1,135 research outputs found
Approximate Hypergraph Coloring under Low-discrepancy and Related Promises
A hypergraph is said to be -colorable if its vertices can be colored
with colors so that no hyperedge is monochromatic. -colorability is a
fundamental property (called Property B) of hypergraphs and is extensively
studied in combinatorics. Algorithmically, however, given a -colorable
-uniform hypergraph, it is NP-hard to find a -coloring miscoloring fewer
than a fraction of hyperedges (which is achieved by a random
-coloring), and the best algorithms to color the hypergraph properly require
colors, approaching the trivial bound of as
increases.
In this work, we study the complexity of approximate hypergraph coloring, for
both the maximization (finding a -coloring with fewest miscolored edges) and
minimization (finding a proper coloring using fewest number of colors)
versions, when the input hypergraph is promised to have the following stronger
properties than -colorability:
(A) Low-discrepancy: If the hypergraph has discrepancy ,
we give an algorithm to color the it with colors.
However, for the maximization version, we prove NP-hardness of finding a
-coloring miscoloring a smaller than (resp. )
fraction of the hyperedges when (resp. ). Assuming
the UGC, we improve the latter hardness factor to for almost
discrepancy- hypergraphs.
(B) Rainbow colorability: If the hypergraph has a -coloring such
that each hyperedge is polychromatic with all these colors, we give a
-coloring algorithm that miscolors at most of the
hyperedges when , and complement this with a matching UG
hardness result showing that when , it is hard to even beat the
bound achieved by a random coloring.Comment: Approx 201
Generalised Pattern Matching Revisited
In the problem of
[STOC'94, Muthukrishnan and Palem], we are given a text of length over
an alphabet , a pattern of length over an alphabet
, and a matching relationship ,
and must return all substrings of that match (reporting) or the number
of mismatches between each substring of of length and (counting).
In this work, we improve over all previously known algorithms for this problem
for various parameters describing the input instance:
* being the maximum number of characters that match a fixed
character,
* being the number of pairs of matching characters,
* being the total number of disjoint intervals of characters
that match the characters of the pattern .
At the heart of our new deterministic upper bounds for and
lies a faster construction of superimposed codes, which solves
an open problem posed in [FOCS'97, Indyk] and can be of independent interest.
To conclude, we demonstrate first lower bounds for . We start by
showing that any deterministic or Monte Carlo algorithm for must
use time, and then proceed to show higher lower bounds
for combinatorial algorithms. These bounds show that our algorithms are almost
optimal, unless a radically new approach is developed
On a generalization of iterated and randomized rounding
We give a general method for rounding linear programs that combines the
commonly used iterated rounding and randomized rounding techniques. In
particular, we show that whenever iterated rounding can be applied to a problem
with some slack, there is a randomized procedure that returns an integral
solution that satisfies the guarantees of iterated rounding and also has
concentration properties. We use this to give new results for several classic
problems where iterated rounding has been useful
An Algorithm for Koml\'os Conjecture Matching Banaszczyk's bound
We consider the problem of finding a low discrepancy coloring for sparse set
systems where each element lies in at most t sets. We give an efficient
algorithm that finds a coloring with discrepancy O((t log n)^{1/2}), matching
the best known non-constructive bound for the problem due to Banaszczyk. The
previous algorithms only achieved an O(t^{1/2} log n) bound. The result also
extends to the more general Koml\'{o}s setting and gives an algorithmic
O(log^{1/2} n) bound
Algorithms to Approximate Column-Sparse Packing Problems
Column-sparse packing problems arise in several contexts in both
deterministic and stochastic discrete optimization. We present two unifying
ideas, (non-uniform) attenuation and multiple-chance algorithms, to obtain
improved approximation algorithms for some well-known families of such
problems. As three main examples, we attain the integrality gap, up to
lower-order terms, for known LP relaxations for k-column sparse packing integer
programs (Bansal et al., Theory of Computing, 2012) and stochastic k-set
packing (Bansal et al., Algorithmica, 2012), and go "half the remaining
distance" to optimal for a major integrality-gap conjecture of Furedi, Kahn and
Seymour on hypergraph matching (Combinatorica, 1993).Comment: Extended abstract appeared in SODA 2018. Full version in ACM
Transactions of Algorithm
A Spectral Bound on Hypergraph Discrepancy
Let be a -regular hypergraph on vertices and edges.
Let be the incidence matrix of and let us denote
. We show that the
discrepancy of is . As a corollary, this
gives us that for every , the discrepancy of a random -regular hypergraph
with vertices and edges is almost surely as
grows. The proof also gives a polynomial time algorithm that takes a hypergraph
as input and outputs a coloring with the above guarantee.Comment: 18 pages. arXiv admin note: substantial text overlap with
arXiv:1811.01491, several changes to the presentatio
Discrepancy Without Partial Colorings
Spencer\u27s theorem asserts that, for any family of n subsets of ground set of size n, the elements of the ground set can be "colored" by the values +1 or -1 such that the sum of every set is O(sqrt(n)) in absolute value. All existing proofs of this result recursively construct "partial colorings", which assign +1 or -1 values to half of the ground set. We devise the first algorithm for Spencer\u27s theorem that directly computes a coloring, without recursively computing partial colorings
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