22 research outputs found
Clustering with Few Disks to Minimize the Sum of Radii
Given a set of points in the Euclidean plane, the -MinSumRadius problem asks to cover this point set using disks with the objective of minimizing the sum of the radii of the disks. After a long line of research on related problems, it was finally discovered that this problem admits a polynomial time algorithm [GKKPV~'12]; however, the running time of this algorithm is , and its relevance is thereby mostly of theoretical nature. A practically and structurally interesting special case of the -MinSumRadius problem is that of small . For the -MinSumRadius problem, a near-quadratic time algorithm with expected running time was given over 30 years ago [Eppstein~'92]. We present the first improvement of this result, namely, a near-linear time algorithm to compute the -MinSumRadius that runs in expected time. We generalize this result to any constant dimension , for which we give an time algorithm. Additionally, we give a near-quadratic time algorithm for -MinSumRadius in the plane that runs in expected time. All of these algorithms rely on insights that uncover a surprisingly simple structure of optimal solutions: we can specify a linear number of lines out of which one separates one of the clusters from the remaining clusters in an optimal solution
Minimum Perimeter-Sum Partitions in the Plane
Let P be a set of n points in the plane. We consider the problem of partitioning P into two subsets P_1 and P_2 such that the sum of the perimeters of CH(P_1) and CH(P_2) is minimized, where CH(P_i) denotes the convex hull of P_i. The problem was first studied by Mitchell and Wynters in 1991 who gave an O(n^2) time algorithm. Despite considerable progress on related problems, no subquadratic time algorithm for this problem was found so far. We present an exact algorithm solving the problem in O(n log^4 n) time and a (1+e)-approximation algorithm running in O(n + 1/e^2 log^4(1/e)) time
Approximation Algorithms for Clustering Problems with Lower Bounds and Outliers
We consider clustering problems with non-uniform lower bounds and outliers, and obtain the first approximation guarantees for these problems. We have a set F of facilities with lower bounds {L_i}_{i in F} and a set D of clients located in a common metric space {c(i,j)}_{i,j in F union D}, and bounds k, m. A feasible solution is a pair (S subseteq F, sigma: D -> S union {out}), where sigma specifies the client assignments, such that |S| = L_i for all i in S, and |sigma^{-1}(out)| <= m. In the lower-bounded min-sum-of-radii with outliers P (LBkSRO) problem, the objective is to minimize sum_{i in S} max_{j in sigma^{-1})i)}, and in the lower-bounded k-supplier with outliers (LBkSupO) problem, the objective is to minimize max_{i in S} max_{j in sigma^{-1})i)} c(i,j).
We obtain an approximation factor of 12.365 for LBkSRO, which improves to 3.83 for the non-outlier version (i.e., m = 0). These also constitute the first approximation bounds for the min-sum-of-radii objective when we consider lower bounds and outliers separately. We apply the primal-dual method to the relaxation where we Lagrangify the |S| <= k constraint. The chief technical contribution and novelty of our algorithm is that, departing from the standard paradigm used for such constrained problems, we obtain an O(1)-approximation despite the fact that we do not obtain a Lagrangian-multiplier-preserving algorithm for the Lagrangian relaxation. We believe that our ideas have broader applicability to other clustering problems with outliers as well.
We obtain approximation factors of 5 and 3 respectively for LBkSupO and its non-outlier version. These are the first approximation results for k-supplier with non-uniform lower bounds
Approximating Fair -Min-Sum-Radii in
The -center problem is a classical clustering problem in which one is
asked to find a partitioning of a point set into clusters such that the
maximum radius of any cluster is minimized. It is well-studied. But what if we
add up the radii of the clusters instead of only considering the cluster with
maximum radius? This natural variant is called the -min-sum-radii problem.
It has become the subject of more and more interest in recent years, inspiring
the development of approximation algorithms for the -min-sum-radii problem
in its plain version as well as in constrained settings.
We study the problem for Euclidean spaces of arbitrary
dimension but assume the number of clusters to be constant. In this case, a
PTAS for the problem is known (see Bandyapadhyay, Lochet and Saurabh, SoCG,
2023). Our aim is to extend the knowledge base for -min-sum-radii to the
domain of fair clustering. We study several group fairness constraints, such as
the one introduced by Chierichetti et al. (NeurIPS, 2017). In this model, input
points have an additional attribute (e.g., colors such as red and blue), and
clusters have to preserve the ratio between different attribute values (e.g.,
have the same fraction of red and blue points as the ground set). Different
variants of this general idea have been studied in the literature. To the best
of our knowledge, no approximative results for the fair -min-sum-radii
problem are known, despite the immense amount of work on the related fair
-center problem.
We propose a PTAS for the fair -min-sum-radii problem in Euclidean spaces
of arbitrary dimension for the case of constant . To the best of our
knowledge, this is the first PTAS for the problem. It works for different
notions of group fairness
Fast Fencing
We consider very natural "fence enclosure" problems studied by Capoyleas,
Rote, and Woeginger and Arkin, Khuller, and Mitchell in the early 90s. Given a
set of points in the plane, we aim at finding a set of closed curves
such that (1) each point is enclosed by a curve and (2) the total length of the
curves is minimized. We consider two main variants. In the first variant, we
pay a unit cost per curve in addition to the total length of the curves. An
equivalent formulation of this version is that we have to enclose unit
disks, paying only the total length of the enclosing curves. In the other
variant, we are allowed to use at most closed curves and pay no cost per
curve.
For the variant with at most closed curves, we present an algorithm that
is polynomial in both and . For the variant with unit cost per curve, or
unit disks, we present a near-linear time algorithm.
Capoyleas, Rote, and Woeginger solved the problem with at most curves in
time. Arkin, Khuller, and Mitchell used this to solve the unit cost
per curve version in exponential time. At the time, they conjectured that the
problem with curves is NP-hard for general . Our polynomial time
algorithm refutes this unless P equals NP
A Weakly-Robust PTAS for Minimum Clique Partition in Unit Disk Graphs
We consider the problem of partitioning the set of vertices of a given unit
disk graph (UDG) into a minimum number of cliques. The problem is NP-hard and
various constant factor approximations are known, with the current best ratio
of 3. Our main result is a {\em weakly robust} polynomial time approximation
scheme (PTAS) for UDGs expressed with edge-lengths, it either (i) computes a
clique partition or (ii) gives a certificate that the graph is not a UDG; for
the case (i) that it computes a clique partition, we show that it is guaranteed
to be within (1+\eps) ratio of the optimum if the input is UDG; however if
the input is not a UDG it either computes a clique partition as in case (i)
with no guarantee on the quality of the clique partition or detects that it is
not a UDG. Noting that recognition of UDG's is NP-hard even if we are given
edge lengths, our PTAS is a weakly-robust algorithm. Our algorithm can be
transformed into an O(\frac{\log^* n}{\eps^{O(1)}}) time distributed PTAS.
We consider a weighted version of the clique partition problem on vertex
weighted UDGs that generalizes the problem. We note some key distinctions with
the unweighted version, where ideas useful in obtaining a PTAS breakdown. Yet,
surprisingly, it admits a (2+\eps)-approximation algorithm for the weighted
case where the graph is expressed, say, as an adjacency matrix. This improves
on the best known 8-approximation for the {\em unweighted} case for UDGs
expressed in standard form.Comment: 21 pages, 9 figure