Integer Linear Programming Instance Research Articles

Automatic Cell Placement for Josephson Transmission Lines in a Cell-Based Layout Design Environment for RSFQ Circuits

Abstract This study proposes a method for automatically tiling appropriate Josephson transmission line (JTL) cells for a given set of paths. Digital RSFQ circuits have been commonly designed in a cell-based design environment. The RSFQ cell library contains pre-designed fragments of JTLs as cells. Appropriate JTL cells are selected and tiled. This method formulates the problem as an instance of integer linear programming (ILP). Subsequently, instances are solved using a solver program. The method generates the tiling of cells according to the solver result. The number of Josephson junctions for JTLs, delay of paths, or weighted sum of such parameters is represented as a linear function and can be optimized. The proposed method performed JTL cell-tiling for several circuits, including a full adder. The runtime was within one second.

A branch-and-price approach to a variant of the cognitive radio resource allocation problem

Radio-frequency portion of the electromagnetic spectrum is a scarce resource. Cognitive radio (CR) technology has emerged as a promising solution to overcome the spectrum scarcity bottleneck. Through this technology, secondary users (SUs) sense the spectrum opportunities free from primary users (PUs), and opportunistically take advantage of these (temporarily) idle portions, known as spectrum holes. In this correspondence, we consider a variant of the cognitive radio resource allocation problem posed by Martinovic et al. in 2017. The distinguishing feature of this version of the problem is that each SU, due to its hardware limitations, imposes the requirement that the to-be-aggregated spectrum holes cannot be arbitrarily far from each other. We call this restriction as the Maximal Aggregation Range (MAR) constraint, and refer to this variant of the problem as the MAR-constrained hole assignment problem. The problem can be formalized as an NP-hard combinatorial optimization problem. We propose a novel binary integer linear programming (ILP) formulation to the problem. The number of constraints in this formulation is the number of spectrum holes plus the number of SUs. On the other hand, the number of binary decision variables in the formulation can be prohibitively large, as for each legitimate spectrum allocation to each SU, one variable is needed. Due to this difficulty, the resulting 0-1 ILP instances may not even be explicitly describable, let alone be solvable by an ILP solver. We propose a branch-and-price (B&P) framework to tackle this challenge. This framework is in fact a branch-and-bound procedure in which at each node of the search tree, we utilize the so-called (delayed) column generation technique for solving the LP relaxation of the corresponding subproblem. As evidenced by the numerical results, the LP relaxation bounds are very tight. This allows for a very effective pruning of the search space. Compared to the previously suggested formulations, the proposed technique can require much less computational effort.

Concise integer linear programming formulation for clique partitioning problems

A Clique Partitioning Problem (CPP) finds an optimal partition of a given edge-weighted undirected graph, such that the sum of the weights is maximized. This general graph problem has a wide range of real-world applications, including correlation clustering, group technology, community detection, and coalition structure generation. Although a CPP is NP-hard, due to the recent advance of Integer Linear Programming (ILP) solvers, we can solve reasonably large problem instances by formulating a CPP as an ILP instance. The first ILP formulation was introduced by Grötschel and Wakabayashi (Mathematical Programming, 45(1-3), 59–96, 1989). Recently, Miyauchi et al. (2018) proposed a more concise ILP formulation that can significantly reduce transitivity constraints as compared to previously introduced models. In this paper, we introduce a series of concise ILP formulations that can reduce even more transitivity constraints. We theoretically evaluate the amount of reduction based on a simple model in which edge signs (positive/negative) are chosen independently. We show that the reduction can be up to 50% (dependent of the ratio of negative edges) and experimentally evaluate the amount of reduction and the performance of our proposed formulation using a variety of graph data sets. Experimental evaluations show that the reduction can exceed 50% (where edge signs can be correlated), and our formulation outperforms the existing state-of-the-art formulations both in terms of memory usage and computational time for most problem instances.

The complexity landscape of decompositional parameters for ILP: Programs with few global variables and constraints

Integer Linear Programming (ILP) has a broad range of applications in various areas of artificial intelligence. Yet in spite of recent advances, we still lack a thorough understanding of which structural restrictions make ILP tractable. Here we study ILP instances consisting of a small number of “global” variables and/or constraints such that the remaining part of the instance consists of small and otherwise independent components; this is captured in terms of a structural measure we call fracture backdoors which generalizes, for instance, the well-studied class of N-fold ILP instances.Our main contributions can be divided into three parts. First, we formally develop fracture backdoors and obtain exact and approximation algorithms for computing these. Second, we exploit these backdoors to develop several new parameterized algorithms for ILP; the performance of these algorithms will naturally scale based on the number of global variables or constraints in the instance. Finally, we complement the developed algorithms with matching lower bounds. Altogether, our results paint a near-complete complexity landscape of ILP with respect to fracture backdoors.1

Stable Divisorial Gonality is in NP

Divisorial gonality and stable divisorial gonality are graph parameters, which have an origin in algebraic geometry. Divisorial gonality of a connected graph G can be defined with help of a chip firing game on G. The stable divisorial gonality of G is the minimum divisorial gonality over all subdivisions of edges of G. In this paper we prove that deciding whether a given connected graph has stable divisorial gonality at most a given integer k belongs to the class NP. Combined with the result that (stable) divisorial gonality is NP-hard by Gijswijt et al., we obtain that stable divisorial gonality is NP-complete. The proof consists of a partial certificate that can be verified by solving an Integer Linear Programming instance. As a corollary, we have that the total number of subdivisions needed for minimum stable divisorial gonality of a graph with m edges is bounded by mO(mn).

Conversion Method of Netlists Consisting of Conventional Logic Gates to RSFQ Logic Circuits Utilizing Special RSFQ Gates

The conversion method of netlists consisting of conventional logic gates to RSFQ circuits is proposed. It treats netlists for CMOS circuits as the design entry and converts them to the netlists of RSFQ circuits, considering the reduction of the number of clocked gates, including D flip-flops (DFFs) for path balancing. It utilizes two kinds of special RSFQ gates for the reduction. One is a confluence buffer that can be utilized for realizing logic-or. The other is a small resettable DFF which can be utilized as an NIMPLY gate with the tuning of the order of pulse arrivals. The detection methods of replaceable gates with those two kinds of gates are proposed. An automatic test pattern generation tool is utilized for detecting replaceable or gates to speed up the previously proposed detection method. To minimize the number of clocked gates considering the cost of replacements, the selection problem of gates for replacements with those special RSFQ gates and the assignment of a logic level for each gate are formulated as an instance of integer linear programming. The proposed conversion method has been evaluated with ISCAS85 circuits, a 32-bit ALU, and a 64-bit ALU. The evaluation results show that the number of clocked gates and the latency in the cycles of circuits is reduced by about 10%. The delay time of the converted circuits has been estimated and the reduction of delay overhead is discussed.

The complexity landscape of decompositional parameters for ILP

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 give an overview of the complexity of ILP w.r.t. decompositional parameters defined on the constraint matrix.

On the path-width of integer linear programming

We consider the feasibility problem of integer linear programming (ILP). We show that solutions of any ILP instance can be naturally represented by an FO-definable class of graphs. For each solution there may be many graphs representing it. However, one of these graphs is of path-width at most 2n, where n is the number of variables in the instance. Since FO is decidable on graphs of bounded path-width, we obtain an alternative decidability result for ILP. The technique we use underlines a common principle to prove decidability which has previously been employed for automata with auxiliary storage. We also show how this new result links to automata theory and program verification.

The Complexity Landscape of Decompositional Parameters for ILP

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.

On the Path-Width of Integer Linear Programming

We consider the feasibility problem of integer linear programming (ILP). We show that solutions of any ILP instance can be naturally represented by an FO-definable class of graphs. For each solution there may be many graphs representing it. However, one of these graphs is of path-width at most 2n, where n is the number of variables in the instance. Since FO is decidable on graphs of bounded path- width, we obtain an alternative decidability result for ILP. The technique we use underlines a common principle to prove decidability which has previously been employed for automata with auxiliary storage. We also show how this new result links to automata theory and program verification.

Efficient reasoning about data trees via integer linear programming

Data trees provide a standard abstraction of XML documents with data values: they are trees whose nodes, in addition to the usual labels, can carry labels from an infinite alphabet (data). Therefore, one is interested in decidable formalisms for reasoning about data trees. While some are known—such as the two-variable logic—they tend to be of very high complexity, and most decidability proofs are highly nontrivial. We are therefore interested in reasonable complexity formalisms as well as better techniques for proving decidability. Here we show that many decidable formalisms for data trees are subsumed—fully or partially—by the power of tree automata together with set constraints and linear constraints on cardinalities of various sets of data values. All these constraints can be translated into instances of integer linear programming, giving us an NP upper bound on the complexity of the reasoning tasks. We prove that this bound, as well as the key encoding technique, remain very robust, and allow the addition of features such as counting of paths and patterns, and even a concise encoding of constraints, without increasing the complexity. The NP bound is tight, as we also show that the satisfiability of a single set constraint is already NP-hard. We then relate our results to several reasoning tasks over XML documents, such as satisfiability of schemas and data dependencies and satisfiability of the two-variable logic. As a final contribution, we describe experimental results based on the implementation of some reasoning tasks using the SMT solver Z3.

Topology Control Schema for Better QoS in Hybrid RF/FSO Mesh Networks

The practical limitations and challenges of radio frequency (RF) based communication networks have become increasingly apparent over the past decade, leading researchers to seek new hybrid communication approaches. One promising strategy that has been the subject of considerable interest is the augmentation of RF technology by Free Space Optics (FSO), using the strength of each communication technology to overcome the limitations of the other. In this article, we introduce a new scheme for controlling the topology in hybrid Radio-Frequency/Free Space Optics (RF/FSO) wireless mesh networks. Our scheme is based on adaptive adjustments to both transmission power (of RF and FSO transmitters) and the optical beam-width (of FSO transmitters) at individual nodes, with the objective of meeting specified Quality of Service (QoS) requirements, specifically end-to-end delay and throughput. We show how one can effectively encode the instantaneous objectives and constraints of the system as an instance of Integer Linear Programming (ILP). We demonstrate that the technique of Lagrangian Relaxation (LR), augmented with iterative repair heuristics, can be used to determine good (albeit sub-optimal) solutions for the ILP problem, making the approach feasible for mid-sized networks. We make the proposed scheme viable for large-scale networks in terms of number of nodes, number of transceivers, and number of source-destination pairs by solving the ILP problem using a Particle Swarm Optimization (PSO) implementation.

USING SAT-BASED TECHNIQUES IN LOW POWER STATE ASSIGNMENT

Power consumption of synchronous sequential circuits can be reduced by careful encoding of the states of the circuit. The idea is to reduce the average number of bit changes per state transition by finding an optimal state assignment. This state assignment problem is NP-hard, and existing techniques rely mainly on heuristic-based methods. The primary goal of this work is to assess the suitability of using complete advanced Boolean Satisfiability and Integer Linear Programming (ILP) methods in finding an optimized solution. We formulate the problem as a 0-1 ILP instance with power minimization being the objective. Using generic and commercial solvers, the proposed approach was tested on sample circuits from the MCNC benchmark suite. Furthermore, in an effort to accelerate the search process, circuits were checked for symmetries and symmetry breaking predicates were added whenever applicable. The experimental results provide a pragmatic insight into the problem and basis for further exploration.

Reachability Analysis of Augmented Marked Graphs via Integer Linear Programming

Augmented marked graphs (AMGs) are extensions of marked graphs that allow resource sharing. It has been shown that AMGs are useful for modeling and analyzing certain types of flexible manufacturing systems (FMSs). To our knowledge, the techniques developed for analyzing AMGs are mostly based upon checking certain Petri net structures such as siphons. This article exploits the integer linear programming approach for the analysis of a subclass of AMGs called decomposable AMGs. We show that reachability between two configurations of a decomposable AMG can be equated with solving an instance of integer linear programming. We further extend our technique to model checking a type of branching time temporal logics. Examples arisen in FMSs are used to demonstrate the application of our technique.

An exact solution to the minimum size test pattern problem

This article addresses the problem of test pattern generation for single stuck-at faults in combinational circuits, under the additional constraint that the number of specified primary input assignments is minimized. This problem has different applications in testing, including the identification of "don't care" conditions to be used in the synthesis of Built-In Self-Test (BIST) logic. The proposed solution is based on an integer linear programming (ILP) formulation which builds on an existing Propositional Satisfiability (SAT) model for test pattern generation. The resulting ILP formulation is linear on the size of the original SAT model for test generation, which is linear on the size of the circuit. Nevertheless, the resulting ILP instances represent complex optimization problems, that require dedicated ILP algorithms. Preliminary results on benchmark circuits validate the practical applicability of the test pattern minimization model and associated ILP algorithm.