133 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
Improved Hardness of Approximation for Geometric Bin Packing
The Geometric Bin Packing (GBP) problem is a generalization of Bin Packing
where the input is a set of -dimensional rectangles, and the goal is to pack
them into unit -dimensional cubes efficiently. It is NP-Hard to obtain a
PTAS for the problem, even when . For general , the best known
approximation algorithm has an approximation guarantee exponential in ,
while the best hardness of approximation is still a small constant
inapproximability from the case when . In this paper, we show that the
problem cannot be approximated within factor unless NP=ZPP.
Recently, -dimensional Vector Bin Packing, a closely related problem to
the GBP, was shown to be hard to approximate within when
is a fixed constant, using a notion of Packing Dimension of set families. In
this paper, we introduce a geometric analog of it, the Geometric Packing
Dimension of set families. While we fall short of obtaining similar
inapproximability results for the Geometric Bin Packing problem when is
fixed, we prove a couple of key properties of the Geometric Packing Dimension
that highlight the difference between Geometric Packing Dimension and Packing
Dimension.Comment: 10 page
d-To-1 Hardness of Coloring 3-Colorable Graphs with O(1) Colors
The d-to-1 conjecture of Khot asserts that it is NP-hard to satisfy an ? fraction of constraints of a satisfiable d-to-1 Label Cover instance, for arbitrarily small ? > 0. We prove that the d-to-1 conjecture for any fixed d implies the hardness of coloring a 3-colorable graph with C colors for arbitrarily large integers C.
Earlier, the hardness of O(1)-coloring a 4-colorable graphs is known under the 2-to-1 conjecture, which is the strongest in the family of d-to-1 conjectures, and the hardness for 3-colorable graphs is known under a certain "fish-shaped" variant of the 2-to-1 conjecture
Adversarial Generation of Natural Language
Generative Adversarial Networks (GANs) have gathered a lot of attention from
the computer vision community, yielding impressive results for image
generation. Advances in the adversarial generation of natural language from
noise however are not commensurate with the progress made in generating images,
and still lag far behind likelihood based methods. In this paper, we take a
step towards generating natural language with a GAN objective alone. We
introduce a simple baseline that addresses the discrete output space problem
without relying on gradient estimators and show that it is able to achieve
state-of-the-art results on a Chinese poem generation dataset. We present
quantitative results on generating sentences from context-free and
probabilistic context-free grammars, and qualitative language modeling results.
A conditional version is also described that can generate sequences conditioned
on sentence characteristics.Comment: 11 pages, 3 figures, 5 table
Baby PIH: Parameterized Inapproximability of Min CSP
The Parameterized Inapproximability Hypothesis (PIH) is the analog of the PCP
theorem in the world of parameterized complexity. It asserts that no FPT
algorithm can distinguish a satisfiable 2CSP instance from one which is only
-satisfiable (where the parameter is the number of variables)
for some constant .
We consider a minimization version of CSPs (Min-CSP), where one may assign
values to each variable, and the goal is to ensure that every constraint is
satisfied by some choice among the pairs of values assigned to its
variables (call such a CSP instance -list-satisfiable). We prove the
following strong parameterized inapproximability for Min CSP: For every , it is W[1]-hard to tell if a 2CSP instance is satisfiable or is not even
-list-satisfiable. We refer to this statement as "Baby PIH", following the
recently proved Baby PCP Theorem (Barto and Kozik, 2021). Our proof adapts the
combinatorial arguments underlying the Baby PCP theorem, overcoming some basic
obstacles that arise in the parameterized setting. Furthermore, our reduction
runs in time polynomially bounded in both the number of variables and the
alphabet size, and thus implies the Baby PCP theorem as well
Permutation Strikes Back: The Power of Recourse in Online Metric Matching
In the classical Online Metric Matching problem, we are given a metric space
with servers. A collection of clients arrive in an online fashion, and upon
arrival, a client should irrevocably be matched to an as-yet-unmatched server.
The goal is to find an online matching which minimizes the total cost, i.e.,
the sum of distances between each client and the server it is matched to. We
know deterministic algorithms~\cite{KP93,khuller1994line} that achieve a
competitive ratio of , and this bound is tight for deterministic
algorithms. The problem has also long been considered in specialized metrics
such as the line metric or metrics of bounded doubling dimension, with the
current best result on a line metric being a deterministic
competitive algorithm~\cite{raghvendra2018optimal}. Obtaining (or refuting)
-competitive algorithms in general metrics and constant-competitive
algorithms on the line metric have been long-standing open questions in this
area.
In this paper, we investigate the robustness of these lower bounds by
considering the Online Metric Matching with Recourse problem where we are
allowed to change a small number of previous assignments upon arrival of a new
client. Indeed, we show that a small logarithmic amount of recourse can
significantly improve the quality of matchings we can maintain. For general
metrics, we show a simple \emph{deterministic} -competitive
algorithm with -amortized recourse, an exponential improvement over
the lower bound when no recourse is allowed. We next consider the line
metric, and present a deterministic algorithm which is -competitive and has
-recourse, again a substantial improvement over the best known
-competitive algorithm when no recourse is allowed
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