3,452 research outputs found
Improved Hardness of Approximating Chromatic Number
We prove that for sufficiently large K, it is NP-hard to color K-colorable
graphs with less than 2^{K^{1/3}} colors. This improves the previous result of
K versus K^{O(log K)} in Khot [14]
From Gap-ETH to FPT-Inapproximability: Clique, Dominating Set, and More
We consider questions that arise from the intersection between the areas of
polynomial-time approximation algorithms, subexponential-time algorithms, and
fixed-parameter tractable algorithms. The questions, which have been asked
several times (e.g., [Marx08, FGMS12, DF13]), are whether there is a
non-trivial FPT-approximation algorithm for the Maximum Clique (Clique) and
Minimum Dominating Set (DomSet) problems parameterized by the size of the
optimal solution. In particular, letting be the optimum and be
the size of the input, is there an algorithm that runs in
time and outputs a solution of size
, for any functions and that are independent of (for
Clique, we want )?
In this paper, we show that both Clique and DomSet admit no non-trivial
FPT-approximation algorithm, i.e., there is no
-FPT-approximation algorithm for Clique and no
-FPT-approximation algorithm for DomSet, for any function
(e.g., this holds even if is the Ackermann function). In fact, our results
imply something even stronger: The best way to solve Clique and DomSet, even
approximately, is to essentially enumerate all possibilities. Our results hold
under the Gap Exponential Time Hypothesis (Gap-ETH) [Dinur16, MR16], which
states that no -time algorithm can distinguish between a satisfiable
3SAT formula and one which is not even -satisfiable for some
constant .
Besides Clique and DomSet, we also rule out non-trivial FPT-approximation for
Maximum Balanced Biclique, Maximum Subgraphs with Hereditary Properties, and
Maximum Induced Matching in bipartite graphs. Additionally, we rule out
-FPT-approximation algorithm for Densest -Subgraph although this
ratio does not yet match the trivial -approximation algorithm.Comment: 43 pages. To appear in FOCS'1
-Approximation Algorithm for Directed Steiner Tree: A Tight Quasi-Polynomial-Time Algorithm
In the Directed Steiner Tree (DST) problem we are given an -vertex
directed edge-weighted graph, a root , and a collection of terminal
nodes. Our goal is to find a minimum-cost arborescence that contains a directed
path from to every terminal. We present an -approximation algorithm for DST that runs in
quasi-polynomial-time. By adjusting the parameters in the hardness result of
Halperin and Krauthgamer, we show the matching lower bound of
for the class of quasi-polynomial-time
algorithms. This is the first improvement on the DST problem since the
classical quasi-polynomial-time approximation algorithm by
Charikar et al. (The paper erroneously claims an approximation due
to a mistake in prior work.)
Our approach is based on two main ingredients. First, we derive an
approximation preserving reduction to the Label-Consistent Subtree (LCST)
problem. The LCST instance has quasi-polynomial size and logarithmic height. We
remark that, in contrast, Zelikovsky's heigh-reduction theorem used in all
prior work on DST achieves a reduction to a tree instance of the related Group
Steiner Tree (GST) problem of similar height, however losing a logarithmic
factor in the approximation ratio. Our second ingredient is an LP-rounding
algorithm to approximately solve LCST instances, which is inspired by the
framework developed by Rothvo{\ss}. We consider a Sherali-Adams lifting of a
proper LP relaxation of LCST. Our rounding algorithm proceeds level by level
from the root to the leaves, rounding and conditioning each time on a proper
subset of label variables. A small enough (namely, polylogarithmic) number of
Sherali-Adams lifting levels is sufficient to condition up to the leaves
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