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1. The Maximum Exposure Problem

   Kumar, N ; Sintos, S ; Suri, S. ( September 2019 , Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques)

   Given a set of points P and axis-aligned rectangles R in the plane, a point p ∈ P is called exposed if it lies outside all rectangles in R. In the max-exposure problem, given an integer parameter k, we want to delete k rectangles from R so as to maximize the number of exposed points. We show that the problem is NP-hard and assuming plausible complexity conjectures is also hard to approximate even when rectangles in R are translates of two fixed rectangles. However, if R only consists of translates of a single rectangle, we present a polynomial-time approximation scheme. For general rectangle range space, we present a simple O(k) bicriteria approximation algorithm; that is by deleting O(k2) rectangles, we can expose at least Ω(1/k) of the optimal number of points.
2. NP-Hardness of Reed-Solomon Decoding and the Prouhet-Tarry-Escott Problem

   https://doi.org/10.1109/FOCS.2016.86  

   Gandikota, Venkata ; Ghazi, Badih ; Grigorescu, Elena ( October 2016 , FOCS)

   Establishing the complexity of {\em Bounded Distance Decoding} for Reed-Solomon codes is a fundamental open problem in coding theory, explicitly asked by Guruswami and Vardy (IEEE Trans. Inf. Theory, 2005). The problem is motivated by the large current gap between the regime when it is NP-hard, and the regime when it is efficiently solvable (i.e., the Johnson radius). We show the first NP-hardness results for asymptotically smaller decoding radii than the maximum likelihood decoding radius of Guruswami and Vardy. Specifically, for Reed-Solomon codes of length $N$ and dimension $K=O(N)$, we show that it is NP-hard to decode more than $ N-K- c\frac{\log N}{\log\log N}$ errors (with $c>0$ an absolute constant). Moreover, we show that the problem is NP-hard under quasipolynomial-time reductions for an error amount $> N-K- c\log{N}$ (with $c>0$ an absolute constant). An alternative natural reformulation of the Bounded Distance Decoding problem for Reed-Solomon codes is as a {\em Polynomial Reconstruction} problem. In this view, our results show that it is NP-hard to decide whether there exists a degree $K$ polynomial passing through $K+ c\frac{\log N}{\log\log N}$ points from a given set of points $(a_1, b_1), (a_2, b_2)\ldots, (a_N, b_N)$. Furthermore, it is NP-hard under quasipolynomial-time reductions to decidemore »whether there is a degree $K$ polynomial passing through $K+c\log{N}$ many points. These results follow from the NP-hardness of a generalization of the classical Subset Sum problem to higher moments, called {\em Moments Subset Sum}, which has been a known open problem, and which may be of independent interest. We further reveal a strong connection with the well-studied Prouhet-Tarry-Escott problem in Number Theory, which turns out to capture a main barrier in extending our techniques. We believe the Prouhet-Tarry-Escott problem deserves further study in the theoretical computer science community.« less
3. Quasi-polynomial time approximation schemes for the Maximum Weight Independent Set Problem in H-free graphs

   https://doi.org/10.1137/1.9781611975994.139  

   Chudnovsky, Maria. ; Pillipczuk, Marcin. ; Pillipczuk, Mihal ; and Thomasse, Stephan. ( April 2020 , Proceedings of the annual ACMSIAM symposium on discrete algorithms)

   In the Maximum Independent Set problem we are asked to find a set of pairwise nonadjacent vertices in a given graph with the maximum possible cardinality. In general graphs, this classical problem is known to be NP-hard and hard to approximate within a factor of n1−ε for any ε > 0. Due to this, investigating the complexity of Maximum Independent Set in various graph classes in hope of finding better tractability results is an active research direction. In H-free graphs, that is, graphs not containing a fixed graph H as an induced subgraph, the problem is known to remain NP-hard and APX-hard whenever H contains a cycle, a vertex of degree at least four, or two vertices of degree at least three in one connected component. For the remaining cases, where every component of H is a path or a subdivided claw, the complexity of Maximum Independent Set remains widely open, with only a handful of polynomial-time solvability results for small graphs H such as P5, P6, the claw, or the fork. We prove that for every such “possibly tractable” graph H there exists an algorithm that, given an H-free graph G and an accuracy parameter ε > 0, findsmore »an independent set in G of cardinality within a factor of (1 – ε) of the optimum in time exponential in a polynomial of log | V(G) | and ε−1. That is, we show that for every graph H for which Maximum Independent Set is not known to be APX-hard in H-free graphs, the problem admits a quasi-polynomial time approximation scheme in this graph class. Our algorithm works also in the more general weighted setting, where the input graph is supplied with a weight function on vertices and we are maximizing the total weight of an independent set.« less
4. Application Provisioning in FOG Computing-enabled Internet-of-Things: A Network Perspective

   https://doi.org/10.1109/INFOCOM.2018.8486269  

   Yu, Ruozhou ; Xue, Guoliang ; Zhang, Xiang ( April 2018 , IEEE INFOCOM'2018)

   The emergence of the Internet-of-Things (IoT) has inspired numerous new applications. However, due to the limited resources in current IoT infrastructures and the stringent quality-of-service requirements of the applications, providing computing and communication supports for the applications is becoming increasingly difficult. In this paper, we consider IoT applications that receive continuous data streams from multiple sources in the network, and study joint application placement and data routing to support all data streams with both bandwidth and delay guarantees. We formulate the application provisioning problem both for a single application and for multiple applications, with both cases proved to be NP-hard. For the case with a single application, we propose a fully polynomial-time approximation scheme. For the multi-application scenario, if the applications can be parallelized among multiple distributed instances, we propose a fully polynomial-time approximation scheme; for general non-parallelizable applications, we propose a randomized algorithm and analyze its performance. Simulations show that the proposed algorithms greatly improve the quality-of-service of the IoT applications compared to the heuristics.
5. Robust Quadratic Programming with Mixed-Integer Uncertainty

   https://doi.org/10.1287/ijoc.2019.0901  

   Mittal, Areesh ; Gokalp, Can ; Hanasusanto, Grani A. ( September 2019 , INFORMS Journal on Computing)

   We study robust convex quadratic programs where the uncertain problem parameters can contain both continuous and integer components. Under the natural boundedness assumption on the uncertainty set, we show that the generic problems are amenable to exact copositive programming reformulations of polynomial size. These convex optimization problems are NP-hard but admit a conservative semidefinite programming (SDP) approximation that can be solved efficiently. We prove that the popular approximate S-lemma method—which is valid only in the case of continuous uncertainty—is weaker than our approximation. We also show that all results can be extended to the two-stage robust quadratic optimization setting if the problem has complete recourse. We assess the effectiveness of our proposed SDP reformulations and demonstrate their superiority over the state-of-the-art solution schemes on instances of least squares, project management, and multi-item newsvendor problems.