Time-approximation trade-offs for inapproximable problems

In this paper we focus on problems which do not admit a constant-factor approximation in polynomial time and explore how quickly their approximability improves as the allowed running time is gradually increased from polynomial to (sub-)exponential. We tackle a number of problems: For Min Independent Dominating Set, Max Induced Path, Forest and Tree, for any r(n), a simple, known scheme gives an approximation ratio of r in time roughly rn/r. We show that, for most values of r, if this running time could be significantly improved the ETH would fail. For Max Minimal Vertex Cover we give a nontrivial √r-approximation in time 2n/r. We match this with a similarly tight result. We also give a log r-approximation for Min ATSP in time 2n/r and an r-approximation for Max Grundy Coloring in time rn/r. Furthermore, we show that Min Set Cover exhibits a curious behavior in this superpolynomial setting: for any δ > 0 it admits an mδ-approximation, where m is the number of sets, in just quasi-polynomial time. We observe that if such ratios could be achieved in polynomial time, the ETH or the Projection Games Conjecture would fail. © Édouard Bonnet, Michael Lampis and Vangelis Th. Paschos; licensed under Creative Commons License CC-BY

From Gap-ETH to FPT-Inapproximability: Clique, Dominating Set, and More

We consider questions that arise from the intersection between the areas of polynomial-time approximation algorithms, subexponential-time algorithms, and fixed-parameter tractable algorithms. The questions, which have been asked several times (e.g., [Marx08, FGMS12, DF13]), are whether there is a non-trivial FPT-approximation algorithm for the Maximum Clique (Clique) and Minimum Dominating Set (DomSet) problems parameterized by the size of the optimal solution. In particular, letting $\text{OPT}$ be the optimum and $N$ be the size of the input, is there an algorithm that runs in $t(\text{OPT})\text{poly}(N)$ time and outputs a solution of size $f(\text{OPT})$, for any functions $t$ and $f$ that are independent of $N$ (for Clique, we want $f(\text{OPT})=\omega(1)$)? In this paper, we show that both Clique and DomSet admit no non-trivial FPT-approximation algorithm, i.e., there is no $o(\text{OPT})$-FPT-approximation algorithm for Clique and no $f(\text{OPT})$-FPT-approximation algorithm for DomSet, for any function $f$ (e.g., this holds even if $f$ is the Ackermann function). In fact, our results imply something even stronger: The best way to solve Clique and DomSet, even approximately, is to essentially enumerate all possibilities. Our results hold under the Gap Exponential Time Hypothesis (Gap-ETH) [Dinur16, MR16], which states that no $2^{o(n)}$-time algorithm can distinguish between a satisfiable 3SAT formula and one which is not even $(1 - \epsilon)$-satisfiable for some constant $\epsilon > 0$. Besides Clique and DomSet, we also rule out non-trivial FPT-approximation for Maximum Balanced Biclique, Maximum Subgraphs with Hereditary Properties, and Maximum Induced Matching in bipartite graphs. Additionally, we rule out $k^{o(1)}$-FPT-approximation algorithm for Densest $k$-Subgraph although this ratio does not yet match the trivial $O(k)$-approximation algorithm.Comment: 43 pages. To appear in FOCS'1

LQG Control and Sensing Co-Design

We investigate a Linear-Quadratic-Gaussian (LQG) control and sensing co-design problem, where one jointly designs sensing and control policies. We focus on the realistic case where the sensing design is selected among a finite set of available sensors, where each sensor is associated with a different cost (e.g., power consumption). We consider two dual problem instances: sensing-constrained LQG control, where one maximizes control performance subject to a sensor cost budget, and minimum-sensing LQG control, where one minimizes sensor cost subject to performance constraints. We prove no polynomial time algorithm guarantees across all problem instances a constant approximation factor from the optimal. Nonetheless, we present the first polynomial time algorithms with per-instance suboptimality guarantees. To this end, we leverage a separation principle, that partially decouples the design of sensing and control. Then, we frame LQG co-design as the optimization of approximately supermodular set functions; we develop novel algorithms to solve the problems; and we prove original results on the performance of the algorithms, and establish connections between their suboptimality and control-theoretic quantities. We conclude the paper by discussing two applications, namely, sensing-constrained formation control and resource-constrained robot navigation.Comment: Accepted to IEEE TAC. Includes contributions to submodular function optimization literature, and extends conference paper arXiv:1709.0882

Parameterized Exact and Approximation Algorithms for Maximum kk-Set Cover and Related Satisfiability Problems

Given a family of subsets $\mathcal S$ over a set of elements~$X$ and two integers~$p$ and~$k$, Max k-Set Cover consists of finding a subfamily~$\mathcal T \subseteq \mathcal S$ of cardinality at most~$k$, covering at least~$p$ elements of~$X$. This problem is W[2]-hard when parameterized by~$k$, and FPT when parameterized by $p$. We investigate the parameterized approximability of the problem with respect to parameters~$k$ and~$p$. Then, we show that Max Sat-k, a satisfiability problem generalizing Max k-Set Cover, is also FPT with respect to parameter~$p$.Comment: Accepted in RAIRO - Theoretical Informatics and Application

(In)approximability of Maximum Minimal FVS

We study the approximability of the NP-complete \textsc{Maximum Minimal Feedback Vertex Set} problem. Informally, this natural problem seems to lie in an intermediate space between two more well-studied problems of this type: \textsc{Maximum Minimal Vertex Cover}, for which the best achievable approximation ratio is $\sqrt{n}$, and \textsc{Upper Dominating Set}, which does not admit any $n^{1-\epsilon}$ approximation. We confirm and quantify this intuition by showing the first non-trivial polynomial time approximation for \textsc{Max Min FVS} with a ratio of $O(n^{2/3})$, as well as a matching hardness of approximation bound of $n^{2/3-\epsilon}$, improving the previous known hardness of $n^{1/2-\epsilon}$. The approximation algorithm also gives a cubic kernel when parameterized by the solution size. Along the way, we also obtain an $O(\Delta)$-approximation and show that this is asymptotically best possible, and we improve the bound for which the problem is NP-hard from $\Delta\ge 9$ to $\Delta\ge 6$. Having settled the problem's approximability in polynomial time, we move to the context of super-polynomial time. We devise a generalization of our approximation algorithm which, for any desired approximation ratio $r$, produces an $r$-approximate solution in time $n^{O(n/r^{3/2})}$. This time-approximation trade-off is essentially tight: we show that under the ETH, for any ratio $r$ and $\epsilon>0$, no algorithm can $r$-approximate this problem in time $n^{O((n/r^{3/2})^{1-\epsilon})}$, hence we precisely characterize the approximability of the problem for the whole spectrum between polynomial and sub-exponential time, up to an arbitrarily small constant in the second exponent.Comment: 31 pages, 2 figures, ISAAC 2020, Preprint submitted to Journal of Computer and System Science

Balancing Utility and Fairness in Submodular Maximization (Technical Report)

Submodular function maximization is central in numerous data science applications, including data summarization, influence maximization, and recommendation. In many of these problems, our goal is to find a solution that maximizes the \emph{average} of the utilities for all users, each measured by a monotone submodular function. When the population of users is composed of several demographic groups, another critical problem is whether the utility is fairly distributed across groups. In the context of submodular optimization, we seek to improve the welfare of the \emph{least well-off} group, i.e., to maximize the minimum utility for any group, to ensure fairness. Although the \emph{utility} and \emph{fairness} objectives are both desirable, they might contradict each other, and, to our knowledge, little attention has been paid to optimizing them jointly. In this paper, we propose a novel problem called \emph{Bicriteria Submodular Maximization} (BSM) to strike a balance between utility and fairness. Specifically, it requires finding a fixed-size solution to maximize the utility function, subject to the value of the fairness function not being below a threshold. Since BSM is inapproximable within any constant factor in general, we propose efficient data-dependent approximation algorithms for BSM by converting it into other submodular optimization problems and utilizing existing algorithms for the converted problems to obtain solutions to BSM. Using real-world and synthetic datasets, we showcase applications of our framework in three submodular maximization problems, namely maximum coverage, influence maximization, and facility location.Comment: 13 pages, 7 figures, under revie