Polynomial-time Approximation Algorithm Research Articles

On flexible and robust parameter assignment for periodic real-time components

In order to increase the flexibility of design process, we consider a component-based system in which some components may be changed, repaired, or upgraded. We assume that the worst-case execution time (WCET) of each task that is implemented by a component is known. Our first goal is to find the smallest set of periods that make the task set schedulable by EDF or by RM. We call these periods safe periods because then any period assignment which is larger than the safe periods will be schedulable. Having safe periods, system designer will be able to use any optimization criteria of his choice to assign periods of the tasks without a need to incorporate the response time analysis techniques, which are usually NP-Hard problems, into his optimization constraints. Instead, the schedulability constraint will be reduced to verifying that the assigned periods are larger than the safe periods. Our solution for safe periods for the RM, is based on finding the smallest set of harmonic periods with utilization 1. Here we use our recently developed polynomial-time approximation algorithm with bounded error of 2 for finding these safe periods for RM. As the second part of our contribution, we consider the robustness property. First we explain how to find safe periods such that a certain level of robustness (based on potential changes of each component) is guaranteed. Doing this, if one of the components changes in the future, other components can still run using their previous periods, and hence, we isolate the effect of future changes from parameters of the other components. Finally, we obtain the robustness factor as a function of the available spare capacity of the system, i.e., the unused utilization. We determine to what extend the WCET of any of the components can be increased without violating the safe periods.

First-Order Methods for Fast Feasibility Pursuit of Non-convex QCQPs

Quadratically constrained quadratic programming (QCQP) is NP-hard in its general non-convex form, yet it frequently arises in various engineering applications. Several polynomial-time approximation algorithms exist for non-convex QCQP problems (QCQPs), but their success hinges upon the ability to find at least one feasible point—which is also hard for a general problem instance. In this paper, we present a heuristic framework for computing feasible points of general non-convex QCQPs using simple first-order methods. Our approach features low computational and memory requirements, which makes it well suited for application to large-scale problems. While a priori it may appear that these benefits come at the expense of technical sophistication, rendering our approach too simple to even merit consideration for a non-convex and NP-hard problem, we provide compelling empirical evidence to the contrary. Experiments on synthetic as well as real-world instances of non-convex QCQPs reveal the surprising effectiveness of first-order methods compared to more established and sophisticated alternatives.

Scheduling Non-Preemptible Jobs to Minimize Peak Demand

This paper examines an important problem in smart grid energy scheduling; peaks in power demand are proportionally more expensive to generate and provision for. The issue is exacerbated in local microgrids that do not benefit from the aggregate smoothing experienced by large grids. Demand-side scheduling can reduce these peaks by taking advantage of the fact that there is often flexibility in job start times. We focus attention on the case where the jobs are non-preemptible, meaning once started, they run to completion. The associated optimization problem is called the peak demand minimization problem, and has been previously shown to be NP-hard. Our results include an optimal fixed-parameter tractable algorithm, a polynomial-time approximation algorithm, as well as an effective heuristic that can also be used in an online setting of the problem. Simulation results show that these methods can reduce peak demand by up to 50% versus on-demand scheduling for household power jobs.

Maximizing Capacity in Cognitive Radio Networks Under Physical Interference Model

A fundamental problem in cognitive radio networks (CRN) is the following capacity maximization in CRN (CM-CRN) problem: given a set of primary links with a common transmitter, together with a set of secondary links, select a maximum cardinality subset of the links that can concurrently transmit successfully under the constraint that all primary links are selected. This problem is intrinsically different from the well-known link scheduling (LS) problem in wireless mesh networks, which does not have the constraint to select all primary links. In this paper, we make both theoretical and practical contributions to the CM-CRN problem. To achieve deep theoretical understanding of the problem, we show that CM-CRN is NP-hard and design a polynomial time approximation algorithm with a constant approximation ratio. In addition, we extend the designed algorithm to find approximate solutions to two variations of CM-CRN, one with the objective of maximizing the number of selected secondary links and the other with multiple primary users. To achieve good performance in practice, we design a simple but effective heuristic algorithm based on a greedy strategy. We also design an optimal algorithm based on integer linear programming, which serves as a benchmark for evaluating the performance of the approximation algorithm and heuristic algorithm, for problem instances of small sizes. Extensive evaluations show that our proved constant ratio of the approximation algorithm is considerably conservative and our heuristic algorithm produces results that are very close to the optimal solution. Our approximation algorithm for CM-CRN is motivated by and can be viewed as a non-trivial extension of the elegant approximation algorithm for the LS problem by Wan et al. to CRNs.

Approximation algorithms for optimization of real-valued general conjugate complex forms

Complex polynomial optimization has recently gained more attention in both theory and practice. In this paper, we study optimization of a real-valued general conjugate complex form over various popular constraint sets including the m-th roots of complex unity, the complex unit circle, and the complex unit sphere. A real-valued general conjugate complex form is a homogenous polynomial function of complex variables as well as their conjugates, and always takes real values. General conjugate form optimization is a wide class of complex polynomial optimization models, which include many homogenous polynomial optimization in the real domain with either discrete or continuous variables, and Hermitian quadratic form optimization as well as its higher degree extensions. All the problems under consideration are NP-hard in general and we focus on polynomial-time approximation algorithms with worst-case performance ratios. These approximation ratios improve previous results when restricting our problems to some special classes of complex polynomial optimization, and improve or equate previous results when restricting our problems to some special classes of polynomial optimization in the real domain. The algorithms are based on tensor relaxation and random sampling. Our novel technical contributions are to establish the first set of probability lower bounds for random sampling over the m-th root of unity, the complex unit circle, and the complex unit sphere, and to propose the first polarization formula linking general conjugate forms and complex multilinear forms. Some preliminary numerical experiments are conducted to show good performance of the proposed algorithms.

Burning a graph is hard

Graph burning is a model for the spread of social contagion. The burning number is a graph parameter associated with graph burning that measures the speed of the spread of contagion in a graph; the lower the burning number, the faster the contagion spreads. We prove that the corresponding graph decision problem is NP-complete when restricted to acyclic graphs with maximum degree three, spider graphs and path-forests. We provide polynomial time algorithms for finding the burning number of spider graphs and path-forests if the number of arms and components, respectively, are fixed. Finally, we describe a polynomial time approximation algorithm with approximation factor 3 for general graphs.

On fault-tolerant path optimization under QoS constraint in multi-channel wireless networks

In multi-channel wireless networks, a fundamental problem is to find node-disjoint paths minimising global cost or the maximum individual path cost, under the constraint that each path operates on a separate channel to maximise the fault tolerance and robustness against channel instability and malicious attacks. Meanwhile, the quality of service (QoS) requirement (e.g., in terms of end-to-end delay) needs to be satisfied on each path. In this paper, we provide a comprehensive formulation and analysis on this multi-path optimization problem by casting it to the problem k-disjoint path with different colours (k-DPDC). We further formulate the Restricted MinSum k-DPDC and Restricted MinMax k-DPDC to denote the problems of finding multiple node- and channel-disjoint paths minimising the global cost and the maximum individual path cost under the QoS constraint on the path end-to-end delay. Given the NP-hardness of both problems, we focus on directed acyclic graphs and propose fully polynomial-time approximation algorithms for both problems.

SkyGraph

Several services today are annotated with points of interest (PoIs) such as "coffee shop", "park", etc. A region of interest (RoI) is a neighborhood that contains PoIs relevant to the user. In this paper, we study the scenario where a user wants to identify the best RoI in a city. The user expresses relevance through a set of keywords denoting PoIs. Ideally, the RoI should be small enough in size such that the user can conveniently explore the PoIs. On the other hand, it should be as relevant as possible. How does one balance the importance of size versus relevance? To a user exploring the RoI on foot, size is more critical. However, for a user equipped with a vehicle, relevance is a more important factor. In this paper, we solve this dilemma through skyline subgraph queries on keyword-embedded road networks. Skyline subgraphs subsume the choice of optimization function for an RoI since the optimal RoI for any rational user is necessarily a part of the skyline set. Our analysis reveals that the problem of computing the skyline set is NP-hard. We overcome the computational bottleneck by proposing a polynomial-time approximation algorithm called SkyGraph. To further expedite the running time, we develop an index structure, Partner Index , that drastically prunes the search space and provides up to 3 orders of magnitude speed-up on real road networks over the baseline approach. The datasets and executables are available at http://www.cse.iitd.ac.in/~sayan/software.html.

Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials

We show a new way of constructing deterministic polynomial-time approximation algorithms for computing complex-valued evaluations of a large class of graph polynomials on bounded degree graphs. Our approach works for the Tutte polynomial, the independence polynomial, and partition functions of complex-valued spin- and edge-coloring models.

A generalized approximation framework for fractional network flow and packing problems

We generalize the fractional packing framework of Garg and Koenemann (SIAM J Comput 37(2):630–652, 2007) to the case of linear fractional packing problems over polyhedral cones. More precisely, we provide approximation algorithms for problems of the form $$\max \{c^T x : Ax \le b, x \in C \}$$ , where the matrix A contains no negative entries and C is a cone that is generated by a finite set S of non-negative vectors. While the cone is allowed to require an exponential-sized representation, we assume that we can access it via one of three types of oracles. For each of these oracles, we present positive results for the approximability of the packing problem. In contrast to other frameworks, the presented one allows the use of arbitrary linear objective functions and can be applied to a large class of packing problems without much effort. In particular, our framework instantly allows to derive fast and simple fully polynomial-time approximation algorithms (FPTASs) for a large set of network flow problems, such as budget-constrained versions of traditional network flows, multicommodity flows, or generalized flows. Some of these FPTASs represent the first ones of their kind, while others match existing results but offer a much simpler proof.

Two-stage no-wait hybrid flowshop scheduling with inter-stage flexibility

This paper addresses the performance of scheduling algorithms for a two-stage no-wait hybrid flowshop environment with inter-stage flexibility, where there exist several parallel machines at each stage. Each job, composed of two operations, must be processed from start to completion without any interruption either on or between the two stages. For each job, the total processing time of its two operations is fixed, and the stage-1 operation is divided into two sub-parts: an obligatory part and an optional part (which is to be determined by a solution), with a constraint that no optional part of a job can be processed in parallel with an idleness of any stage-2 machine. The objective is to minimize the makespan. We prove that even for the special case with only one machine at each stage, this problem is strongly NP-hard. For the case with one machine at stage 1 and m machines at stage 2, we propose two polynomial time approximation algorithms with worst case ratio of $$3-\frac{2}{m+1}$$ and $$2-\frac{1}{m+1}$$ , respectively. For the case with m machines at stage 1 and one machine at stage 2, we propose a polynomial time approximation algorithm with worst case ratio of 2. We also prove that all the worst case ratios are tight.

Covering Arrays on Product Graphs

Two vectors x, y in $$\mathbb {Z}_g^n$$ are qualitatively independent if for all pairs $$(a,b)\in \mathbb {Z}_g\times \mathbb {Z}_g$$ , there exists $$i\in \{1,2,\ldots ,n\}$$ such that $$(x_i,y_i)=(a,b)$$ . A covering array on a graph G, denoted by CA(n, G, g), is a $$|V(G)|\times n$$ array on $$\mathbb {Z}_g$$ with the property that any two rows which correspond to adjacent vertices in G are qualitatively independent. The number of columns in such array is called its size. Given a graph G, a covering array on G with minimum size is called optimal. Our primary concern in this paper is with constructions that make optimal covering arrays on large graphs that are obtained from product of smaller graphs. We consider four most extensively studied graph products in the literature and give upper and lower bounds on the size of covering arrays on product graphs. We find families of graphs for which the size of covering array on the Cartesian product graphs achieves the lower bound. Finally, we present a polynomial time approximation algorithm with approximation ratio $$\left\lceil \log (\frac{|V|}{2^{k-1}})\right\rceil $$ for constructing covering array on graph $$G=(V,E)$$ with $$k>1$$ prime factors with respect to the Cartesian product.

Approximation algorithms for nonnegative polynomial optimization problems over unit spheres

We consider approximation algorithms for nonnegative polynomial optimization problems over unit spheres. These optimization problems have wide applications e.g., in signal and image processing, high order statistics, and computer vision. Since these problems are NP-hard, we are interested in studying on approximation algorithms. In particular, we propose some polynomial-time approximation algorithms with new approximation bounds. In addition, based on these approximation algorithms, some efficient algorithms are presented and numerical results are reported to show the efficiency of our proposed algorithms.

Approximation algorithms for the scaffolding problem and its generalizations

Scaffolding is one of the main stages in genome assembly. During this stage, we want to merge contigs assembled from the paired-end reads into bigger chains called scaffolds. For this purpose, the following graph-theoretical problem has been proposed: Given an edge-weighted complete graph G and a perfect matching D of G, we wish to find a Hamiltonian path P in G such that all edges of D appear in P and the total weight of edges in P but not in D is maximized. This problem is NP-hard and the previously best polynomial-time approximation algorithm for it achieves a ratio of 12. In this paper, we design a new polynomial-time approximation algorithm achieving a ratio of 5−5ϵ9−8ϵ for any constant 0<ϵ<1. Several generalizations of the problem have also been introduced in the literature and we present polynomial-time approximation algorithms for them that achieve better approximation ratios than the previous bests. In particular, one of the algorithms answers an open question.

Scheduling deteriorating jobs with machine availability constraints to minimize the total completion time

In this study, we explored the scheduling problems with non-resumable deteriorating jobs and machine availability constraints. It is assumed that jobs have a different basic processing time and same deterioration rate, and the machines may be unavailable in some specified time intervals. The objective is to minimize the total completion time. We presented a pseudo-polynomial dynamic programming algorithm and a fully polynomial-time approximation scheme for the single machine version. For the parallel machine version, we first presented a pseudo-polynomial dynamic programming algorithm and then showed that there exists no polynomial time approximation algorithm with a constant worst-case bound unless P = NP.

Adversarial patrolling with spatially uncertain alarm signals

When securing complex infrastructures or large environments, constant surveillance of every area is not affordable. To cope with this issue, a common countermeasure is the usage of cheap but wide-ranged sensors, able to detect suspicious events that occur in large areas, supporting patrollers to improve the effectiveness of their strategies. However, such sensors are commonly affected by uncertainty. In the present paper, we focus on spatially uncertain alarm signals. That is, the alarm system is able to detect an attack but it is uncertain on the exact position where the attack is taking place. This is common when the area to be secured is wide, such as in border patrolling and fair site surveillance. We propose, to the best of our knowledge, the first Patrolling Security Game where a Defender is supported by a spatially uncertain alarm system, which non-deterministically generates signals once a target is under attack. We show that finding the optimal strategy is FNP-hard even in tree graphs and APX-hard in arbitrary graphs. We provide two (exponential time) exact algorithms and two (polynomial time) approximation algorithms. Finally, we show that, without false positives and missed detections, the best patrolling strategy reduces to stay in a place, wait for a signal, and respond to it at best. This strategy is optimal even with non-negligible missed detection rates, which, unfortunately, affect every commercial alarm system. We evaluate our methods in simulation, assessing both quantitative and qualitative aspects.

On the most imbalanced orientation of a graph

We study the problem of orienting the edges of a graph such that the minimum over all the vertices of the absolute difference between the outdegree and the indegree of a vertex is maximized. We call this minimum the imbalance of the orientation, i.e. the higher it gets, the more imbalanced the orientation is. The studied problem is denoted by $${{\mathrm{\textsc {MaxIm}}}}$$ . We first characterize graphs for which the optimal objective value of $${{\mathrm{\textsc {MaxIm}}}}$$ is zero. Next we show that $${{\mathrm{\textsc {MaxIm}}}}$$ is generally NP-hard and cannot be approximated within a ratio of $$\frac{1}{2}+\varepsilon $$ for any constant $$\varepsilon >0$$ in polynomial time unless $$\texttt {P}=\texttt {NP}$$ even if the minimum degree of the graph $$\delta $$ equals 2. Then we describe a polynomial-time approximation algorithm whose ratio is almost equal to $$\frac{1}{2}$$ . An exact polynomial-time algorithm is also derived for cacti. Finally, two mixed integer linear programming formulations are presented. Several valid inequalities are exhibited with the related separation algorithms. The performance of the strengthened formulations is assessed through several numerical experiments.

Algorithms for Max-Min Share Fair Allocation of Indivisible Chores

We consider Max-min Share (MmS) fair allocations of indivisible chores (items with negative utilities). We show that allocation of chores and classical allocation of goods (items with positive utilities) have some fundamental connections but also differences which prevent a straightforward application of algorithms for goods in the chores setting and vice-versa. We prove that an MmS allocation does not need to exist for chores and computing an MmS allocation - if it exists - is strongly NP-hard. In view of these non-existence and complexity results, we present a polynomial-time 2-approximation algorithm for MmS fairness for chores. We then introduce a new fairness concept called optimal MmS that represents the best possible allocation in terms of MmS that is guaranteed to exist. We use connections to parallel machine scheduling to give (1) a polynomial-time approximation scheme for computing an optimal MmS allocation when the number of agents is fixed and (2) an effective and efficient heuristic with an ex-post worst-case analysis.

Competitive algorithms for multistage online scheduling

We study an online flow shop scheduling problem where each job consists of several tasks that have to be completed in t different stages and the goal is to maximize the total weight of accepted jobs. The set of tasks of a job contains one task for each stage and each stage has a dedicated set of identical parallel machines corresponding to it that can only process tasks of this stage. In order to gain the weight (profit) associated with a job j, each of its tasks has to be executed between a task-specific release date and deadline subject to the constraint that all tasks of job j from stages 1,⋯,i−1 have to be completed before the task of the ith stage can be started. In the online version, jobs arrive over time and all information about the tasks of a job becomes available at the release date of its first task. This model can be used to describe production processes in supply chains when customer orders arrive online.Even the basic version of the offline problem with a single machine in each stage, unit weights, unit processing times, and fixed execution times for all tasks (i.e., deadline minus release date equals processing time) is APX-hard and we show that the approximation ratio of any polynomial-time approximation algorithm for this basic version of the problem must depend on the number t of stages.For the online version of the basic problem, we provide a (2t−1)-competitive deterministic online algorithm and a matching lower bound. Moreover, we provide several (sometimes tight) upper and lower bounds on the competitive ratios of online algorithms for several generalizations of the basic problem involving different weights, arbitrary release dates and deadlines, different processing times of tasks, and several identical machines per stage.

An Approximation Algorithm for the Maximum Traveling Salesman Problem

The question considered in the travelling salesman problem is the following. For a given list of cities and the distances between each pair of cities, to construct the shortest possible path that visits each city exactly once and returns to the origin city. The problem is an NP-hard problem in combinatorial optimization and is important in operations research. The traveling salesman problem is one of the most famous and heavily researched problems in theoretical computer science. We consider the version, which is the Symmetric Maximum Traveling Salesman Problem. The article describes an approximation algorithm for the maximum traveling salesman problem, based on two known polynomial time approximation algorithms. The accuracy of this algorithm is 25/33.