Minimum power partial multi-cover on a line

Highlights

• We studied the minimum power partial multi-cover problem on a line (MinPowPMC-Line).

• We show that MinPowMC-Line (without partial cover requirement) is polynomial-time solvable.

• A dynamic programming is presented which runs in polynomial time when the maximum covering requirement is a constant.

Abstract

This paper studies the minimum power partial multi-cover problem on a line (MinPowPMC-Line), the goal of which is to find an assignment of powers to sensors such that at least a required number of points are covered up to their covering requirements. We first present an LP method to show that the minimum power multi-cover problem on a line (without partial covering requirement) is solvable in polynomial time. But this method no longer works when facing partial covering requirement. We turn to dynamic programming method to find an optimal solution for MinPowPMC-Line in time $O(n^4{m}^{1+2(c{r}_{max})})$, where $n,m$ are the number of points and the number of sensors, respectively, and $c{r}_{max}$ denotes the maximum covering requirement of elements. So, this problem is polynomial-time solvable when $c{r}_{\max }$ is upper bounded by a constant.

Introduction

Wireless sensor network has been used widely in real-world applications such as traffic control, environment monitoring. One of the most fundamental task of wireless sensor network is to provide coverage of the deployment region. In many applications when sensors are deployed on the ground, the deployment region can be abstracted as a plane, and the sensing region of a sensor s is regarded as a disk $D_s$ centered at s with radius $r_s$, where $r_s$ is the sensing radius of s. A point p is said to be covered by sensor s if $p\in D_s$.

When a sensor can adjust its power, its sensing range is changed accordingly. The sensing radius $r_s$ of sensor s is determined by its power $p_s$ as follows: $p_s=\alpha r_s^{\kappa }$, where κ is the attenuation factor of power and α is a constant. In a minimum power cover problem (MinPowC), the goal is to find an assignment of powers to sensors such that all points are covered and the total power consumed is as small as possible.

Suppose that power can be adjusted continuously. Then there are uncountable number of powers to be considered. However, a MinPowC instance can be discretized as a special case of the minimum weight disk cover problem (MinWDC), the goal of which is to select a set of disks of the smallest total weight to cover all points. This can be observed by noticing that in an optimal solution to a MinPowC instance, every sensor s with positive power must have a point on its boundary (otherwise, one can shrink $D_s$ until some point touches its boundary, which will not affect feasibility but reduce the total power). So, if $\mathcal{P}$ is the set of points, $\mathcal{S}$ is the set of sensors, we may consider a disk set $\mathcal{D}=\{D_{s,p}:s\in \mathcal{S},p\in \mathcal{P}\}$, where $D_{s,p}$ is the disk with center s and radius $\Vert sp\Vert$ (the Euclidean distance between s and p), and let the weight $w_{D_{s,p}}=\alpha {\Vert sp\Vert }^{\kappa }$. Assigning powers to sensors is equivalent to selecting disks from $\mathcal{D}$, and the total power equals the total weight of selected disks.

More abstractly, a MinWDC problem is a special case the minimum weight set cover problem (MinWSC), the goal of which is to select the minimum weight collection of sets to cover all elements. If every set is the set of points covered by a disk, then it is the MinWDC problem. Because of the geometry possessed by the MinWDC problem, it admits much better approximation than the general MinWSC problem. In fact, the best approximation ratio for MinWSC is $\ln \mathrm{\Delta }$ where Δ is the size of the largest set, and the ratio is tight unless $P=NP$. While MinWDC admits constant approximation [9]. Similarly, the speciality of MinPowC also leads to better approximation than MinWDC. In fact, Bilò et al. [7] presented a PTAS for MinPowC.

There are a lot of variants of the coverage problem, including the multi-cover problem and partial cover problem. In a multi-cover problem, it is required that some elements are covered more than once. Such a requirement is to guarantee fault-tolerance of monitoring. In a partial cover problem, it is only required to cover a partial number of elements. Such a requirement arises because it often happens in the real world that a one-hundred percentage of perfection may cost too much resources, while a reasonable percentage of feasibility is already satisfactorily good. Combining these two requirements, we have the following problem.

Definition 1.1 Minimum power partial multi-cover problem (MinPowPMC)

Suppose $\mathcal{P}$ is a set of points and $\mathcal{S}$ is a set of sensors on the plane, each point $p\in \mathcal{P}$ has a covering requirement $c{r}_p\in {\mathbb{Z}}^{+}$. Given a quota $q\le |\mathcal{P}|$, the goal of MinPowPMC is to assign powers to the sensors such that the total power is as small as possible under the condition that at least q points are fully covered, where a point p is fully covered if p belongs to at least $c{r}_p$ disks determined by the power assignment.

Similar to the above argument, we shall show that the MinPowPMC problem can be transformed into a series of minimum weight disk partial multi-cover problem (MinWDPMC). A question is: can the speciality of the MinPowPMC problem bring better solution than the MinWDPMC problem?

This paper considers the MinPowPMC problem on a line, that is, both the points and the sensors are deployed on a line. Denote such a problem as MinPowPMC-Line. Such a setting is not rare in the real world. For example, when considering monitoring traffic on a highway, or the boundary of a seaside, the deployment region can be regarded as a line.

The minimum power partial multi-cover problem (MinPowPMC) is a combination of the minimum power (full) multi-cover problem (MinPowMC) and the minimum power partial (single) cover problem (MinPowPC).

For the uniform-MinPowMC problem, that is, when all points have the same covering requirement k, Abu-Affash et al. [1] considered the special case of $\kappa =2$, and gave an $O(k)$-approximation. Bar-Yehuda and Rawitz [3] gave an algorithm that achieves approximation factor $O(k)$ for any κ. Bhowmick et al. [5] gave an $O(1)$-approximation for $\kappa =2$, thus obtaining a guarantee that is independent of the covering requirement. Notice that this result holds for the MinPowMC problem in which different points may have different covering requirements. The algorithm in [5] was generalized in [6] to obtain an approximation guarantee of $4\cdot {(27\sqrt{2})}^{\kappa }$ for any $\kappa \ge 1$, and then further generalized to any metric space in [4] to obtain an approximation ratio of $2\cdot {(16\cdot 9)}^{\kappa }$.

As far as we know, there are only two previous papers studying MinPowPC, that is, $c{r}_p=1$ for all points p. One is [11], in which Freund and Rawitz obtained a $(12+\epsilon )$-approximation for the special case of $\kappa =2$, where ε is an arbitrary constant greater than zero. The other is [19], in which Li et al. proposed a primal-dual algorithm to obtain approximation ratio $3^{\kappa }$ for any κ.

Minimum power cover problems are special minimum disk cover problems (MinDC).

For the cardinality version of MinDC, the goal is to find the minimum number of disks covering all points, Hochbaum [13] proposed a powerful technique called partition and shifting, which yields a PTAS for the special case when the disks are uniform and can be placed anywhere on the plane. For a general case in which disks may have different sizes and their locations are prefixed, Mustafa and Ray [21] proposed a local search method to obtain a PTAS. This PTAS was generalized by Roy et al. [24] to non-piercing regions including pseudo-disks.

Considering weight, Varadarajan [27] presented a clever quasi-uniform sampling technique, which was improved by Chan et al. [9] to yield a constant approximation for the minimum weight disk cover problem (MinWDC). This method was further generalized by Bansal and Pruhs [2] to obtain a constant approximation for the minimum weight disk multi-cover problem (MinWDMC) in which every point has to be covered multiple times. Using a separator framework, Mustafa et al. [20] obtained a quasi-PTAS for MinWDC.

To our knowledge, there are two papers explicitly claiming to be dedicated to the geometric minimum partial set cover problem. The first paper is [12], in which Gandhi et al. presented a PTAS for the minimum (cardinality) partial unit disk cover problem under the assumption that a unit disk can be placed anywhere on the plane. Another paper is [15] in which Inamdar and Varadarajan obtained a constant approximation for the minimum weight partial disk cover problem (MinWPDC). In fact, their approximation ratio is $2(\beta +1)$, where β is the integrality gap of the natural linear program for the minimum weight (full) set cover problem. Neither the algorithm nor the analysis use any geometry explicitly, the only place where geometry is used is that “since MinWDC has a constant approximation by [9], so is its partial version MinWPDC”. So, ratio $2(\beta +1)$ holds for general partial set cover problem. Chandra et al. [10] further improved this ratio to $\frac{e}{e-1}(\beta +1)$.

The MinPowPMC problem is a special case of the minimum weight partial set multi-cover problem (MinWPSMC). In a general setting, MinWPSMC is very hard, which cannot be approximated within factor $O(n^{\frac{1}{2{(\log \log n)}^c}})$ under the ETH assumption [22]. For the positive side, Ran et al. [23] proposed an approximation algorithm with ratio $b+\sqrt{n\cdot b}$, where b is a parameter related with frequency of elements and n is the number of elements. Shi et al. [26] designed a randomized $(b/q\epsilon ,1-\epsilon )$-bicriteria algorithm where q is the percentage of elements required to be fully covered and the feasibility might be violated by a factor of $1-\epsilon$.

As far as we know, there is no work studying the minimum power cover problem combining both the multi-cover requirement and partial cover requirement. And its generalization, namely the MinWPSMC problem, is very hard. So, the question is whether special setting might make the situation better? As a starting step, we consider the special setting when both points and sensors are deployed on a line.

We discretize a MinPowPMC instance as a minimum weight group disk partial multi-cover instance (MinWGDPMC), in which disks are partitioned into groups, and at most one disk can be picked from each group. In order to avoid choosing more than one disk from a group, we further transform the MinWGDPMC instance into $|\mathcal{P}||\mathcal{S}|$ unconstrained MinWDPMC instances, where every group has size one. Such a transformation enables us to focus on algorithms for unconstrained MinWDPMC-Line.

Using totally unimodular theory, we show that unconstrained MinWDMC-Line (without partial cover requirement) can be solved in polynomial time. It should be noticed that a direct implementation of totally unimodular theory does not work on a natural linear program for MinWDMC-Line, for which we shall give a counter example. When adding partial cover requirement, totally unimodular theory does not work too, which will be supported by another counter example. Hence, we continue to propose a dynamic programming method for unconstrained MinWDPMC-Line with running time $O(|\mathcal{P}{|}^3|\mathcal{S}{|}^{2c{r}_{max}})$, where $c{r}_{\max }=\max \{c{r}_p:p\in \mathcal{P}\}$ is the maximum covering requirement of a point.

The remaining part of this paper is organized as follows. Sect. 2 uses totally unimodular theory to solve the MinPowMC-Line problem. Dynamic program methods are given in Sect. 3. For an easier understanding of the ideas, we first present the dynamic program algorithm for MinPowMC-Line, and then generalize it to the MinPowPMC-Line problem. Strict analysis on correctness and time complexity is given. Section 4 concludes the paper.