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Register a new userImproved Approximations for Min Sum Vertex Cover and Generalized Min Sum Set Cover
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Added by
Nikhil Bansal
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We study the generalized min sum set cover (GMSSC) problem, wherein given a collection of hyperedges $E$ with arbitrary covering requirements $k_e$, the goal is to find an ordering of the vertices to minimize the total cover time of the hyperedges; a hyperedge $e$ is considered covered by the first time when $k_e$ many of its vertices appear in the ordering. We give a $4.642$ approximation algorithm for GMSSC, coming close to the best possible bound of $4$, already for the classical special case (with all $k_e=1$) of min sum set cover (MSSC) studied by Feige, Lov{a}sz and Tetali, and improving upon the previous best known bound of $12.4$ due to Im, Sviridenko and van der Zwaan. Our algorithm is based on transforming the LP solution by a suitable kernel and applying randomized rounding. This also gives an LP-based $4$ approximation for MSSC. As part of the analysis of our algorithm, we also derive an inequality on the lower tail of a sum of independent Bernoulli random variables, which might be of independent interest and broader utility. Another well-known special case is the min sum vertex cover (MSVC) problem, in which the input hypergraph is a graph and $k_e = 1$, for every edge. We give a $16/9$ approximation for MSVC, and show a matching integrality gap for the natural LP relaxation. This improves upon the previous best $1.999946$ approximation of Barenholz, Feige and Peleg. (The claimed $1.79$ approximation result of Iwata, Tetali and Tripathi for the MSVC turned out have an unfortunate, seemingly unfixable, mistake in it.) Finally, we revisit MSSC and consider the $ell_p$ norm of cover-time of the hyperedges. Using a dual fitting argument, we show that the natural greedy algorithm achieves tight, up to NP-hardness, approximation guarantees of $(p+1)^{1+1/p}$, for all $pge 1$. For $p=1$, this gives yet another proof of the $4$ approximation for MSSC.
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We consider the online Min-Sum Set Cover (MSSC), a natural and intriguing generalization of the classical list update problem. In Online MSSC, the algorithm maintains a permutation on $n$ elements based on subsets $S_1, S_2, ldots$ arriving online. The algorithm serves each set $S_t$ upon arrival, using its current permutation $pi_{t}$, incurring an access cost equal to the position of the first element of $S_t$ in $pi_{t}$. Then, the algorithm may update its permutation to $pi_{t+1}$, incurring a moving cost equal to the Kendall tau distance of $pi_{t}$ to $pi_{t+1}$. The objective is to minimize the total access and moving cost for serving the entire sequence. We consider the $r$-uniform version, where each $S_t$ has cardinality $r$. List update is the special case where $r = 1$. We obtain tight bounds on the competitive ratio of deterministic online algorithms for MSSC against a static adversary, that serves the entire sequence by a single permutation. First, we show a lower bound of $(r+1)(1-frac{r}{n+1})$ on the competitive ratio. Then, we consider several natural generalizations of successful list update algorithms and show that they fail to achieve any interesting competitive guarantee. On the positive side, we obtain a $O(r)$-competitive deterministic algorithm using ideas from online learning and the multiplicative weight updates (MWU) algorithm. Furthermore, we consider efficient algorithms. We propose a memoryless online algorithm, called Move-All-Equally, which is inspired by the Double Coverage algorithm for the $k$-server problem. We show that its competitive ratio is $Omega(r^2)$ and $2^{O(sqrt{log n cdot log r})}$, and conjecture that it is $f(r)$-competitive. We also compare Move-All-Equally against the dynamic optimal solution and obtain (almost) tight bounds by showing that it is $Omega(r sqrt{n})$ and $O(r^{3/2} sqrt{n})$-competitive.
We investigate the polynomial-time approximability of the multistage version of Min-Sum Set Cover ($mathrm{DSSC}$), a natural and intriguing generalization of the classical List Update problem. In $mathrm{DSSC}$, we maintain a sequence of permutations $(pi^0, pi^1, ldots, pi^T)$ on $n$ elements, based on a sequence of requests $(R^1, ldots, R^T)$. We aim to minimize the total cost of updating $pi^{t-1}$ to $pi^{t}$, quantified by the Kendall tau distance $mathrm{D}_{mathrm{KT}}(pi^{t-1}, pi^t)$, plus the total cost of covering each request $R^t$ with the current permutation $pi^t$, quantified by the position of the first element of $R^t$ in $pi^t$. Using a reduction from Set Cover, we show that $mathrm{DSSC}$ does not admit an $O(1)$-approximation, unless $mathrm{P} = mathrm{NP}$, and that any $o(log n)$ (resp. $o(r)$) approximation to $mathrm{DSSC}$ implies a sublogarithmic (resp. $o(r)$) approximation to Set Cover (resp. where each element appears at most $r$ times). Our main technical contribution is to show that $mathrm{DSSC}$ can be approximated in polynomial-time within a factor of $O(log^2 n)$ in general instances, by randomized rounding, and within a factor of $O(r^2)$, if all requests have cardinality at most $r$, by deterministic rounding.
A famous conjecture of Tuza states that the minimum number of edges needed to cover all the triangles in a graph is at most twice the maximum number of edge-disjoint triangles. This conjecture was couched in a broader setting by Aharoni and Zerbib who proposed a hypergraph version of this conjecture, and also studied its implied fraction
We study the classic set cover problem from the perspective of sub-linear algorithms. Given access to a collection of $m$ sets over $n$ elements in the query model, we show that sub-linear algorithms derived from existing techniques have almost tight query complexities. On one hand, first we show an adaptation of the streaming algorithm presented in Har-Peled et al. [2016] to the sub-linear query model, that returns an $alpha$-approximate cover using $tilde{O}(m(n/k)^{1/(alpha-1)} + nk)$ queries to the input, where $k$ denotes the value of a minimum set cover. We then complement this upper bound by proving that for lower values of $k$, the required number of queries is $tilde{Omega}(m(n/k)^{1/(2alpha)})$, even for estimating the optimal cover size. Moreover, we prove that even checking whether a given collection of sets covers all the elements would require $Omega(nk)$ queries. These two lower bounds provide strong evidence that the upper bound is almost tight for certain values of the parameter $k$. On the other hand, we show that this bound is not optimal for larger values of the parameter $k$, as there exists a $(1+varepsilon)$-approximation algorithm with $tilde{O}(mn/kvarepsilon^2)$ queries. We show that this bound is essentially tight for sufficiently small constant $varepsilon$, by establishing a lower bound of $tilde{Omega}(mn/k)$ query complexity.
The CONNECTED VERTEX COVER problem asks for a vertex cover in a graph that induces a connected subgraph. The problem is known to be fixed-parameter tractable (FPT), and is unlikely to have a polynomial sized kernel (under complexity theoretic assumptions) when parameterized by the solution size. In a recent paper, Lokshtanov et al.[STOC 2017], have shown an $alpha$-approximate kernel for the problem for every $alpha > 1$, in the framework of approximate or lossy kernelization. In this work, we exhibit lossy kernels and FPT algorithms for CONNECTED VERTEX COVER for parameters that are more natural and functions of the input, and in some cases, smaller than the solution size. The parameters we consider are the sizes of a split deletion set, clique deletion set, clique cover, cluster deletion set and chordal deletion set.
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