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Register a new userThe Non-Uniform k-Center Problem
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Added by
Prachi Goyal
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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In this paper, we introduce and study the Non-Uniform k-Center problem (NUkC). Given a finite metric space $(X,d)$ and a collection of balls of radii ${r_1geq cdots ge r_k}$, the NUkC problem is to find a placement of their centers on the metric space and find the minimum dilation $alpha$, such that the union of balls of radius $alphacdot r_i$ around the $i$th center covers all the points in $X$. This problem naturally arises as a min-max vehicle routing problem with fleets of different speeds. The NUkC problem generalizes the classic $k$-center problem when all the $k$ radii are the same (which can be assumed to be $1$ after scaling). It also generalizes the $k$-center with outliers (kCwO) problem when there are $k$ balls of radius $1$ and $ell$ balls of radius $0$. There are $2$-approximation and $3$-approximation algorithms known for these problems respectively; the former is best possible unless P=NP and the latter remains unimproved for 15 years. We first observe that no $O(1)$-approximation is to the optimal dilation is possible unless P=NP, implying that the NUkC problem is more non-trivial than the above two problems. Our main algorithmic result is an $(O(1),O(1))$-bi-criteria approximation result: we give an $O(1)$-approximation to the optimal dilation, however, we may open $Theta(1)$ centers of each radii. Our techniques also allow us to prove a simple (uni-criteria), optimal $2$-approximation to the kCwO problem improving upon the long-standing $3$-factor. Our main technical contribution is a connection between the NUkC problem and the so-called firefighter problems on trees which have been studied recently in the TCS community.
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The Non-Uniform $k$-center (NUkC) problem has recently been formulated by Chakrabarty, Goyal and Krishnaswamy [ICALP, 2016] as a generalization of the classical $k$-center clustering problem. In NUkC, given a set of $n$ points $P$ in a metric space and non-negative numbers $r_1, r_2, ldots , r_k$, the goal is to find the minimum dilation $alpha$ and to choose $k$ balls centered at the points of $P$ with radius $alphacdot r_i$ for $1le ile k$, such that all points of $P$ are contained in the union of the chosen balls. They showed that the problem is NP-hard to approximate within any factor even in tree metrics. On the other hand, they designed a bi-criteria constant approximation algorithm that uses a constant times $k$ balls. Surprisingly, no true approximation is known even in the special case when the $r_i$s belong to a fixed set of size 3. In this paper, we study the NUkC problem under perturbation resilience, which was introduced by Bilu and Linial [Combinatorics, Probability and Computing, 2012]. We show that the problem under 2-perturbation resilience is polynomial time solvable when the $r_i$s belong to a constant sized set. However, we show that perturbation resilience does not help in the general case. In particular, our findings imply that even with perturbation resilience one cannot hope to find any good approximation for the problem.
In this paper we initiate the study of the heterogeneous capacitated $k$-center problem: given a metric space $X = (F cup C, d)$, and a collection of capacities. The goal is to open each capacity at a unique facility location in $F$, and also to assign clients to facilities so that the number of clients assigned to any facility is at most the capacity installed; the objective is then to minimize the maximum distance between a client and its assigned facility. If all the capacities $c_i$s are identical, the problem becomes the well-studied uniform capacitated $k$-center problem for which constant-factor approximations are known. The additional choice of determining which capacity should be installed in which location makes our problem considerably different from this problem, as well the non-uniform generalizations studied thus far in literature. In fact, one of our contributions is in relating the heterogeneous problem to special-cases of the classical Santa Claus problem. Using this connection, and by designing new algorithms for these special cases, we get the following results: (a)A quasi-polynomial time $O(log n/epsilon)$-approximation where every capacity is violated by $1+varepsilon$, (b) A polynomial time $O(1)$-approximation where every capacity is violated by an $O(log n)$ factor. We get improved results for the {em soft-capacities} version where we can place multiple facilities in the same location.
We consider the generalized $k$-server problem on uniform metrics. We study the power of memoryless algorithms and show tight bounds of $Theta(k!)$ on their competitive ratio. In particular we show that the textit{Harmonic Algorithm} achieves this competitive ratio and provide matching lower bounds. This improves the $approx 2^{2^k}$ doubly-exponential bound of Chiplunkar and Vishwanathan for the more general setting of uniform metrics with different weights.
We study the Capacitated k-Median problem, for which all the known constant factor approximation algorithms violate either the number of facilities or the capacities. While the standard LP-relaxation can only be used for algorithms violating one of the two by a factor of at least two, Shi Li [SODA15, SODA16] gave algorithms violating the number of facilities by a factor of 1+{epsilon} exploring properties of extended relaxations. In this paper we develop a constant factor approximation algorithm for Uniform Capacitated k-Median violating only the capacities by a factor of 1+{epsilon}. The algorithm is based on a configuration LP. Unlike in the algorithms violating the number of facilities, we cannot simply open extra few facilities at selected locations. Instead, our algorithm decides about the facility openings in a carefully designed dependent rounding process.
The Densest $k$-Subgraph (D$k$S) problem, and its corresponding minimization problem Smallest $p$-Edge Subgraph (S$p$ES), have come to play a central role in approximation algorithms. This is due both to their practical importance, and their usefulness as a tool for solving and establishing approximation bounds for other problems. These two problems are not well understood, and it is widely believed that they do not an admit a subpolynomial approximation ratio (although the best known hardness results do not rule this out). In this paper we generalize both D$k$S and S$p$ES from graphs to hypergraphs. We consider the Densest $k$-Subhypergraph problem (given a hypergraph $(V, E)$, find a subset $Wsubseteq V$ of $k$ vertices so as to maximize the number of hyperedges contained in $W$) and define the Minimum $p$-Union problem (given a hypergraph, choose $p$ of the hyperedges so as to minimize the number of vertices in their union). We focus in particular on the case where all hyperedges have size 3, as this is the simplest non-graph setting. For this case we provide an $O(n^{4(4-sqrt{3})/13 + epsilon}) leq O(n^{0.697831+epsilon})$-approximation (for arbitrary constant $epsilon > 0$) for Densest $k$-Subhypergraph and an $tilde O(n^{2/5})$-approximation for Minimum $p$-Union. We also give an $O(sqrt{m})$-approximation for Minimum $p$-Union in general hypergraphs. Finally, we examine the interesting special case of interval hypergraphs (instances where the vertices are a subset of the natural numbers and the hyperedges are intervals of the line) and prove that both problems admit an exact polynomial time solution on these instances.
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