. This lets us show that certain natural classes are not homogenizable: 1) the class of locally consistent systems of linear equations over the two-element field or any finite Abelian group, and 2) the class of finite structures that forbid homomorphisms from a specific MSO-definable class of structures of treewidth two. In combination with known results, the first example shows that, up to pp-interpretability, the CSPs that are solvable by local consistency methods are distinguished from the rest by the fact that their classes of locally consistent instances are homogenizable. The second example shows that, for MSO-definable classes of forbidden patterns, treewidth one versus two is the dividing line to homogenizability
QCSP on partially reflexive forests
We study the (non-uniform) quantified constraint satisfaction problem QCSP(H)
as H ranges over partially reflexive forests. We obtain a complexity-theoretic
dichotomy: QCSP(H) is either in NL or is NP-hard. The separating condition is
related firstly to connectivity, and thereafter to accessibility from all
vertices of H to connected reflexive subgraphs. In the case of partially
reflexive paths, we give a refinement of our dichotomy: QCSP(H) is either in NL
or is Pspace-complete
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