11,463 research outputs found
On largest volume simplices and sub-determinants
We show that the problem of finding the simplex of largest volume in the
convex hull of points in can be approximated with a factor
of in polynomial time. This improves upon the previously best
known approximation guarantee of by Khachiyan. On the other hand,
we show that there exists a constant such that this problem cannot be
approximated with a factor of , unless . % This improves over the
inapproximability that was previously known. Our hardness result holds
even if , in which case there exists a \bar c\,^{d}-approximation
algorithm that relies on recent sampling techniques, where is again a
constant. We show that similar results hold for the problem of finding the
largest absolute value of a subdeterminant of a matrix
Pre-Reduction Graph Products: Hardnesses of Properly Learning DFAs and Approximating EDP on DAGs
The study of graph products is a major research topic and typically concerns
the term , e.g., to show that . In this paper, we
study graph products in a non-standard form where is a
"reduction", a transformation of any graph into an instance of an intended
optimization problem. We resolve some open problems as applications.
(1) A tight -approximation hardness for the minimum
consistent deterministic finite automaton (DFA) problem, where is the
sample size. Due to Board and Pitt [Theoretical Computer Science 1992], this
implies the hardness of properly learning DFAs assuming (the
weakest possible assumption).
(2) A tight hardness for the edge-disjoint paths (EDP)
problem on directed acyclic graphs (DAGs), where denotes the number of
vertices.
(3) A tight hardness of packing vertex-disjoint -cycles for large .
(4) An alternative (and perhaps simpler) proof for the hardness of properly
learning DNF, CNF and intersection of halfspaces [Alekhnovich et al., FOCS 2004
and J. Comput.Syst.Sci. 2008]
Improved approximation for 3-dimensional matching via bounded pathwidth local search
One of the most natural optimization problems is the k-Set Packing problem,
where given a family of sets of size at most k one should select a maximum size
subfamily of pairwise disjoint sets. A special case of 3-Set Packing is the
well known 3-Dimensional Matching problem. Both problems belong to the Karp`s
list of 21 NP-complete problems. The best known polynomial time approximation
ratio for k-Set Packing is (k + eps)/2 and goes back to the work of Hurkens and
Schrijver [SIDMA`89], which gives (1.5 + eps)-approximation for 3-Dimensional
Matching. Those results are obtained by a simple local search algorithm, that
uses constant size swaps.
The main result of the paper is a new approach to local search for k-Set
Packing where only a special type of swaps is considered, which we call swaps
of bounded pathwidth. We show that for a fixed value of k one can search the
space of r-size swaps of constant pathwidth in c^r poly(|F|) time. Moreover we
present an analysis proving that a local search maximum with respect to O(log
|F|)-size swaps of constant pathwidth yields a polynomial time (k + 1 +
eps)/3-approximation algorithm, improving the best known approximation ratio
for k-Set Packing. In particular we improve the approximation ratio for
3-Dimensional Matching from 3/2 + eps to 4/3 + eps.Comment: To appear in proceedings of FOCS 201
On the Combinatorial Complexity of Approximating Polytopes
Approximating convex bodies succinctly by convex polytopes is a fundamental
problem in discrete geometry. A convex body of diameter
is given in Euclidean -dimensional space, where is a constant. Given an
error parameter , the objective is to determine a polytope of
minimum combinatorial complexity whose Hausdorff distance from is at most
. By combinatorial complexity we mean the
total number of faces of all dimensions of the polytope. A well-known result by
Dudley implies that facets suffice, and a dual
result by Bronshteyn and Ivanov similarly bounds the number of vertices, but
neither result bounds the total combinatorial complexity. We show that there
exists an approximating polytope whose total combinatorial complexity is
, where conceals a
polylogarithmic factor in . This is a significant improvement
upon the best known bound, which is roughly .
Our result is based on a novel combination of both old and new ideas. First,
we employ Macbeath regions, a classical structure from the theory of convexity.
The construction of our approximating polytope employs a new stratified
placement of these regions. Second, in order to analyze the combinatorial
complexity of the approximating polytope, we present a tight analysis of a
width-based variant of B\'{a}r\'{a}ny and Larman's economical cap covering.
Finally, we use a deterministic adaptation of the witness-collector technique
(developed recently by Devillers et al.) in the context of our stratified
construction.Comment: In Proceedings of the 32nd International Symposium Computational
Geometry (SoCG 2016) and accepted to SoCG 2016 special issue of Discrete and
Computational Geometr
Learning to Approximate a Bregman Divergence
Bregman divergences generalize measures such as the squared Euclidean
distance and the KL divergence, and arise throughout many areas of machine
learning. In this paper, we focus on the problem of approximating an arbitrary
Bregman divergence from supervision, and we provide a well-principled approach
to analyzing such approximations. We develop a formulation and algorithm for
learning arbitrary Bregman divergences based on approximating their underlying
convex generating function via a piecewise linear function. We provide
theoretical approximation bounds using our parameterization and show that the
generalization error for metric learning using our framework
matches the known generalization error in the strictly less general Mahalanobis
metric learning setting. We further demonstrate empirically that our method
performs well in comparison to existing metric learning methods, particularly
for clustering and ranking problems.Comment: 19 pages, 4 figure
Fast and Deterministic Approximations for k-Cut
In an undirected graph, a k-cut is a set of edges whose removal breaks the graph into at least k connected components. The minimum weight k-cut can be computed in n^O(k) time, but when k is treated as part of the input, computing the minimum weight k-cut is NP-Hard [Goldschmidt and Hochbaum, 1994]. For poly(m,n,k)-time algorithms, the best possible approximation factor is essentially 2 under the small set expansion hypothesis [Manurangsi, 2017]. Saran and Vazirani [1995] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed via O(k) minimum cuts, which implies a O~(km) randomized running time via the nearly linear time randomized min-cut algorithm of Karger [2000]. Nagamochi and Kamidoi [2007] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed deterministically in O(mn + n^2 log n) time. These results prompt two basic questions. The first concerns the role of randomization. Is there a deterministic algorithm for 2-approximate k-cuts matching the randomized running time of O~(km)? The second question qualitatively compares minimum cut to 2-approximate minimum k-cut. Can 2-approximate k-cuts be computed as fast as the minimum cut - in O~(m) randomized time?
We give a deterministic approximation algorithm that computes (2 + eps)-minimum k-cuts in O(m log^3 n / eps^2) time, via a (1 + eps)-approximation for an LP relaxation of k-cut
- …