82 research outputs found
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
Improved hardness for H-colourings of G-colourable graphs
We present new results on approximate colourings of graphs and, more
generally, approximate H-colourings and promise constraint satisfaction
problems.
First, we show NP-hardness of colouring -colourable graphs with
colours for every . This improves
the result of Bul\'in, Krokhin, and Opr\v{s}al [STOC'19], who gave NP-hardness
of colouring -colourable graphs with colours for , and the
result of Huang [APPROX-RANDOM'13], who gave NP-hardness of colouring
-colourable graphs with colours for sufficiently large .
Thus, for , we improve from known linear/sub-exponential gaps to
exponential gaps.
Second, we show that the topology of the box complex of H alone determines
whether H-colouring of G-colourable graphs is NP-hard for all (non-bipartite,
H-colourable) G. This formalises the topological intuition behind the result of
Krokhin and Opr\v{s}al [FOCS'19] that 3-colouring of G-colourable graphs is
NP-hard for all (3-colourable, non-bipartite) G. We use this technique to
establish NP-hardness of H-colouring of G-colourable graphs for H that include
but go beyond , including square-free graphs and circular cliques (leaving
and larger cliques open).
Underlying all of our proofs is a very general observation that adjoint
functors give reductions between promise constraint satisfaction problems.Comment: Mention improvement in Proposition 2.5. SODA 202
A Strongly Polynomial-Time Algorithm for Weighted General Factors with Three Feasible Degrees
General factors are a generalization of matchings. Given a graph with a
set of feasible degrees, called a degree constraint, for each vertex
of , the general factor problem is to find a (spanning) subgraph of
such that for every of . When all
degree constraints are symmetric -matroids, the problem is solvable in
polynomial time. The weighted general factor problem is to find a general
factor of the maximum total weight in an edge-weighted graph. Strongly
polynomial-time algorithms are only known for weighted general factor problems
that are reducible to the weighted matching problem by gadget constructions.
In this paper, we present the first strongly polynomial-time algorithm for a
type of weighted general factor problems with real-valued edge weights that is
provably not reducible to the weighted matching problem by gadget
constructions
Linearly ordered colourings of hypergraphs
A linearly ordered (LO) -colouring of an -uniform hypergraph assigns an
integer from to every vertex so that, in every edge, the
(multi)set of colours has a unique maximum. Equivalently, for , if two
vertices in an edge are assigned the same colour, then the third vertex is
assigned a larger colour (as opposed to a different colour, as in classic
non-monochromatic colouring). Barto, Battistelli, and Berg [STACS'21] studied
LO colourings on -uniform hypergraphs in the context of promise constraint
satisfaction problems (PCSPs). We show two results.
First, given a 3-uniform hypergraph that admits an LO -colouring, one can
find in polynomial time an LO -colouring with .
Second, given an -uniform hypergraph that admits an LO -colouring, we
establish NP-hardness of finding an LO -colouring for every constant
uniformity . In fact, we determine relationships between
polymorphism minions for all uniformities , which reveals a key
difference between and and which may be of independent
interest. Using the algebraic approach to PCSPs, we actually show a more
general result establishing NP-hardness of finding an LO -colouring for LO
-colourable -uniform hypergraphs for and .Comment: Full version (with stronger both tractability and intractability
results) of an ICALP 2022 pape
On rainbow-free colourings of uniform hypergraphs
We study rainbow-free colourings of -uniform hypergraphs; that is,
colourings that use colours but with the property that no hyperedge attains
all colours. We show that is the threshold function for
the existence of a rainbow-free colouring in a random -uniform hypergraph
Boolean symmetric vs. functional PCSP dichotomy
Given a 3-uniform hypergraph that is promised to admit a
-colouring such that every edge contains exactly one , can one find
a -colouring such that for every
? This can be cast as a promise constraint satisfaction problem (PCSP)
of the form , where
defines the relation , and is an example of
, where (and thus wlog
also ) is symmetric. The computational complexity of such problems
is understood for and on Boolean domains by the work
of Ficak, Kozik, Ol\v{s}\'{a}k, and Stankiewicz [ICALP'19].
As our first result, we establish a dichotomy for
, where is Boolean and
symmetric and is functional (on a domain of any size); i.e, all
but one element of any tuple in a relation in determine the last
element. This includes PCSPs of the form
, where is functional,
thus making progress towards a classification of
, which were studied by Barto,
Battistelli, and Berg [STACS'21] for on three-element domains.
As our second result, we show that for
, where contains a
single Boolean symmetric relation and is arbitrary (and thus not
necessarily functional), the combined basic linear programmin relaxation (BLP)
and the affine integer programming relaxation (AIP) of Brakensiek et al.
[SICOMP'20] is no more powerful than the (in general strictly weaker) AIP
relaxation of Brakensiek and Guruswami [SICOMP'21]
Point-width and Max-CSPs
International audienceThe complexity of (unbounded-arity) Max-CSPs under structural restrictions is poorly understood. The two most general hypergraph properties known to ensure tractability of Max-CSPs, β-acyclicity and bounded (incidence) MIM-width, are incomparable and lead to very different algorithms. We introduce the framework of point decompositions for hypergraphs and use it to derive a new sufficient condition for the tractability of (structurally restricted) Max-CSPs, which generalises both bounded MIM-width and β-acyclicity. On the way, we give a new characterisation of bounded MIM-width and discuss other hypergraph properties which are relevant to the complexity of Max-CSPs, such as β-hypertreewidth
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
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
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