Concise integer linear programming formulation for clique partitioning problems

We consider the placement of FPGA designs with multiple I/O standards on modern FPGAs that support multiple I/O standards. We propose an efficient approach to solve the constrained I/O placement problem by 0-1 integer linear programming within a high performance placement flow. We derive an elegant 0-1 integer linear program formulation which is applicable not only for devices with symmetric I/O banks but also for devices with asymmetric I/O banks (i.e., different banks may have different sizes and/or support different subsets of I/O standards). Moreover, it is capable of handling user's prelocked I/Os. We also show that additional restrictions such as conditional usage of V/sub ref/ pins can be easily incorporated. Our formulation involves only a small number of 0-1 integer variables independent of the device size or the number of I/O objects, hence our approach can comfortably handle very large problem instances. Extensive experimentation showed that the 0-1 integer linear program corresponding to a feasible instance of the constrained I/O placement problem can be solved in seconds.

Integer Linear Programming (ILP) is commonly used in high level and system level synthesis. It is an NP-Complete problem (in general cases). There exists some tools that give an optimal solution for small ILP formulations. Nevertheless, these tools may not give solutions for complex formulations. In this paper, we present a solution to overcome the problem of complexity in ILP formulations. We propose a partitioning methodology based on a constraint graph representing all the constraints included in any ILP formulation. To direct the partitioning, the constraint graph nodes are grouped to represent Data Flow Graph (DFG) nodes. This reduced constraint graph can be used to partition any ILP formulation based on DFG. We illustrate this method on ILP formulation for scheduling. Results on ILP scheduling formulations are promising.

In this paper we propose new simple integer linear programs (ILPs) formulations for minimizing capacity (in wavelength link) utilization in survivable WDM network. The study examines the performance of shared based protection schemes, such as path protection scheme and link protection scheme under single fiber failure. The numerical results obtained show a reduction in capacity utilization using random traffic compared to the reported ILP formulation. We also present the results using Poisson’s traffic to identify the frequently used links for the widely used NSF network. The proposed work not only reduces the wavelength consumption in different traffic scenarios but also efficient in terms of simulation time. In this paper we propose new simple integer linear programs (ILPs) formulations for minimizing capacity (in wavelength link) utilization in survivable WDM network. The study examines the performance of shared based protection schemes, such as path protection scheme and link protection scheme under single fiber failure. The numerical results obtained show a reduction in capacity utilization using random traffic compared to the reported ILP formulation. We also present the results using Poisson’s traffic to identify the frequently used links for the widely used NSF network. The proposed work not only reduces the wavelength consumption in different traffic scenarios but also efficient in terms of simulation time.

Integer Linear Programming (ILP) can be seen as the archetypical problem for NP-complete optimization problems, and a wide range of problems in artificial intelligence are solved in practice via a translation to ILP. Despite its huge range of applications, only few tractable fragments of ILP are known, probably the most prominent of which is based on the notion of total unimodularity. Using entirely different techniques, we identify new tractable fragments of ILP by studying structural parameterizations of the constraint matrix within the framework of parameterized complexity. In particular, we show that ILP is fixed-parameter tractable when parameterized by the treedepth of the constraint matrix and the maximum absolute value of any coefficient occurring in the ILP instance. Together with matching hardness results for the more general parameter treewidth, we draw a detailed complexity landscape of ILP w.r.t. decompositional parameters defined on the constraint matrix.

The Fiber Installation Problem (FIP) in Wavelength Division Multiplexing (WDM) optical networks consists in routing a set of lightpaths (all-optical connections) such that the cost of the optical devices necessary to operate the network is minimized. Each of these devices is worth hundreds of thousands of dollars. In consequence, any improvement in the lightpath routing may save millions of dollars for the network operator. All the works in the literature for solving this problem are based on greedy heuristics and genetic algorithms. No information is known on how good are the solutions provided by these heuristics compared to the optimal solution. Besides, no proof that the problem is NP-Hard can be found. In this paper, we prove that FIP is NP-Hard and also present an Integer Linear Programming (ILP) formulation for the problem. In addition, we propose an implementation of the Iterated Local Search (ILS) metaheuristic to solve large instances of the problem. Computational experiments carried out on 21 realistic instances showed that the CPLEX solver running with our ILP formulation was able to solve 11 out of the 21 instances to optimality in less than two minutes. These results also showed that the ILS heuristic has an average optimality gap of 1% on the instances for which the optimal solution is known. For the other instances, the results showed that the proposed heuristic outperforms the best heuristic in the literature by 7%.

Recently, a biochip called MEDA (Micro Electrode Dot Array) has been studied in the field of biochemistry. During conducting experiments using a MEDA biochip, we need to move a droplet to a target position. We can split a droplet to make it smaller, and/or change its shape to go through a narrow path on a MEDA biochip. Because the velocity of a droplet is different depending on its shape and the direction of the movement, it should be complex to find an optimal routing for a droplet (i.e., to determine where to split the droplet, which path should be taken, etc.). Thus, existing methods utilize ILP (Integer Linear Programming) formulation to find an optimal routing; existing methods cannot treat large-scale problems because ILP formulation is inherently an exponential algorithm to the problem size. This paper proposes a completely new approach from the ILP formulation; our method constructs a weighted graph so that we can utilize Dijkstra's algorithm to find an optimal routing in polynomial time with respect to the chip size. Our experimental result confirms that the proposed method reduces computation time by more than 99% compared to existing ILP-based methods, and we can now find an optimal routing for a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$100 \times 100$</tex>-sized chip, which is totally impossible by existing ILP-based methods.

Digital micro fluidic biochips have become one of the most promising technologies in many biomedical fields. In a digital micro fluidic biochip, a group of (adjacent) cells in the micro fluidic array can be configured to work as storage, functional operations, as well as for transporting fluid droplets dynamically. There has been proposed a very efficient Integer Linear Programming (ILP)-based method to schedule operations with considering this dynamic reconfigurability. However, the method cannot directly deal with a case having an operation with multiple successors, which is necessary in some bioassays, e.g., Sample preparation applications. Sample preparation is a crucial preprocessing step to produce droplets of the desired target concentrations from input reagents. In every optimized application graphs for sample preparation, there are many operations with more than one successors. Thus, the previous method cannot be used to get an optimal scheduling, and thus we propose a new framework to use another ILP formulation for scheduling operations in a digital micro fluidic biochips. Our formulation can successfully deal with a case where there is a node with multiple successors. Also, our method can find a binding solution at the same time unlike the above-mentioned previous one thanks to our new ILP formulation. Although our ILP formulation seems to be a bit more complex than the previous one, a preliminary experiment demonstrates that the proposed ILP formulation can successfully find the optimal scheduling for practical sample preparation applications within a reasonable time.

This paper presents a technique for memory optimization for a class of computations that arises in the field of correlated electronic structure methods such as coupled cluster and configuration interaction methods in quantum chemistry. In this class of computations, loop computations perform a multi-dimensional sum of product of input arrays. There are many different ways to get the same final results that differ in the required number of arithmetic operations required. In addition, for a given number of arithmetic operations, different expressions of the loop have different memory requirements. Loop fusion is a plausible solution for reducing memory usage. By fusing loops between producer loop nest and consumer loop nest, the required storage of intermediate array is reduced by the range of the fused loop. Because resultant loops have to be legal after fusion, some loops can not be fused at the same time. In this paper, we have developed a novel integer linear programming (ILP) formulation that is shown to be highly effective on a number of test cases producing the optimal solutions using very small execution times. The main idea in the ILP formulation is the encoding of legality rules for loop fusion of a special class of loops using logical constraints over binary decision variables and a highly effective approximation of memory usage

Summary form only given. We present a way to automatically select, within an architectural synthesis tool, the best operand and operator number systems, in order to find the best speed/area tradeoff. This implies the use of different number systems (redundant and non-redundant) for the same design: this is what we call mixed arithmetics. We present an integer linear programming (ILP) formulation to solve a scheduling problem.

In general, multicast routing and wavelength assignment (MC-RWA) can be subdivided in routing and wavelength assignment issues in wavelength-division multiplexing (WDM) mesh networks. Previous studies on WDM multicast have mainly focused on WDM multicast routing. The multicast wavelength assignment problem is studied in this paper. A unicast routing path can be established by a lightpath in an all-optical network. However, in the multicasting case, a multicast routing tree can be established by a single light-tree or several lightpaths, or a combination of several light-trees and lightpaths. We propose a wavelength assignment algorithm for finding an optimal combination of lightpaths and light-trees to construct a newly required multicast session. First of all, two cost functions are given to evaluate the establishing cost for each feasible wavelength, and then find a set of wavelengths that covers all destinations with the minimal cost using Integer Linear Programming (ILP) formulation. We focus on maximizing the total number of users served in a multicast session and the network capacity. The simulation results show that the proposed algorithm can improve system resource utilization and reduce the blocking probability compared with the First-Fit algorithm.

Coarse-grained reconfigurable array (CGRA) overlays can improve dataflow kernel throughput by an order of magnitude over Vivado HLS. This is possible with a combination of carefully floorplanned high-frequency (645-768 MHz) design and a scalable, communication-aware compiler. Our 2D torus CGRA architecture supports versatile processing element (PE) functionality and a configurable number of communication channels to match application demands. Compared to recent FPGA overlays like CGRA-ME's ADRES and HyCUBE implementations, our design operates at 1.8-3.4× faster clock frequency, while scaling to an orders-of-magnitude larger array size of 19×69 on Xilinx Alveo U280. Our communication-aware compiler targets HLS objectives such as initiation interval (II) and minimizes communication cost using an integer linear programming (ILP) formulation. Unlike SDC schedulers in FPGA HLS tools, we treat data movement as a first-class citizen by encoding the space and time resources communication network of the overlay in the ILP formulation. Given the same constraints on operational resources as Vivado HLS, we can retain our target II and achieve up to 9.2× higher frequency. Our ILP scheduler outperforms a PathFinder space-time router implementation in quality of result by up to nearly 2×.

Recently a number of algorithmic results have appeared which show the tractability of Integer Linear Programming (ILP) instances under strong restrictions on variable domains and/or coefficients (AAAI 2016, AAAI 2017, IJCAI 2017). In this paper, we target ILPs where neither the variable domains nor the coefficients are restricted by a fixed constant or parameter; instead, we only require that our instances can be encoded in unary. We provide new algorithms and lower bounds for such ILPs by exploiting the structure of their variable interactions, represented as a graph. Our first set of results focuses on solving ILP instances through the use of a graph parameter called clique-width, which can be seen as an extension of treewidth which also captures well-structured dense graphs. In particular, we obtain a polynomial-time algorithm for instances of bounded clique-width whose domain and coefficients are polynomially bounded by the input size, and we complement this positive result by a number of algorithmic lower bounds. Afterwards, we turn our attention to ILPs with acyclic variable interactions. In this setting, we obtain a complexity map for the problem with respect to the graph representation used and restrictions on the encoding.

The complexity landscape of decompositional parameters for ILP

This paper addresses the problem of counting models in integer linear programming (ILP) using Boolean Satisfiability (SAT) techniques, and proposes two approaches to solve this problem. The first approach consists of encoding ILP instances into pseudo-Boolean (PB) instances. Moreover, the paper introduces a model counter for PB constraints, which can be used for counting models in PB as well as in ILP. A second alternative approach consists of encoding instances of ILP into instances of SAT. A two-step procedure is proposed, consisting of first mapping the ILP instance into PB constraints and then encoding the PB constraints into SAT. One key observation is that not all existing PB to SAT encodings can be used for counting models. The paper provides conditions for PB to SAT encodings that can be safely used for model counting, and proves that some of the existing encodings are safe for model counting while others are not. Finally, the paper provides experimental results, comparing the PB and SAT approaches, as well as existing alternative solutions.

The Prize-collecting Scheduling Problem with Deadlines