A Fast Parallel Implementation of a PTAS for Fractional Packing and Covering Linear Programs

Abstract

We present a parallel implementation of the randomized $(1+\varepsilon )$ approximation algorithm for packing and covering linear programs presented by Koufogiannakis and Young (2007). Their approach builds on ideas of the sublinear time algorithm of Grigoriadis and Khachiyan’s (Oper Res Lett 18(2):53–58, 1995) and Garg and Könemann’s (SIAM J Comput 37(2):630–652, 2007) non-uniform-increment amortization scheme. With high probability it computes a feasible primal and dual solution whose costs are within a factor of $1+\varepsilon$ of the optimal cost. In order to make their algorithm more parallelizable we also implemented a deterministic version of the algorithm, i.e. instead of updating a single random entry at each iteration we updated deterministically many entries at once. This slowed down a single iteration of the algorithm but allowed for larger step-sizes which lead to fewer iterations. We use NVIDIA’s parallel computing architecture CUDA for the parallel environment. We report a speedup between one and two orders of magnitude over the times reported by Koufogiannakis and Young (2007).

Appendices

Appendix 1: Evaluation of the Randomized Parallel Algorithm

Measurements in Tables 6 and 7 are done on 0/1 and rational matrices, respectively, where $m$ and $n$ denote the number of rows and columns, respectively, and $d$ denotes the density, i.e. the probability of 1 is $1/2^d$. Moreover, *_pp is time for preprocessing, *_cp denotes time for copying data structures from host to device and back, *_comp is time for the main while loop, *_pdg is the primal-dual gap, and ALG1, ALG2 and GUR denotes the total running time for three different approaches, namely Algorithms 1, 2 and GUROBI’s barrier method.

Table 6 Experiments on $0-1$ matrices

Table 7 Experiments on rational matrices

Appendix 2: Evaluation of the Derandomized Parallel Algorithm

Measurements in Tables 8 and 9 are done on 0/1 and rational matrices, respectively, where $m$ and $n$ denote the number of rows and columns, respectively, and $d$ denotes the density, i.e. the probability of 1 is $1/2^d$. Moreover, *_numiter is the number of iterations, *_pdg is the primal-dual gap, and ALG1, ALG3 and GUR denotes the total running time for three different approaches, namely Algorithms 1, 3 and GUROBI’s barrier method.

Table 8 Experiments on $0-1$ matrices

Table 9 Experiments on rational matrices

Appendix 3: Terrain Guarding Instances

In this section we will explain all the steps needed to generate arbitrary terrain guarding instances. First, we introduce some technicalities. For two points $p=(p_x,p_y)$ and $q=(q_x,q_y)$ in ${\mathbb {R}}^2$ we say that $q$ is to the left (right) of $p$, which is denoted by $q\le p$ ($q\ge p$), if $q_x\le p_x$ ($q_x\ge p_x$). Similarly, we say that $q$ is strictly to the left (right) of $p$ if $q_x < p_x$ ($q_x > p_x$), which is denoted by $q<p$ ($q>p$). Let $T$ be a polygonal line that is determined by points $P_1P_2\ldots P_n$, where $P_i=(x_i,y_i)\in {\mathbb {R}}^2$ for $i=1,\ldots ,n$. We say that $T$ is a $1.5D$ terrain of complexity $n$ if $T$ i $x$-monotone, i.e. $P_i<P_j$ for all $1\le i<j\le n$. We say that two points $p$ and $q$ on the $1.5D$ terrain $T$ are visible by each other, i.e. $p\sim q$, if the line segment $\overline{pq}$ does not intersect the terrain $T$. Otherwise, we say that $p$ and $q$ are not visible by each other, i.e. $p \not \sim q$.

For a given $1.5D$ terrain $T$ and a set of guards $g\in G\subset T$ and a set of clients $p\in N\subset T$, we label guards in $G$ by $1,\ldots ,n$ and clients in $N$ by $1,\ldots , m$. An incidence between guards $G$ and clients $N$ can be described with the matrix $A$ whose elements are given by:

$$\begin{aligned} A_{ij}= {\left\{ \begin{array}{ll} \mathtt 1 , &{} j\sim i \\ \mathtt 0 , &{} j \not \sim i \end{array}\right. } \end{aligned}$$

(20)

Visibility between $g\in G$ and $p\in N$ can be efficiently computed by determining the sign of the determinant that is formed by the vectors $p, v$ and $q$ (Fig. 8 ).

Fig. 8

Since $p\sim q$ the area of the triangle $\varDelta _{gvp}$ is positive, for all $v\in T$ such that $g<v<p$. On the other hand, since $p\not \sim g'$ there mush exist some $v'\in T, p<v'<g'$, such that the area of the triangle $\varDelta _{pv'g'}$ is negative. Note that such $v'$ can be easily determined by inspecting appropriate vertices of $T$

Our goal is to find the subset $X\subset G$ with minimum number of guards such that all clients in $N$ are visible by at least one guard in $X$. This problem can be formulated as a set cover problem. An integer programming formulation is given by

$$\begin{aligned} \min _{x\in \{0,1\}^{n}}\{x:Ax\ge {\mathbf {1}}\}. \end{aligned}$$

(21)

where matrix $A$ is defined in (20). A relaxation of the integrality constraints in (21) gives a fractional covering problem

$$\begin{aligned} \min _{x\in {\mathbb {R}}^{n}}\{x:Ax\ge {\mathbf {1}}, x\ge 0\}. \end{aligned}$$