117 research outputs found
Sticky Brownian Rounding and its Applications to Constraint Satisfaction Problems
Semidefinite programming is a powerful tool in the design and analysis of
approximation algorithms for combinatorial optimization problems. In
particular, the random hyperplane rounding method of Goemans and Williamson has
been extensively studied for more than two decades, resulting in various
extensions to the original technique and beautiful algorithms for a wide range
of applications. Despite the fact that this approach yields tight approximation
guarantees for some problems, e.g., Max-Cut, for many others, e.g., Max-SAT and
Max-DiCut, the tight approximation ratio is still unknown. One of the main
reasons for this is the fact that very few techniques for rounding semidefinite
relaxations are known.
In this work, we present a new general and simple method for rounding
semi-definite programs, based on Brownian motion. Our approach is inspired by
recent results in algorithmic discrepancy theory. We develop and present tools
for analyzing our new rounding algorithms, utilizing mathematical machinery
from the theory of Brownian motion, complex analysis, and partial differential
equations. Focusing on constraint satisfaction problems, we apply our method to
several classical problems, including Max-Cut, Max-2SAT, and MaxDiCut, and
derive new algorithms that are competitive with the best known results. To
illustrate the versatility and general applicability of our approach, we give
new approximation algorithms for the Max-Cut problem with side constraints that
crucially utilizes measure concentration results for the Sticky Brownian
Motion, a feature missing from hyperplane rounding and its generalization
Adding cardinality constraints to integer programs with applications to maximum satisfiability
Max-SAT-CC is the following optimization problem: Given a formula in CNF and a bound k, find an assignment with at most k variables being set to true that maximizes the number of satisfied clauses among all such assignments. If each clause is restricted to have at most ā literals, we obtain the problem Max-āSAT-CC. Sviridenko [Algorithmica 30 (3) (2001) 398ā405] designed a (1āeā1)-approximation algorithm for Max-SAT-CC. This result is tight unless P=NP [U. Feige, J. ACM 45 (4) (1998) 634ā652]. Sviridenko asked if it is possible to achieve a better approximation ratio in the case of Max-āSAT-CC. We answer this question in the affirmative by presenting a randomized approximation algorithm whose approximation ratio is 1-(1-1/ā)ā-Īµ. To do this, we develop a general technique for adding a cardinality constraint to certain integer programs. Our algorithm can be derandomized using pairwise independent random variables with small probability space
Sticky Brownian rounding and its applications to constraint satisfaction problems
Semidefinite programming is a powerful tool in the design and analysis of approximation algorithms for combinatorial optimization problems. In particular, the random hyperplane rounding method of Goemans and Williamson [23] has been extensively studied for more than two decades, resulting in various extensions to the original technique and beautiful algorithms for a wide range of applications. Despite the fact that this approach yields tight approximation guarantees for some problems, e.g., Max-Cut, for many others, e.g., Max-SAT and Max-DiCut, the tight approximation ratio is still unknown. One of the main reasons for this is the fact that very few techniques for rounding semidefinite relaxations are known. In this work, we present a new general and simple method for rounding semi-definite programs, based on Brownian motion. Our approach is inspired by recent results in algorithmic discrepancy theory. We develop and present tools for analyzing our new rounding algorithms, utilizing mathematical machinery from the theory of Brownian motion, complex analysis, and partial differential equations. Focusing on constraint satisfaction problems, we apply our method to several classical problems, including Max-Cut, Max-2SAT, and Max-DiCut, and derive new algorithms that are competitive with the best known results. To illustrate the versatility and general applicability of our approach, we give new approximation algorithms for the Max-Cut problem with side constraints that crucially utilizes measure concentration results for the Sticky Brownian Motion, a feature missing from hyperplane rounding and its generalizations
Global Cardinality Constraints Make Approximating Some Max-2-CSPs Harder
Assuming the Unique Games Conjecture, we show that existing approximation algorithms for some Boolean Max-2-CSPs with cardinality constraints are optimal. In particular, we prove that Max-Cut with cardinality constraints is UG-hard to approximate within ~~0.858, and that Max-2-Sat with cardinality constraints is UG-hard to approximate within ~~0.929. In both cases, the previous best hardness results were the same as the hardness of the corresponding unconstrained Max-2-CSP (~~0.878 for Max-Cut, and ~~0.940 for Max-2-Sat).
The hardness for Max-2-Sat applies to monotone Max-2-Sat instances, meaning that we also obtain tight inapproximability for the Max-k-Vertex-Cover problem
Stochastic Budget Optimization in Internet Advertising
Internet advertising is a sophisticated game in which the many advertisers
"play" to optimize their return on investment. There are many "targets" for the
advertisements, and each "target" has a collection of games with a potentially
different set of players involved. In this paper, we study the problem of how
advertisers allocate their budget across these "targets". In particular, we
focus on formulating their best response strategy as an optimization problem.
Advertisers have a set of keywords ("targets") and some stochastic information
about the future, namely a probability distribution over scenarios of cost vs
click combinations. This summarizes the potential states of the world assuming
that the strategies of other players are fixed. Then, the best response can be
abstracted as stochastic budget optimization problems to figure out how to
spread a given budget across these keywords to maximize the expected number of
clicks.
We present the first known non-trivial poly-logarithmic approximation for
these problems as well as the first known hardness results of getting better
than logarithmic approximation ratios in the various parameters involved. We
also identify several special cases of these problems of practical interest,
such as with fixed number of scenarios or with polynomial-sized parameters
related to cost, which are solvable either in polynomial time or with improved
approximation ratios. Stochastic budget optimization with scenarios has
sophisticated technical structure. Our approximation and hardness results come
from relating these problems to a special type of (0/1, bipartite) quadratic
programs inherent in them. Our research answers some open problems raised by
the authors in (Stochastic Models for Budget Optimization in Search-Based
Advertising, Algorithmica, 58 (4), 1022-1044, 2010).Comment: FINAL versio
Metro-Line Crossing Minimization: Hardness, Approximations, and Tractable Cases
Crossing minimization is one of the central problems in graph drawing.
Recently, there has been an increased interest in the problem of minimizing
crossings between paths in drawings of graphs. This is the metro-line crossing
minimization problem (MLCM): Given an embedded graph and a set L of simple
paths, called lines, order the lines on each edge so that the total number of
crossings is minimized. So far, the complexity of MLCM has been an open
problem. In contrast, the problem variant in which line ends must be placed in
outermost position on their edges (MLCM-P) is known to be NP-hard. Our main
results answer two open questions: (i) We show that MLCM is NP-hard. (ii) We
give an -approximation algorithm for MLCM-P
Optimal Constant-Time Approximation Algorithms and (Unconditional) Inapproximability Results for Every Bounded-Degree CSP
Raghavendra (STOC 2008) gave an elegant and surprising result: if Khot's
Unique Games Conjecture (STOC 2002) is true, then for every constraint
satisfaction problem (CSP), the best approximation ratio is attained by a
certain simple semidefinite programming and a rounding scheme for it. In this
paper, we show that similar results hold for constant-time approximation
algorithms in the bounded-degree model. Specifically, we present the
followings: (i) For every CSP, we construct an oracle that serves an access, in
constant time, to a nearly optimal solution to a basic LP relaxation of the
CSP. (ii) Using the oracle, we give a constant-time rounding scheme that
achieves an approximation ratio coincident with the integrality gap of the
basic LP. (iii) Finally, we give a generic conversion from integrality gaps of
basic LPs to hardness results. All of those results are \textit{unconditional}.
Therefore, for every bounded-degree CSP, we give the best constant-time
approximation algorithm among all. A CSP instance is called -far from
satisfiability if we must remove at least an -fraction of constraints
to make it satisfiable. A CSP is called testable if there is a constant-time
algorithm that distinguishes satisfiable instances from -far
instances with probability at least . Using the results above, we also
derive, under a technical assumption, an equivalent condition under which a CSP
is testable in the bounded-degree model
- ā¦