On the complexity of graph embeddings

We consider the classic scheduling problem of minimizing the total weighted flow-time on a single machine (min-WPFT), when preemption is allowed. In this problem, we are given a set of $n$ jobs, each job having a release time $r_j$, a processing time $p_j$, and a weight $w_j$. The flow-time of a job is defined as the amount of time the job spends in the system before it completes; that is, $F_j = C_j - r_j$, where $C_j$ is the completion time of job. The objective is to minimize the total weighted flow-time of jobs. This NP-hard problem has been studied quite extensively for decades. In a recent breakthrough, Batra, Garg, and Kumar presented a {\em pseudo-polynomial} time algorithm that has an $O(1)$ approximation ratio. The design of a truly polynomial time algorithm, however, remained an open problem. In this paper, we show a transformation from pseudo-polynomial time algorithms to polynomial time algorithms in the context of min-WPFT. Our result combined with the result of Batra, Garg, and Kumar settles the long standing conjecture that there is a polynomial time algorithm with $O(1)$-approximation for min-WPFT.

Planarity, bipartiteness and (directed) acyclicity are basic graph properties with classic linear time recognition algorithms. However, the problems of testing whether a given (di)graph has k vertices whose deletion makes it planar, bipartite or a directed acyclic graph (DAG) are all fundamental NP-complete problems when k is part of the input. As a result, a significant amount of research has been devoted to understanding whether, for every fixed k, these problems admit a polynomial time algorithm (where the exponent in the polynomial is independent of k) and in particular, whether they admit linear time algorithms. While we now know that for any fixed k, we can test in linear time whether a graph is k vertices away from being planar [FOCS 2009, SODA 2014] or bipartite [SODA 2014, SICOMP 2016], the best known algorithms in the case of directed acyclicity are the algorithm of Garey and Tarjan [IPL 78] which runs in time O(nk–1m) and the algorithm of Chen, Liu, Lu, O'Sullivan and Razgon [JACM 2008] which runs in time O(k!4kk4nm). In other words, it has remained open whether it is possible to recognize in linear time, a graph which is 2 vertices away from being acyclic! In this paper, we settle this question by giving an algorithm that decides whether a given graph is k vertices away from being acyclic, in time O(k!4kk5(n + m)). That is, for every fixed k, our algorithm runs in time O(m + n), thus mirroring the case for planarity and bipartiteness. Our algorithm is designed via a general methodology that shaves off a factor of n from some algorithms that use the powerful technique of iterative compression. The two main features of our methodology are: (i) This is the first generic technique for designing linear time algorithms for directed cut-problems and (ii) it can be used in combination with future improvements in algorithms for the compression version of other well-studied cut-problems such as Multicut and Directed Subset Feedback Vertex Set.

A faster FPTAS for knapsack problem with cardinality constraint

Chudnovsky-type algorithms for the multiplication in finite extensions of finite fields are well-known for having a good bilinear complexity, both asymptotically and at finite distance. More precisely, for every degree n of the extension, the existence of a family of algorithms with linear bilinear complexity in n has been proved using the original method applied to an explicit recursive tower of function fields. However, there is currently no method to build these algorithms in polynomial time. Nevertheless, one can construct in polynomial time a Chudnovsky-type algorithm over the projective line for the multiplication in any extension degree, with a quasi-linear bilinear complexity. In this paper, we prove that we can obtain algorithms both constructible in polynomial time and having a linear bilinear complexity by mixing up these two strategies.

The closest string and substring problems find applications in PCR primer design, genetic probe design, motif finding, and antisense drug design. For their importance, the two problems have been extensively studied recently in computational biology. Unfortunately both problems are NP-complete. Researchers have developed both fixed-parameter algorithms and approximation algorithms for the two problems. In terms of fixed-parameter, when the radius d is the parameter, the best-known fixed-parameter algorithm for closest string has time complexity O(ndd+1), which is still superpolynomial even if d = O(log n). In this paper we provide an O(n|Σ|O(d)) algorithm where Σ is the alphabet. This gives a polynomial time algorithm when d = O(log n) and Σ has constant size. Using the same technique, we additionally provide a more efficient subexponential time algorithm for the closest substring problem. In terms of approximation, both closest string and closest substring problems admit polynomial time approximation schemes (PTAS). The best known time complexity of the PTAS is O(nO(Ɛ-2 log 1/Ɛ)). In this paper we present a PTAS with time complexity O(nO(Ɛ-2)). At last, we prove that a restricted version of the closest substring has the same parameterized complexity as closest substring, answering an open question in the literature.

How to Choose Between Exponentially Many Strategies In “Solving Zero-Sum Games Using Best-Response Oracles with Applications to Search Games,” L. Hellerstein, T. Lidbetter, and D. Pirutinsky consider zero-sum games between a minimizer and a maximizer, where the number of pure strategies of the minimizer is exponential in the number of pure strategies of the maximizer. Such games are frequent in the search-games literature. Solving them with standard algorithms typically takes exponential time. The authors show how to compute (approximate) solutions in polynomial time, provided that there is a polynomial time (approximate) algorithm solving the best-response problem: given a mixed (randomized) strategy of the maximizer, what is a best response of the minimizer? The paper presents both a learning approach using weight updates and an approach of solely theoretical interest based on the ellipsoid algorithm. The learning approach performs well experimentally compared with approaches in the literature. The results are applied to obtain new algorithms for solving specific search games.

We study the integer maximum flow problem on wireless sensor networks with energy constraint. In this problem, sensor nodes gather data and then relay them to a base station, before they run out of battery power. Packets are considered as integral units and not splittable. The problem is to find the maximum data flow in the sensor network subject to the energy constraint of the sensors. We show that this integral version of the problem is strongly NP-complete and in fact APX-hard. It follows that the problem is unlikely to have a polynomial time approximation scheme. Even when restricted to graphs with concrete geometrically defined connectivity and transmission costs, the problem is still strongly NP-complete. We provide some interesting polynomial time algorithms that give good approximations for the general case nonetheless. For networks of bounded treewidth greater than two, we show that the problem is weakly NP-complete and provide pseudo-polynomial time algorithms. For a special case of graphs with treewidth two, we give a polynomial time algorithm.

Maximum Weight Stable Set on graphs without claw and co-claw (and similar graph classes) can be solved in linear time

We present the first polynomial time algorithm to learn nontrivial classes of languages of infinite trees. Specifically, our algorithm uses membership and equivalence queries to learn classes of omega-tree languages derived from weak regular omega-word languages in polynomial time. The method is a general polynomial time reduction of learning a class of derived omega-tree languages to learning the underlying class of omega-word languages, for any class of omega-word languages recognized by a deterministic Büchi acceptor. Our reduction, combined with the polynomial time learning algorithm of Maler and Pnueli [Maler and Pneuli, Inform. Comput., 1995] for the class of weak regular omega-word languages yields the main result. We also show that subset queries that return counterexamples can be implemented in polynomial time using subset queries that return no counterexamples for deterministic or non-deterministic finite word acceptors, and deterministic or non-deterministic Büchi omega-word acceptors. A previous claim of an algorithm to learn regular omega-trees due to Jayasrirani, Begam and Thomas [Jayasrirani et al., ICGI, 2008] is unfortunately incorrect, as shown in [Angluin, YALEU/DCS/TR-1528, 2016].

A graph G = {V, E}, with a finite number of vertices V and a finite number of edges E, has a property of a possibly infinite number of paths where a path is a sequence of vertices one walks along edges from one vertex to another. In this paper, we describe a graph as a function in the Enterprise-Participant (EP) data model and outline that a graph G, i.e., the function representing G, is Probably Approximately Correct (PAC) learnable, i.e., there is a learning algorithm in polynomial time that constructs edge representations in an EP database as a program P from a set of sampled paths and EP’s built-in reduction rules enables the reasoning of global transitive relations of the graph: there is a path from A to C if there are two paths A to B and B to C. Such a reasoning is also called a prediction, where predicting that A to C is not a path is an error when A to C is actually a path and an edge along the path A to B or B to C was not fed to the algorithm as part of a sample path. We generalize this learnability to a class of bounded functions. It is a reminder that data can be better represented in an EP database than in a graph before being embedded in distributional semantic learning models.

As a contention‐free channel access protocol, Time Division Multiple Access (TDMA) is widely applied in Energy Harvesting Wireless Sensor Networks (EH‐WSNs) due to ease of implementation. TDMA scheduling in EH‐WSNs faces the problem of low channel utilization since time slots assigned to some nodes may not be used by them due to shortage of energy or data. It is important to design TDMA scheduling schemes that enhance channel utilization. In this paper, the Fixed Frame Size Scheme (FFSS) and the Adaptive Frame Size Scheme (AFSS) are proposed to improve channel utilization of EH‐WSNs. The FFSS aims at the optimal TDMA slot assignment for the TDMA scheduling with fixed frame size, and the AFSS considers the TDMA with variable frame size and targets both the optimal frame size and the optimal slot assignment. The optimization problems maximizing channel utilization are formulated for the FFSS and the AFSS, respectively, which take the upcoming energy and data into account. The optimization problems are transformed into assignment‐like problems and solved by the Hungarian‐based algorithm in polynomial time. Simulation results indicate that the proposed FFSS and AFSS can considerably improve channel utilization in the EH‐WSNs compared with the existing TDMA scheduling schemes.

We propose a rate-distortion optimal algorithm for sinusoidal modeling of audio and speech. The algorithm determines, for a pre-specified target bit-rate, the optimal (variable-length) time segmentation, the optimal distribution of sinusoidal components over the segments and the optimal (scalar) quantizers for quantizing the sinusoid parameters. The optimization is done by jointly optimizing the segment lengths, number of sinusoids and quantizers using high-resolution quantization theory and dynamic programming techniques, which makes it possible to solve the algorithm in polynomial time. A particular advantage of the proposed method is that, given a target bit-rate, it solves the problem of finding the optimal balance between total number of sinusoids and number of bits per sinusoid.

Most existing or currently developing Internet of Things (IoT) communication standards are based on the assumption that the IoT services only require low data rate transmission and therefore can be supported by limited resources such as narrow-band channels. This assumption rules out those IoT services with burst traffic, critical missions, and low latency requirements. In this paper, we propose to utilize the idle devices in mission-critical IoT networks to boost the transmission data rate for critical tasks through multiple concurrent transmissions. This approach virtually expands the existing narrow-band IoT protocols to break the bandwidth limitation in order to provide low latency services for critical tasks. In this approach, we propose the task-balance method and the first-link descending order to determine the relay order and data partition in a given relay set. We theoretically prove that the optimal relay configuration that minimizes the uploading latency in single source scenario can be derived by the proposed algorithms in polynomial time when we have sufficient number of available channels. We also propose a greedy algorithm to approximate the optimal solution within a 1/2 performance lower bound in general scenarios. The simulation results shows that the proposed approach can reduce the latency of critical tasks up to 76% comparing with traditional approaches.

We are concerned with optimally grouping active mobile users in a two-user-based cooperative diversity system to maximize the cooperative diversity energy gain in a radio cell. The optimization problem is formulated as a non-bipartite weighted-matching problem in a static network setting. The weighted-matching problem can be solved using maximum weighted (MW) matching algorithm in polynomial time O(n3). To reduce the implementation and computational complexity, we develop a Worst-Link-First (WLF) matching algorithm, which gives the user with the worse channel condition and the higher energy consumption rate a higher priority to choose its partner. The computational complexity of the proposed WLF algorithm is O(n) while the achieved average energy gain is only slightly lower than that of the optimal maximum weighted- matching algorithm and similar to that of the 1/2-approximation Greedy matching algorithm (with computational complexity of O(n2 log n)) for a static-user network. We further investigate the optimal matching problem in mobile networks. By intelligently applying user mobility information in the matching algorithm, high cooperative diversity energy gain with moderate overhead is possible. In mobile networks, the proposed WLF matching algorithm, being less complex than the MW and the Greedy matching algorithms, yields performance characteristics close to those of the MW matching algorithm and better than the Greedy matching algorithm.

The matrix sensing problem is an important low-rank optimization problem that has found a wide range of applications, such as matrix completion, phase synchornization/retrieval, robust principal component analysis (PCA), and power system state estimation. In this work, we focus on the general matrix sensing problem with linear measurements that are corrupted by random noise. We investigate the scenario where the search rank r is equal to the true rank [Formula: see text] of the unknown ground truth (the exact parametrized case), as well as the scenario where r is greater than [Formula: see text] (the overparametrized case). We quantify the role of the restricted isometry property (RIP) in shaping the landscape of the nonconvex factorized formulation and assisting with the success of local search algorithms. First, we develop a global guarantee on the maximum distance between an arbitrary local minimizer of the nonconvex problem and the ground truth under the assumption that the RIP constant is smaller than [Formula: see text]. We then present a local guarantee for problems with an arbitrary RIP constant, which states that any local minimizer is either considerably close to the ground truth or far away from it. More importantly, we prove that this noisy, overparametrized problem exhibits the strict saddle property, which leads to the global convergence of perturbed gradient descent algorithm in polynomial time. The results of this work provide a comprehensive understanding of the geometric landscape of the matrix sensing problem in the noisy and overparametrized regime. Funding: This work was supported by grants from the National Science Foundation, Office of Naval Research, Air Force Office of Scientific Research, and Army Research Office.