29 research outputs found
A Note on Two-Colorability of Nonuniform Hypergraphs
For a hypergraph , let denote the expected number of monochromatic
edges when the color of each vertex in is sampled uniformly at random from
the set of size 2. Let denote the minimum size of an edge in .
Erd\H{o}s asked in 1963 whether there exists an unbounded function such
that any hypergraph with and is two
colorable. Beck in 1978 answered this question in the affirmative for a
function . We improve this result by showing that, for
an absolute constant , a version of random greedy coloring procedure
is likely to find a proper two coloring for any hypergraph with
and
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
Applications of the Lovász Local Lemma and related methods
V této práci se zabýváme aplikacemi Lovászova lokálního lemmatu a s ním souvisejících metod. Popíšeme postupný vývoj těchto metod a ukážeme konkrétní příklady jejich užití na příkladech z oblasti výzkumu nezávislých transverzál a hypergrafů.ObhájenoIn this thesis we investigate applications of the Lovász local lemma and its related methods. We are going to describe the gradual development of these methods and show the specific examples of its use in the field of research on independent transversals and hypergraphs
Improved Approximation Algorithms for Projection Games
The projection games (aka Label-Cover) problem is of great importance to the field of approximation algorithms, since most of the NP-hardness of approximation results we know today are reductions from Label-Cover. In this paper we design several approximation algorithms for projection games:
1. A polynomial-time approximation algorithm that improves on the previous best approximation by Charikar, Hajiaghayi and Karloff [7].
2. A sub-exponential time algorithm with much tighter approximation for the case of smooth projection games.
3. A PTAS for planar graphs.National Science Foundation (U.S.) (Grant 1218547
Homogeneity and Homogenizability: Hard Problems for the Logic SNP
We show that the question whether a given SNP sentence defines a
homogenizable class of finite structures is undecidable, even if the sentence
comes from the connected Datalog fragment and uses at most binary relation
symbols. As a byproduct of our proof, we also get the undecidability of some
other properties for Datalog programs, e.g., whether they can be rewritten in
MMSNP, whether they solve some finite-domain CSP, or whether they define the
age of a reduct of a homogeneous Ramsey structure in a finite relational
signature. We subsequently show that the closely related problem of testing the
amalgamation property for finitely bounded classes is EXPSPACE-hard or
PSPACE-hard, depending on whether the input is specified by a universal
sentence or a set of forbidden substructures.Comment: 34 pages, 3 figure
EXPLORING DIFFERENT MODELS OF QUERY COMPLEXITY AND COMMUNICATION COMPLEXITY
Ph.DDOCTOR OF PHILOSOPH