3 research outputs found
Rainbow Coloring Hardness via Low Sensitivity Polymorphisms
A k-uniform hypergraph is said to be r-rainbow colorable if there is an r-coloring of its vertices such that every hyperedge intersects all r color classes. Given as input such a hypergraph, finding a r-rainbow coloring of it is NP-hard for all k >= 3 and r >= 2. Therefore, one settles for finding a rainbow coloring with fewer colors (which is an easier task). When r=k (the maximum possible value), i.e., the hypergraph is k-partite, one can efficiently 2-rainbow color the hypergraph, i.e., 2-color its vertices so that there are no monochromatic edges. In this work we consider the next smaller value of r=k-1, and prove that in this case it is NP-hard to rainbow color the hypergraph with q := ceil[(k-2)/2] colors. In particular, for k <=6, it is NP-hard to 2-color (k-1)-rainbow colorable k-uniform hypergraphs.
Our proof follows the algebraic approach to promise constraint satisfaction problems. It proceeds by characterizing the polymorphisms associated with the approximate rainbow coloring problem, which are rainbow colorings of some product hypergraphs on vertex set [r]^n. We prove that any such polymorphism f: [r]^n -> [q] must be C-fixing, i.e., there is a small subset S of C coordinates and a setting a in [q]^S such that fixing x_{|S} = a determines the value of f(x). The key step in our proof is bounding the sensitivity of certain rainbow colorings, thereby arguing that they must be juntas. Armed with the C-fixing characterization, our NP-hardness is obtained via a reduction from smooth Label Cover
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
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