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Register a new userHardness and Approximation of Submodular Minimum Linear Ordering Problems
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Added by
Majid Farhadi
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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The minimum linear ordering problem (MLOP) seeks to minimize an aggregated cost $f(cdot)$ due to an ordering $sigma$ of the items (say $[n]$), i.e., $min_{sigma} sum_{iin [n]} f(E_{i,sigma})$, where $E_{i,sigma}$ is the set of items that are mapped by $sigma$ to indices at most $i$. This problem has been studied in the literature for various special cases of the cost function $f$, and in a general setting for a submodular or supermodular cost $f$ [ITT2012]. Though MLOP was known to be NP-hard for general submodular functions, it was unknown whether the special case of graphic matroid MLOP (with $f$ being the matroid rank function of a graph) was polynomial-time solvable. Following this motivation, we explore related classes of linear ordering problems, including symmetric submodular MLOP, minimum latency vertex cover, and minimum sum vertex cover. We show that the most special cases of our problem, graphic matroid MLOP and minimum latency vertex cover, are both NP-hard. We further expand the toolkit for approximating MLOP variants: using the theory of principal partitions, we show a $2-frac{1+ell_{f}}{1+|E|}$ approximation to monotone submodular MLOP, where $ell_{f}=frac{f(E)}{max_{xin E}f({x})}$ satisfies $1 leq ell_f leq |E|$. Thus our result improves upon the best known bound of $2-frac{2}{1+|E|}$ by Iwata, Tetali, and Tripathi [ITT2012]. This leads to a $2-frac{1+r(E)}{1+|E|}$ approximation for the matroid MLOP, corresponding to the case when $r$ is the rank function of a given matroid. Finally, we show that MLVC can be $4/3$ approximated, matching the integrality gap of its vanilla LP relaxation.
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The problem of solving linear systems is one of the most fundamental problems in computer science, where given a satisfiable linear system $(A,b)$, for $A in mathbb{R}^{n times n}$ and $b in mathbb{R}^n$, we wish to find a vector $x in mathbb{R}^n$ such that $Ax = b$. The current best algorithms for solving dense linear systems reduce the problem to matrix multiplication, and run in time $O(n^{omega})$. We consider the problem of finding $varepsilon$-approximate solutions to linear systems with respect to the $L_2$-norm, that is, given a satisfiable linear system $(A in mathbb{R}^{n times n}, b in mathbb{R}^n)$, find an $x in mathbb{R}^n$ such that $||Ax - b||_2 leq varepsilon||b||_2$. Our main result is a fine-grained reduction from computing the rank of a matrix to finding $varepsilon$-approximate solutions to linear systems. In particular, if the best known $O(n^omega)$ time algorithm for computing the rank of $n times O(n)$ matrices is optimal (which we conjecture is true), then finding an $varepsilon$-approximate solution to a dense linear system also requires $tilde{Omega}(n^{omega})$ time, even for $varepsilon$ as large as $(1 - 1/text{poly}(n))$. We also prove (under some modified conjectures for the rank-finding problem) optimal hardness of approximation for sparse linear systems, linear systems over positive semidefinite matrices, well-conditioned linear systems, and approximately solving linear systems with respect to the $L_p$-norm, for $p geq 1$. At the heart of our results is a novel reduction from the rank problem to a decision version of the approximate linear systems problem. This reduction preserves properties such as matrix sparsity and bit complexity.
There has been a resurgence of interest in lower bounds whose truth rests on the conjectured hardness of well known computational problems. These conditional lower bounds have become important and popular due to the painfully slow progress on proving strong unconditional lower bounds. Nevertheless, the long term goal is to replace these conditional bounds with unconditional ones. In this paper we make progress in this direction by studying the cell probe complexity of two conjectured to be hard problems of particular importance: matrix-vector multiplication and a version of dynamic set disjointness known as Patrascus Multiphase Problem. We give improved unconditional lower bounds for these problems as well as introducing new proof techniques of independent interest. These include a technique capable of proving strong threshold lower bounds of the following form: If we insist on having a very fast query time, then the update time has to be slow enough to compute a lookup table with the answer to every possible query. This is the first time a lower bound of this type has been proven.
An ordering constraint satisfaction problem (OCSP) is given by a positive integer $k$ and a constraint predicate $Pi$ mapping permutations on ${1,ldots,k}$ to ${0,1}$. Given an instance of OCSP$(Pi)$ on $n$ variables and $m$ constraints, the goal is to find an ordering of the $n$ variables that maximizes the number of constraints that are satisfied, where a constraint specifies a sequence of $k$ distinct variables and the constraint is satisfied by an ordering on the $n$ variables if the ordering induced on the $k$ variables in the constraint satisfies $Pi$. OCSPs capture natural problems including Maximum acyclic subgraph (MAS) and Betweenness. In this work we consider the task of approximating the maximum number of satisfiable constraints in the (single-pass) streaming setting, where an instance is presented as a stream of constraints. We show that for every $Pi$, OCSP$(Pi)$ is approximation-resistant to $o(n)$-space streaming algorithms. This space bound is tight up to polylogarithmic factors. In the case of MAS our result shows that for every $epsilon>0$, MAS is not $1/2+epsilon$-approximable in $o(n)$ space. The previous best inapproximability result only ruled out a $3/4$-approximation in $o(sqrt n)$ space. Our results build on recent works of Chou, Golovnev, Sudan, Velingker, and Velusamy who show tight, linear-space inapproximability results for a broad class of (non-ordering) constraint satisfaction problems over arbitrary (finite) alphabets. We design a family of appropriate CSPs (one for every $q$) from any given OCSP, and apply their work to this family of CSPs. We show that the hard instances from this earlier work have a particular small-set expansion property. By exploiting this combinatorial property, in combination with the hardness results of the resulting families of CSPs, we give optimal inapproximability results for all OCSPs.
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$ $In many optimization problems, a feasible solution induces a multi-dimensional cost vector. For example, in load-balancing a schedule induces a load vector across the machines. In $k$-clustering, opening $k$ facilities induces an assignment cost vector across the clients. In this paper we consider the following minimum norm optimization problem : Given an arbitrary monotone, symmetric norm, find a solution which minimizes the norm of the induced cost-vector. This generalizes many fundamental NP-hard problems. We give a general framework to tackle the minimum norm problem, and illustrate its efficacy in the unrelated machine load balancing and $k$-clustering setting. Our concrete results are the following. $bullet$ We give constant factor approximation algorithms for the minimum norm load balancing problem in unrelated machines, and the minimum norm $k$-clustering problem. To our knowledge, our results constitute the first constant-factor approximations for such a general suite of objectives. $bullet$ In load balancing with unrelated machines, we give a $2$-approximation for the problem of finding an assignment minimizing the sum of the largest $ell$ loads, for any $ell$. We give a $(2+varepsilon)$-approximation for the so-called ordered load-balancing problem. $bullet$ For $k$-clustering, we give a $(5+varepsilon)$-approximation for the ordered $k$-median problem significantly improving the constant factor approximations from Byrka, Sornat, and Spoerhase (STOC 2018) and Chakrabarty and Swamy (ICALP 2018). $bullet$ Our techniques also imply $O(1)$ approximations to the best simultaneous optimization factor for any instance of the unrelated machine load-balancing and the $k$-clustering setting. To our knowledge, these are the first positive simultaneous optimization results in these settings.
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