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Register a new userMemoryless Algorithms for the Generalized $k$-server Problem on Uniform Metrics
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Added by
Grigorios Koumoutsos
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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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.
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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.
We introduce and study a discrete multi-period extension of the classical knapsack problem, dubbed generalized incremental knapsack. In this setting, we are given a set of $n$ items, each associated with a non-negative weight, and $T$ time periods with non-decreasing capacities $W_1 leq dots leq W_T$. When item $i$ is inserted at time $t$, we gain a profit of $p_{it}$; however, this item remains in the knapsack for all subsequent periods. The goal is to decide if and when to insert each item, subject to the time-dependent capacity constraints, with the objective of maximizing our total profit. Interestingly, this setting subsumes as special cases a number of recently-studied incremental knapsack problems, all known to be strongly NP-hard. Our first contribution comes in the form of a polynomial-time $(frac{1}{2}-epsilon)$-approximation for the generalized incremental knapsack problem. This result is based on a reformulation as a single-machine sequencing problem, which is addressed by blending dynamic programming techniques and the classical Shmoys-Tardos algorithm for the generalized assignment problem. Combined with further enumeration-based self-reinforcing ideas and newly-revealed structural properties of nearly-optimal solutions, we turn our basic algorithm into a quasi-polynomial time approximation scheme (QPTAS). Hence, under widely believed complexity assumptions, this finding rules out the possibility that generalized incremental knapsack is APX-hard.
Constraint satisfaction problems (CSPs) models many important intractable NP-hard problems such as propositional satisfiability problem (SAT). Algorithms with non-trivial upper bounds on running time for restricted SAT with bounded clause length k (k-SAT) can be classified into three styles: DPLL-like, PPSZ-like and Local Search, with local search algorithms having already been generalized to CSP with bounded constraint arity k (k-CSP). We generalize a DPLL-like algorithm in its simplest form and a PPSZ-like algorithm from k-SAT to k-CSP. As far as we know, this is the first attempt to use PPSZ-like strategy to solve k-CSP, and before little work has been focused on the DPLL-like or PPSZ-like strategies for k-CSP.
The three-in-a-tree problem asks for an induced tree of the input graph containing three mandatory vertices. In 2006, Chudnovsky and Seymour [Combinatorica, 2010] presented the first polynomial time algorithm for this problem, which has become a critical subroutine in many algorithms for detecting induced subgraphs, such as beetles, pyramids, thetas, and even and odd-holes. In 2007, Derhy and Picouleau [Discrete Applied Mathematics, 2009] considered the natural generalization to $k$ mandatory vertices, proving that, when $k$ is part of the input, the problem is $mathsf{NP}$-complete, and ask what is the complexity of four-in-a-tree. Motivated by this question and the relevance of the original problem, we study the parameterized complexity of $k$-in-a-tree. We begin by showing that the problem is $mathsf{W[1]}$-hard when jointly parameterized by the size of the solution and minimum clique cover and, under the Exponential Time Hypothesis, does not admit an $n^{o(k)}$ time algorithm. Afterwards, we use Courcelles Theorem to prove fixed-parameter tractability under cliquewidth, which prompts our investigation into which parameterizations admit single exponential algorithms; we show that such algorithms exist for the unrelated parameterizations treewidth, distance to cluster, and distance to co-cluster. In terms of kernelization, we present a linear kernel under feedback edge set, and show that no polynomial kernel exists under vertex cover nor distance to clique unless $mathsf{NP} subseteq mathsf{coNP}/mathsf{poly}$. Along with other remarks and previous work, our tractability and kernelization results cover many of the most commonly employed parameters in the graph parameter hierarchy.
Given a graph $G=(V,E)$ with two distinguished vertices $s,tin V$ and an integer parameter $L>0$, an {em $L$-bounded cut} is a subset $F$ of edges (vertices) such that the every path between $s$ and $t$ in $Gsetminus F$ has length more than $L$. The task is to find an $L$-bounded cut of minimum cardinality. Though the problem is very simple to state and has been studied since the beginning of the 70s, it is not much understood yet. The problem is known to be $cal{NP}$-hard to approximate within a small constant factor even for $Lgeq 4$ (for $Lgeq 5$ for the vertex cuts). On the other hand, the best known approximation algorithm for general graphs has approximation ratio only $mathcal{O}({n^{2/3}})$ in the edge case, and $mathcal{O}({sqrt{n}})$ in the vertex case, where $n$ denotes the number of vertices. We show that for planar graphs, it is possible to solve both the edge- and the vertex-version of the problem optimally in time $mathcal{O}(L^{3L}n)$. That is, the problem is fixed parameter tractable (FPT) with respect to $L$ on planar graphs. Furthermore, we show that the problem remains FPT even for bounded genus graphs, a super class of planar graphs. Our second contribution deals with approximations of the vertex version of the problem. We describe an algorithm that for a given a graph $G$, its tree decomposition of treewidth $tau$ and vertices $s$ and $t$ computes a $tau$-approximation of the minimum $L$-bounded $s-t$ vertex cut; if the decomposition is not given, then the approximation ratio is $mathcal{O}(tau sqrt{log tau})$. For graphs with treewidth bounded by $mathcal{O}(n^{1/2-epsilon})$ for any $epsilon>0$, but not by a constant, this is the best approximation in terms of~$n$ that we are aware of.
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