211,556 research outputs found
Optimally Repurposing Existing Algorithms to Obtain Exponential-Time Approximations
The goal of this paper is to understand how exponential-time approximation
algorithms can be obtained from existing polynomial-time approximation
algorithms, existing parameterized exact algorithms, and existing parameterized
approximation algorithms. More formally, we consider a monotone subset
minimization problem over a universe of size (e.g., Vertex Cover or
Feedback Vertex Set). We have access to an algorithm that finds an
-approximate solution in time if a solution of
size exists (and more generally, an extension algorithm that can
approximate in a similar way if a set can be extended to a solution with
further elements). Our goal is to obtain a time
-approximation algorithm for the problem with as small as possible.
That is, for every fixed , we would like to determine
the smallest possible that can be achieved in a model where our
problem-specific knowledge is limited to checking the feasibility of a solution
and invoking the -approximate extension algorithm. Our results
completely resolve this question:
(1) For every fixed , a simple algorithm
(``approximate monotone local search'') achieves the optimum value of .
(2) Given , we can efficiently compute the optimum
up to any precision .
Earlier work presented algorithms (but no lower bounds) for the special case
[Fomin et al., J. ACM 2019] and for the special case
[Esmer et al., ESA 2022]. Our work generalizes these
results and in particular confirms that the earlier algorithms are optimal in
these special cases.Comment: 80 pages, 5 figure
Vertex Sparsifiers: New Results from Old Techniques
Given a capacitated graph and a set of terminals ,
how should we produce a graph only on the terminals so that every
(multicommodity) flow between the terminals in could be supported in
with low congestion, and vice versa? (Such a graph is called a
flow-sparsifier for .) What if we want to be a "simple" graph? What if
we allow to be a convex combination of simple graphs?
Improving on results of Moitra [FOCS 2009] and Leighton and Moitra [STOC
2010], we give efficient algorithms for constructing: (a) a flow-sparsifier
that maintains congestion up to a factor of , where , (b) a convex combination of trees over the terminals that maintains
congestion up to a factor of , and (c) for a planar graph , a
convex combination of planar graphs that maintains congestion up to a constant
factor. This requires us to give a new algorithm for the 0-extension problem,
the first one in which the preimages of each terminal are connected in .
Moreover, this result extends to minor-closed families of graphs.
Our improved bounds immediately imply improved approximation guarantees for
several terminal-based cut and ordering problems.Comment: An extended abstract appears in the 13th International Workshop on
Approximation Algorithms for Combinatorial Optimization Problems (APPROX),
2010. Final version to appear in SIAM J. Computin
Subexponential LPs Approximate Max-Cut
We show that for every , the degree-
Sherali-Adams linear program (with variables
and constraints) approximates the maximum cut problem within a factor of
, for some . Our
result provides a surprising converse to known lower bounds against all linear
programming relaxations of Max-Cut, and hence resolves the extension complexity
of approximate Max-Cut for approximation factors close to (up to
the function ). Previously, only semidefinite
programs and spectral methods were known to yield approximation factors better
than for Max-Cut in time . We also show that
constant-degree Sherali-Adams linear programs (with variables
and constraints) can solve Max-Cut with approximation factor close to on
graphs of small threshold rank: this is the first connection of which we are
aware between threshold rank and linear programming-based algorithms.
Our results separate the power of Sherali-Adams versus Lov\'asz-Schrijver
hierarchies for approximating Max-Cut, since it is known that
approximation of Max Cut requires
rounds in the Lov\'asz-Schrijver hierarchy.
We also provide a subexponential time approximation for Khot's Unique Games
problem: we show that for every the degree- Sherali-Adams linear program distinguishes instances of Unique Games
of value from instances of value , for
some , where is the alphabet size. Such
guarantees are qualitatively similar to those of previous subexponential-time
algorithms for Unique Games but our algorithm does not rely on semidefinite
programming or subspace enumeration techniques
Optimal Approximation for Submodular and Supermodular Optimization with Bounded Curvature
We design new approximation algorithms for the problems of optimizing submodular and supermodular functions subject to a single matroid constraint. Specifically, we consider the case in which we wish to maximize a monotone increasing submodular function or minimize a monotone decreasing supermodular function with a bounded total curvature c. Intuitively, the parameter c represents how nonlinear a function f is: when c = 0, f is linear, while for c = 1, f may be an arbitrary monotone increasing submodular function. For the case of submodular maximization with total curvature c, we obtain a (1 ā c/e)-approximationāthe first improvement over the greedy algorithm of of Conforti and CornuĆ©jols from 1984, which holds for a cardinality constraint, as well as a recent analogous result for an arbitrary matroid constraint. Our approach is based on modifications of the continuous greedy algorithm and nonoblivious local search, and allows us to approximately maximize the sum of a nonnegative, monotone increasing submodular function and a (possibly negative) linear function. We show how to reduce both submodular maximization and supermodular minimization to this general problem when the objective function has bounded total curvature. We prove that the approximation results we obtain are the best possible in the value oracle model, even in the case of a cardinality constraint. We define an extension of the notion of curvature to general monotone set functions and show a (1 ā c)-approximation for maximization and a 1/(1 ā c)-approximation for minimization cases. Finally, we give two concrete applications of our results in the settings of maximum entropy sampling, and the column-subset selection problem
Submodular Maximization Meets Streaming: Matchings, Matroids, and More
We study the problem of finding a maximum matching in a graph given by an
input stream listing its edges in some arbitrary order, where the quantity to
be maximized is given by a monotone submodular function on subsets of edges.
This problem, which we call maximum submodular-function matching (MSM), is a
natural generalization of maximum weight matching (MWM), which is in turn a
generalization of maximum cardinality matching (MCM). We give two incomparable
algorithms for this problem with space usage falling in the semi-streaming
range---they store only edges, using working memory---that
achieve approximation ratios of in a single pass and in
passes respectively. The operations of these algorithms
mimic those of Zelke's and McGregor's respective algorithms for MWM; the
novelty lies in the analysis for the MSM setting. In fact we identify a general
framework for MWM algorithms that allows this kind of adaptation to the broader
setting of MSM.
In the sequel, we give generalizations of these results where the
maximization is over "independent sets" in a very general sense. This
generalization captures hypermatchings in hypergraphs as well as independence
in the intersection of multiple matroids.Comment: 18 page
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