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Added by
Sebastien Bougleux
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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We consider the problem of transforming a set of elements into another by a sequence of elementary edit operations, namely substitutions, removals and insertions of elements. Each possible edit operation is penalized by a non-negative cost and the cost of a transformation is measured by summing the costs of its operations. A solution to this problem consists in defining a transformation having a minimal cost, among all possible transformations. To compute such a solution, the classical approach consists in representing removal and insertion operations by augmenting the two sets so that they get the same size. This allows to express the problem as a linear sum assignment problem (LSAP), which thus finds an optimal bijection (or permutation, perfect matching) between the two augmented sets. While the LSAP is known to be efficiently solvable in polynomial time complexity, for instance with the Hungarian algorithm, useless time and memory are spent to treat the elements which have been added to the initial sets. In this report, we show that the problem can be formalized as an extension of the LSAP which considers only one additional element in each set to represent removal and insertion operations. A solution to the problem is no longer represented as a bijection between the two augmented sets. We show that the considered problem is a binary linear program (BLP) very close to the LSAP. While it can be solved by any BLP solver, we propose an adaptation of the Hungarian algorithm which improves the time and memory complexities previously obtained by the approach based on the LSAP. The importance of the improvement increases as the size of the two sets and their absolute difference increase. Based on the analysis of the problem presented in this report, other classical algorithms can be adapted.
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In the Subset Sum problem we are given a set of $n$ positive integers $X$ and a target $t$ and are asked whether some subset of $X$ sums to $t$. Natural parameters for this problem that have been studied in the literature are $n$ and $t$ as well as the maximum input number $rm{mx}_X$ and the sum of all input numbers $Sigma_X$. In this paper we study the dense case of Subset Sum, where all these parameters are polynomial in $n$. In this regime, standard pseudo-polynomial algorithms solve Subset Sum in polynomial time $n^{O(1)}$. Our main question is: When can dense Subset Sum be solved in near-linear time $tilde{O}(n)$? We provide an essentially complete dichotomy by designing improved algorithms and proving conditional lower bounds, thereby determining essentially all settings of the parameters $n,t,rm{mx}_X,Sigma_X$ for which dense Subset Sum is in time $tilde{O}(n)$. For notational convenience we assume without loss of generality that $t ge rm{mx}_X$ (as larger numbers can be ignored) and $t le Sigma_X/2$ (using symmetry). Then our dichotomy reads as follows: - By reviving and improving an additive-combinatorics-based approach by Galil and Margalit [SICOMP91], we show that Subset Sum is in near-linear time $tilde{O}(n)$ if $t gg rm{mx}_X Sigma_X/n^2$. - We prove a matching conditional lower bound: If Subset Sum is in near-linear time for any setting with $t ll rm{mx}_X Sigma_X/n^2$, then the Strong Exponential Time Hypothesis and the Strong k-Sum Hypothesis fail. We also generalize our algorithm from sets to multi-sets, albeit with non-matching upper and lower bounds.
In the classical Subset Sum problem we are given a set $X$ and a target $t$, and the task is to decide whether there exists a subset of $X$ which sums to $t$. A recent line of research has resulted in $tilde{O}(t)$-time algorithms, which are (near-)optimal under popular complexity-theoretic assumptions. On the other hand, the standard dynamic programming algorithm runs in time $O(n cdot |mathcal{S}(X,t)|)$, where $mathcal{S}(X,t)$ is the set of all subset sums of $X$ that are smaller than $t$. Furthermore, all known pseudopolynomial algorithms actually solve a stronger task, since they actually compute the whole set $mathcal{S}(X,t)$. As the aforementioned two running times are incomparable, in this paper we ask whether one can achieve the best of both worlds: running time $tilde{O}(|mathcal{S}(X,t)|)$. In particular, we ask whether $mathcal{S}(X,t)$ can be computed in near-linear time in the output-size. Using a diverse toolkit containing techniques such as color coding, sparse recovery, and sumset estimates, we make considerable progress towards this question and design an algorithm running in time $tilde{O}(|mathcal{S}(X,t)|^{4/3})$. Central to our approach is the study of top-$k$-convolution, a natural problem of independent interest: given sparse polynomials with non-negative coefficients, compute the lowest $k$ non-zero monomials of their product. We design an algorithm running in time $tilde{O}(k^{4/3})$, by a combination of sparse convolution and sumset estimates considered in Additive Combinatorics. Moreover, we provide evidence that going beyond some of the barriers we have faced requires either an algorithmic breakthrough or possibly new techniques from Additive Combinatorics on how to pass from information on restricted sumsets to information on unrestricted sumsets.
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.
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.
A graph $G = (V,E)$ is a double-threshold graph if there exist a vertex-weight function $w colon V to mathbb{R}$ and two real numbers $mathtt{lb}, mathtt{ub} in mathbb{R}$ such that $uv in E$ if and only if $mathtt{lb} le mathtt{w}(u) + mathtt{w}(v) le mathtt{ub}$. In the literature, those graphs are studied as the pairwise compatibility graphs that have stars as their underlying trees. We give a new characterization of double-threshold graphs, which gives connections to bipartite permutation graphs. Using the new characterization, we present a linear-time algorithm for recognizing double-threshold graphs. Prior to our work, the fastest known algorithm by Xiao and Nagamochi [COCOON 2018] ran in $O(n^6)$ time, where $n$ is the number of vertices.
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