7 research outputs found
PTAS for Sparse General-Valued CSPs
We study polynomial-time approximation schemes (PTASes) for constraint
satisfaction problems (CSPs) such as Maximum Independent Set or Minimum Vertex
Cover on sparse graph classes. Baker's approach gives a PTAS on planar graphs,
excluded-minor classes, and beyond. For Max-CSPs, and even more generally,
maximisation finite-valued CSPs (where constraints are arbitrary non-negative
functions), Romero, Wrochna, and \v{Z}ivn\'y [SODA'21] showed that the
Sherali-Adams LP relaxation gives a simple PTAS for all
fractionally-treewidth-fragile classes, which is the most general "sparsity"
condition for which a PTAS is known. We extend these results to general-valued
CSPs, which include "crisp" (or "strict") constraints that have to be satisfied
by every feasible assignment. The only condition on the crisp constraints is
that their domain contains an element which is at least as feasible as all the
others (but possibly less valuable). For minimisation general-valued CSPs with
crisp constraints, we present a PTAS for all Baker graph classes -- a
definition by Dvo\v{r}\'ak [SODA'20] which encompasses all classes where
Baker's technique is known to work, except possibly for
fractionally-treewidth-fragile classes. While this is standard for problems
satisfying a certain monotonicity condition on crisp constraints, we show this
can be relaxed to diagonalisability -- a property of relational structures
connected to logics, statistical physics, and random CSPs
Fully dynamic approximation schemes on planar and apex-minor-free graphs
The classic technique of Baker [J. ACM '94] is the most fundamental approach
for designing approximation schemes on planar, or more generally
topologically-constrained graphs, and it has been applied in a myriad of
different variants and settings throughout the last 30 years. In this work we
propose a dynamic variant of Baker's technique, where instead of finding an
approximate solution in a given static graph, the task is to design a data
structure for maintaining an approximate solution in a fully dynamic graph,
that is, a graph that is changing over time by edge deletions and edge
insertions. Specifically, we address the two most basic problems -- Maximum
Weight Independent Set and Minimum Weight Dominating Set -- and we prove the
following: for a fully dynamic -vertex planar graph , one can:
* maintain a -approximation of the maximum weight of an
independent set in with amortized update time ; and,
* under the additional assumption that the maximum degree of the graph is
bounded at all times by a constant, also maintain a
-approximation of the minimum weight of a dominating set in
with amortized update time .
In both cases, is doubly-exponential in
and the data structure can be initialized in
time . All our results in fact hold in the
larger generality of any graph class that excludes a fixed apex-graph as a
minor.Comment: 37 pages, accepted to SODA '2
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
Hitting Subgraphs in Sparse Graphs and Geometric Intersection Graphs
We investigate a fundamental vertex-deletion problem called (Induced)
Subgraph Hitting: given a graph and a set of forbidden
graphs, the goal is to compute a minimum-sized set of vertices of such
that does not contain any graph in as an (induced)
subgraph. This is a generic problem that encompasses many well-known problems
that were extensively studied on their own, particularly (but not only) from
the perspectives of both approximation and parameterization. We focus on the
design of efficient approximation schemes, i.e., with running time
, which are also of significant
interest to both communities. Technically, our main contribution is a
linear-time approximation-preserving reduction from (Induced) Subgraph Hitting
on any graph class of bounded expansion to the same problem on
bounded degree graphs within . This yields a novel algorithmic
technique to design (efficient) approximation schemes for the problem on very
broad graph classes, well beyond the state-of-the-art. Specifically, applying
this reduction, we derive approximation schemes with (almost) linear running
time for the problem on any graph classes that have strongly sublinear
separators and many important classes of geometric intersection graphs (such as
fat-object graphs, pseudo-disk graphs, etc.). Our proofs introduce novel
concepts and combinatorial observations that may be of independent interest
(and, which we believe, will find other uses) for studies of approximation
algorithms, parameterized complexity, sparse graph classes, and geometric
intersection graphs. As a byproduct, we also obtain the first robust algorithm
for -Subgraph Isomorphism on intersection graphs of fat objects and
pseudo-disks, with running time .Comment: 60 pages, abstract shortened to fulfill the length limi
Treewidth-Pliability and PTAS for Max-CSPs
We identify a sufficient condition, treewidth-pliability, that gives a
polynomial-time approximation scheme (PTAS) for a large class of Max-2-CSPs
parametrised 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 PTASes. One is
Baker's layering technique, which applies to sparse graphs such as planar or
excluded-minor graphs. The other is based on Szemer\'{e}di's regularity lemma
and applies to dense graphs. We extend the applicability of both techniques to
new classes of Max-CSPs.
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
fractionally-treewidth-fragile and has bounded degree