198 research outputs found
On local search and LP and SDP relaxations for k-Set Packing
Set packing is a fundamental problem that generalises some well-known
combinatorial optimization problems and knows a lot of applications. It is
equivalent to hypergraph matching and it is strongly related to the maximum
independent set problem. In this thesis we study the k-set packing problem
where given a universe U and a collection C of subsets over U, each of
cardinality k, one needs to find the maximum collection of mutually disjoint
subsets. Local search techniques have proved to be successful in the search for
approximation algorithms, both for the unweighted and the weighted version of
the problem where every subset in C is associated with a weight and the
objective is to maximise the sum of the weights. We make a survey of these
approaches and give some background and intuition behind them. In particular,
we simplify the algebraic proof of the main lemma for the currently best
weighted approximation algorithm of Berman ([Ber00]) into a proof that reveals
more intuition on what is really happening behind the math. The main result is
a new bound of k/3 + 1 + epsilon on the integrality gap for a polynomially
sized LP relaxation for k-set packing by Chan and Lau ([CL10]) and the natural
SDP relaxation [NOTE: see page iii]. We provide detailed proofs of lemmas
needed to prove this new bound and treat some background on related topics like
semidefinite programming and the Lovasz Theta function. Finally we have an
extended discussion in which we suggest some possibilities for future research.
We discuss how the current results from the weighted approximation algorithms
and the LP and SDP relaxations might be improved, the strong relation between
set packing and the independent set problem and the difference between the
weighted and the unweighted version of the problem.Comment: There is a mistake in the following line of Theorem 17: "As an
induced subgraph of H with more edges than vertices constitutes an improving
set". Therefore, the proofs of Theorem 17, and hence Theorems 19, 23 and 24,
are false. It is still open whether these theorems are tru
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
Higher-order Projected Power Iterations for Scalable Multi-Matching
The matching of multiple objects (e.g. shapes or images) is a fundamental
problem in vision and graphics. In order to robustly handle ambiguities, noise
and repetitive patterns in challenging real-world settings, it is essential to
take geometric consistency between points into account. Computationally, the
multi-matching problem is difficult. It can be phrased as simultaneously
solving multiple (NP-hard) quadratic assignment problems (QAPs) that are
coupled via cycle-consistency constraints. The main limitations of existing
multi-matching methods are that they either ignore geometric consistency and
thus have limited robustness, or they are restricted to small-scale problems
due to their (relatively) high computational cost. We address these
shortcomings by introducing a Higher-order Projected Power Iteration method,
which is (i) efficient and scales to tens of thousands of points, (ii)
straightforward to implement, (iii) able to incorporate geometric consistency,
(iv) guarantees cycle-consistent multi-matchings, and (iv) comes with
theoretical convergence guarantees. Experimentally we show that our approach is
superior to existing methods
Extended Formulation Lower Bounds via Hypergraph Coloring?
Exploring the power of linear programming for combinatorial optimization
problems has been recently receiving renewed attention after a series of
breakthrough impossibility results. From an algorithmic perspective, the
related questions concern whether there are compact formulations even for
problems that are known to admit polynomial-time algorithms.
We propose a framework for proving lower bounds on the size of extended
formulations. We do so by introducing a specific type of extended relaxations
that we call product relaxations and is motivated by the study of the
Sherali-Adams (SA) hierarchy. Then we show that for every approximate
relaxation of a polytope P, there is a product relaxation that has the same
size and is at least as strong. We provide a methodology for proving lower
bounds on the size of approximate product relaxations by lower bounding the
chromatic number of an underlying hypergraph, whose vertices correspond to
gap-inducing vectors.
We extend the definition of product relaxations and our methodology to mixed
integer sets. However in this case we are able to show that mixed product
relaxations are at least as powerful as a special family of extended
formulations. As an application of our method we show an exponential lower
bound on the size of approximate mixed product formulations for the metric
capacitated facility location problem, a problem which seems to be intractable
for linear programming as far as constant-gap compact formulations are
concerned
Matchings, hypergraphs, association schemes, and semidefinite optimization
We utilize association schemes to analyze the quality of semidefinite
programming (SDP) based convex relaxations of integral packing and covering
polyhedra determined by matchings in hypergraphs. As a by-product of our
approach, we obtain bounds on the clique and stability numbers of some regular
graphs reminiscent of classical bounds by Delsarte and Hoffman. We determine
exactly or provide bounds on the performance of Lov\'{a}sz-Schrijver SDP
hierarchy, and illustrate the usefulness of obtaining commutative subschemes
from non-commutative schemes via contraction in this context
Lower Bounds on Query Complexity for Testing Bounded-Degree CSPs
In this paper, we consider lower bounds on the query complexity for testing
CSPs in the bounded-degree model.
First, for any ``symmetric'' predicate except \equ
where , we show that every (randomized) algorithm that distinguishes
satisfiable instances of CSP(P) from instances -far
from satisfiability requires queries where is the
number of variables and is a constant that depends on and
. This breaks a natural lower bound , which is
obtained by the birthday paradox. We also show that every one-sided error
tester requires queries for such . These results are hereditary
in the sense that the same results hold for any predicate such that
. For EQU, we give a one-sided error tester
whose query complexity is . Also, for 2-XOR (or,
equivalently E2LIN2), we show an lower bound for
distinguishing instances between -close to and -far
from satisfiability.
Next, for the general k-CSP over the binary domain, we show that every
algorithm that distinguishes satisfiable instances from instances
-far from satisfiability requires queries. The
matching NP-hardness is not known, even assuming the Unique Games Conjecture or
the -to- Conjecture. As a corollary, for Maximum Independent Set on
graphs with vertices and a degree bound , we show that every
approximation algorithm within a factor d/\poly\log d and an additive error
of requires queries. Previously, only super-constant
lower bounds were known
Efficient Flow-based Approximation Algorithms for Submodular Hypergraph Partitioning via a Generalized Cut-Matching Game
In the past 20 years, increasing complexity in real world data has lead to
the study of higher-order data models based on partitioning hypergraphs.
However, hypergraph partitioning admits multiple formulations as hyperedges can
be cut in multiple ways. Building upon a class of hypergraph partitioning
problems introduced by Li & Milenkovic, we study the problem of minimizing
ratio-cut objectives over hypergraphs given by a new class of cut functions,
monotone submodular cut functions (mscf's), which captures hypergraph expansion
and conductance as special cases.
We first define the ratio-cut improvement problem, a family of local
relaxations of the minimum ratio-cut problem. This problem is a natural
extension of the Andersen & Lang cut improvement problem to the hypergraph
setting. We demonstrate the existence of efficient algorithms for approximately
solving this problem. These algorithms run in almost-linear time for the case
of hypergraph expansion, and when the hypergraph rank is at most .
Next, we provide an efficient -approximation algorithm for finding
the minimum ratio-cut of . We generalize the cut-matching game framework of
Khandekar et. al. to allow for the cut player to play unbalanced cuts, and
matching player to route approximate single-commodity flows. Using this
framework, we bootstrap our algorithms for the ratio-cut improvement problem to
obtain approximation algorithms for minimum ratio-cut problem for all mscf's.
This also yields the first almost-linear time -approximation
algorithms for hypergraph expansion, and constant hypergraph rank.
Finally, we extend a result of Louis & Makarychev to a broader set of
objective functions by giving a polynomial time -approximation algorithm for the minimum ratio-cut problem based on
rounding -metric embeddings.Comment: Comments and feedback welcom
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