Iskander Aliev, Martin Henk
Given an integer mxn matrix A satisfying certain regularity assumptions, a well-known integer programming problem asks to find an integer point in the associated knapsack polytope P(A, b)={x: A x= b, x>=0} or determine that no such point exists. We obtain a LLL-based polynomial time algorithm that solves the problem subject to a constraint on the location of the vector b.
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The main result of the paper shows that the asymptotic growth of the Frobenius number in average is significantly slower than the growth of the maximum Frobenius number.
Given an integer mxn matrix A satisfying certain regularity assumptions, we consider the set F(A) of all integer vectors b such that the associated knapsack polytope P(A,b)={x: Ax=b, x>=0} contains an integer point. When m=1 the set F(A) is known to contain all consecutive integers greater than the Frobenius number associated with A. In this paper we introduce the diagonal Frobenius number g(A) which reflects in an analogous way feasibility properties of the problem and the structure of F(A) in the general case. We give an optimal upper bound for g(A) and also estimate the asymptotic growth of…
Given an integer matrix A satisfying certain regularity assumptions, we consider for a positive integer s the set F_s(A) of all integer vectors b such that the associated knapsack polytope P(A,b)={x: Ax=b, x non-negative} contains at least s integer points. In this paper we investigate the structure of the set F_s(A) sing the concept of s-covering radius. In particular, in a special case we prove an optimal lower bound for the s-Frobenius number.
We obtain optimal lower and upper bounds for the (additive) integrality gaps of integer knapsack problems. In a randomised setting, we show that the integrality gap of a "typical" knapsack problem is drastically smaller than the integrality gap that occurs in a worst case scenario.
Let $A$ be an integral nonnegative $m\times n$ matrix, $b$ be an integral nonnegative vector. It is suggested new method for reduction of integer program $\max \{cx| Ax=b, x\ge 0, x\in\mathbf{Z}^n\}$ to knapsack problem $\max \{c'x| fAx=fb, x\ge 0, x\in\mathbf{Z}^n\}$.
It is important to design separation algorithms of low computational complexity in mixed integer programming. We study the separation problems of the two continuous knapsack polyhedra with divisible capacities. The two polyhedra are the convex hulls of the sets which consist of $ n $ nonnegative integer variables, one unbounded continuous, $ m $ bounded continuous variables, and one linear constraint in either $ \geq $ or $ \leq $ form where the coefficients of integer variables are integer and divisible. Wolsey and Yaman (Math Program 156: 1--20, 2016) have shown that the polyhedra can be des…
Efficient exact sequential lifting algorithm for binary knapsack set
本文提出了一种高效精确顺序提升算法，用于处理具有非整数权重和大容量的二进制背包集问题，提升了计算效率和稳定性。
In the context of branch-and-bound (B&B) for integer programming (IP) problems, a direction along which the polyhedron of the IP has minimum width is termed a thin direction. We demonstrate that a thin direction need not always be a good direction to branch on for solving the problem efficiently. Further, the integer width, which is the number of B&B nodes created when branching on the direction, may also not be an accurate indicator of good branching directions.
A valid inequality α^Tx \ge α_0 for a set covering problem is said to have pitch <= k ( a positive integer) if the k smallest positive α_j sum to at least alpha_0. This paper presents a new, simple derivation of a relaxation for set covering problems whose solutions satisfy all valid inequalities of pitch and is of polynomial size, for each fixed . We also consider the minimum knapsack problem, and show that for each fixed integer p > 0 and 0 < ε< 1 one can separate, within additive tolerance ε, from the relaxation defined by the valid inequalities with coefficients in {0, 1, . . . , p} in tim…
Mixed-Integer Linear Programming Approximations for the Stochastic Knapsack
针对随机背包问题，提出混合整数线性规划近似方法，在静态和动态设置下最大化期望净收益，并证明该方法在正态分布及多元正态分布下的有效性，作为通用分布的有效启发式算法。
We show that in a knapsack feasibility problem an integral vector $p$, which is short, and near parallel to the constraint vector gives a branching direction with small integer width. We use this result to analyze two computationally efficient reformulation techniques on low density knapsack problems. Both reformulations have a constraint matrix with columns reduced in the sense of Lenstra, Lenstra, and Lovász. We prove an upper bound on the integer width along the last variable, which becomes 1, when the density is sufficiently small. In the proof we extract from the transformation matrices a…
When solving large-scale multiobjective optimization problems, solvers can get stuck with the memory or time limit. In such cases, one is left with no information how far is the best feasible solution, found before the optimization process has stopped, to the true Pareto optimal solution. In this work, we show how to provide such information when solving multiobjective multidimensional knapsack problems by a commercial mixed-integer linear solver. We illustrate the proposed approach on biobjective multidimensional knapsack problems derived from singleobjective multidimensional knapsack problem…
We consider binary integer programming problems with the min-max regret objective function under interval objective coefficients. We propose a new heuristic framework, which we call the iterated dual substitution (iDS) algorithm. The iDS algorithm iteratively invokes a dual substitution heuristic and excludes from the search space any solution already checked in previous iterations. In iDS, we use a best-scenario-based lemma to improve performance. We apply iDS to four typical combinatorial optimization problems: the knapsack problem, the multidimensional knapsack problem, the generalized assi…
Lower Bounds on the Complexity of Mixed-Integer Programs for Stable Set and Knapsack
该研究通过扩展信息论方法，证明稳定集与背包问题的多项式规模MIP公式所需整数变量数的下界为Ω(n/log²n)，改进了此前已知的下界，同时给出了对应稳定集多面体近似扩展公式的规模下界。
In this paper, we will shed light on when to pack and use 3D-printers in disaster response operations. For that, we introduce a new type of problem, which we call the two-stage stochastic 3D-printing knapsack problem. We provide a two-stage stochastic programming formulation for this problem, for which both the first and the second stage are NP-hard integer linear programs. We reformulate this formulation to an equivalent integer linear program, which can be efficiently solved by standard solvers. Our numerical results illustrate that for most situations using a 3D-printer is beneficial. Only …
We consider bounded integer knapsacks where the weights and variable upper bounds together form a superincreasing sequence. The elements of this superincreasing knapsack are exactly those vectors that are lexicographically smaller than the greedy solution to optimizing over this knapsack. We describe the convex hull of this $n$-dimensional set with $\mathcal{O}(n)$ facets. We also establish a distributive property by proving that the convex hull of $\le$- and $\ge$-type superincreasing knapsacks can be obtained by intersecting the convex hulls of $\le$- and $\ge$-sets taken individually. Our p…
Solving 0-1 Integer Programs with Unknown Knapsack Constraints Using Membership Oracles
本文针对未知背包约束的0-1整数规划问题，提出基于成员资格预言机和SVM的主动学习框架，改进采样策略与线性分离方法，实验验证其有效性。
A novel canonical duality theory (CDT) is presented for solving general bilevel mixed integer nonlinear optimization governed by linear and quadratic knapsack problems. It shows that the challenging knapsack problems can be solved analytically in term of their canonical dual solutions. The existence and uniqueness of these analytical solutions are proved. NP-Hardness of the knapsack problems is discussed. A powerful CDT algorithm combined with an alternative iteration and a volume reduction method is proposed for solving the NP-hard bilevel knapsack problems. Application is illustrated by a be…
Knapsack with compactness: a semidefinite approach
本文针对带紧致性约束的最小背包问题提出半定规划方法，通过将紧致性融入约束或目标，并利用单一超参数平衡紧致性与准确性，在困难实例上相比混合整数规划展现出更优的有效性和灵活性。
The Sequential Multiple Knapsack Problem is a special case of Multiple knapsack problem in which the items sizes are divisible. A characterization of the optimal solutions of the problem and a description of the convex hull of all the integer solutions are presented. More precisely, it is shown that a new formulation of the problem allows to generate a decomposition approach for enumerating all optimal solutions of the problem. Such a decomposition approach is used for finding the inequalities (defined by an inductive scheme) describing the Sequential Multiple Knapsack polytope.
We consider chance-constrained binary knapsack problems, where the weights of items are independent random variables with the means and standard deviations known. The chance constraint can be reformulated as a second-order cone constraint under some assumptions for the probability distribution of the weights. The problem becomes a second-order cone-constrained binary knapsack problem, which is equivalent to a robust binary knapsack problem with an ellipsoidal uncertainty set. We demonstrate that optimal solutions to robust binary knapsack problems with inner and outer polyhedral approximations…
We propose a very simple preconditioning method for integer programming feasibility problems: replacing the problem b' <= Ax <= b, x \in Z^n with b' <= AUy <= b, y \in Z^n, where U is a unimodular matrix computed via basis reduction, to make the columns of $AU$ short and nearly orthogonal. The reformulation is called rangespace reformulation. It is motivated by the reformulation technique proposed for equality constrained IPs by Aardal, Hurkens and Lenstra. We also study a family of IP instances, called decomposable knapsack problems (DKPs). DKPs generalize the instances proposed by Jeroslow, …
A tight 1/3-approximation algorithm and fully polynomial-time approximation schemes for the Colored Knapsack Problem
针对彩色背包问题，提出首个近似算法（性能比1/3）及两个FPTAS，通过舍入LP解和利润缩放实现多项式时间近似。
On Solving the Knapsack Problem with Conflicts
本文针对带冲突约束的背包问题，采用已知混合整数线性规划模型并用开源求解器CP-SAT求解，实验表明该方法在概念和实现简单的同时，效果可与最先进求解器相当甚至更优。
Simultaneous Group-Envelope Bounds for $Γ$-Robust Multiple-Choice Knapsack Problems
本文针对Γ-鲁棒多选背包问题，提出 simultaneous 组-包络界方法，可同时界定阈值族，大幅加速松弛求解，在实验中实现2.37倍几何平均加速。
Solving a large oral examination timetabling problem using a multidimensional knapsack MILP formulation
本文提出了一种多维背包MILP方法，用于解决法国著名工程师学校口试排程问题，通过将问题建模为多维背包问题，有效处理了大规模考生和复杂约束条件。
In this paper three heuristic algorithms using the Divide-and-Conquer paradigm are developed and assessed for three integer optimizations problems: Multidimensional Knapsack Problem (d-KP), Bin Packing Problem (BPP) and Travelling Salesman Problem (TSP). For each case, the algorithm is introduced, together with the design of numerical experiments, in order to empirically establish its performance from both points of view: its computational time and its numerical accuracy.
We present a new algorithm for computing a truncated Markov basis of a lattice. In general, this new algorithm is faster than existing methods. We then extend this new algorithm so that it solves the linear integer feasibility problem with promising results for equality knapsack problems. We also present a novel Groebner basis approach to solve a particular integer linear program as opposed to previous Groebner basis methods that effectively solved many different integer linear programs simultaneously. Initial results indicate that this optimisation algorithm performs better than previous Groe…
Sequence partition problems arise in many fields, such as sequential data analysis, information transmission, and parallel computing. In this paper, we study the following partition problem variant: given a sequence of $n$ items $1,\ldots,n$, where each item $i$ is associated with weight $w_i$ and another parameter $s_i$, partition the sequence into several consecutive subsequences, so that the total weight of each subsequence is no more than a threshold $w_0$, and the sum of the largest $s_i$ in each subsequence is minimized. This problem admits a straightforward solution based on dynamic pro…
We propose a new exact approach for solving integer linear programming (ILP) problems which we will call projective splitting algorithms (PSAs). Unlike classical methods for solving ILP problems, PSAs conduct the search for the optimal solution by generating candidate solutions tailored to specific values of the objective function. As a consequence of this strategy, the number of variables in the original ILP problem is systematically reduced without adding any additional constraint to the initial formulation. This is the first of a two-part series on PSAs. In this paper we focus on the resolu…
Motivated by their success in the single-objective domain, we propose a very simple linear programming-based matheuristic for tri-objective binary integer programming. To tackle the problem, we obtain lower bound sets by means of the vector linear programming solver Bensolve. Then, simple heuristic approaches, such as rounding and path relinking, are applied to this lower bound set to obtain high-quality approximations of the optimal set of trade-off solutions. The proposed algorithm is compared to a recently suggested algorithm which is, to the best of our knowledge, the only existing matheur…
Inclusion-wise extremal knapsack problems are (mostly) hard to approximate
本文研究两种包含式极值背包问题，证明其大多难以近似，确定了可恢复可近似性的限制条件，并为部分限制情形给出近似方案或改进了已有结果。
Decision making needs to take an uncertain environment into account. Over the last decades, robust optimization has emerged as a preeminent method to produce solutions that are immunized against uncertainty. The main focus in robust discrete optimization has been on the analysis and solution of one- or two-stage problems, where the decision maker has limited options in reacting to additional knowledge gained after parts of the solution have been fixed. Due to its computational difficulty, multistage problems beyond two stages have received less attention. In this paper we argue that multistage…
On contention resolution for the hypergraph matching, knapsack, and $k$-column sparse packing problems
本文将竞争解决框架应用于超图匹配、背包和k列稀疏装箱问题，证明了相关间隙下界并构造了多个单调平衡竞争解决方案，从而改进了k-CS-PIP的近似比。
Learning-Based Hierarchical Approach for Fast Mixed-Integer Optimization
本文提出一种基于学习的分层方法，通过分解混合整数规划问题并利用凸优化技术，实现快速求解高质量解。
We present a simple and at the same time fficient algorithm to compute all nondominated extreme points in the outcome set of multi-objective mixed integer linear programmes in any dimension. The method generalizes the well-known dichotomic scheme to compute the set of nondominated extreme points in the outcome set of a bi-objective programme based on the iterative solution of weighted sum scalarizations. It uses as a main routine a convex hull algorithm. The algorithm is illustrated with, and numerically tested on, instances of multi-objective assignment and knapsack problems. Experimental res…
Reduction in Solving Some Integer Least Squares Problems
针对普通整数最小二乘问题和椭球约束整数最小二乘问题，提出新的约简算法以提高搜索效率，并在数值稳定性上进行了优化。
In large organisations and companies, making investment decisions is a complex and challenging task. In the Australian Department of Defence (Defence), the complexity is even higher because defence capabilities are public goods and do not have a financial return \textit{per se}. In this work we mathematically define Defence's investment portfolio problem as a Set-Union Knapsack Problem (SUKP). We present a practical way to linearise the model as an Integer Linear Programming (ILP) problem. This linear model was developed as the optimisation engine of the New Investments to Risked Options (NITR…
Quantum Algorithms for Bandits with Knapsacks with Improved Regret and Time Complexities
本文首次研究带背包约束赌博机的量子算法，通过量子预言机和量子线性规划求解器，在问题无关和问题相关遗憾界上分别实现因子改进和二次改进，并实现时间复杂度的多项式加速。
This paper introduces a family of valid inequalities, that we term consistency cuts, to be applied to a Dantzig-Wolfe reformulation (or decomposition) with linking variables. We prove that these cuts ensure an integer solution to the corresponding Dantzig-Wolfe relaxation when certain criteria to the structure of the decomposition are met. We implement the cuts and use them to solve a commonly used test set of 200 instances of the temporal knapsack problem. We assess the performance with and without the cuts and compare further to CPLEX and other solution methods that have historically been us…
Convex Minimization with Integer Minima in $\widetilde O(n^4)$ Time
针对整数最小值的凸最小化问题，提出结合快速LLL算法与体积中心切割平面法的新算法，将运行时间从$\widetildeΩ(n^8)$降至$O(n^4\log n)$，并改进子模函数最小化的复杂度。
Counterfactual Explanations for Integer Optimization Problems
本文研究了整数优化问题的反事实解释，证明了构建CE的复杂性，并提出了解决特定情况的算法，通过实验验证了方法的有效性。
In this article, we discuss an exact algorithm for solving mixed integer concave minimization problems. A piecewise inner-approximation of the concave function is achieved using an auxiliary linear program that leads to a bilevel program, which provides a lower bound to the original problem. The bilevel program is reduced to a single level formulation with the help of Karush-Kuhn-Tucker (KKT) conditions. Incorporating the KKT conditions lead to complementary slackness conditions that are linearized using BigM, for which we identify a tight value for general problems. Multiple bilevel programs,…
Decision-Tree Surrogates for Weak Counterfactual Explanations in Integer Optimization
针对整数优化中弱反事实解释计算困难的问题，本文提出用决策树近似可行区域的快速启发式算法，并在背包和设施选址问题上验证了其有效性。
We propose a methodology at the nexus of operations research and machine learning (ML) leveraging generic approximators available from ML to accelerate the solution of mixed-integer linear two-stage stochastic programs. We aim at solving problems where the second stage is highly demanding. Our core idea is to gain large reductions in online solution time while incurring small reductions in first-stage solution accuracy by substituting the exact second-stage solutions with fast, yet accurate supervised ML predictions. This upfront investment in ML would be justified when similar problems are so…
Exact and Heuristic Methods for $Γ$-Robust Min-Max Problems
本文研究带Γ-鲁棒不确定数据的混合整数线性极大极小问题，提出精确与启发式求解方法，经560个背包拦截问题实例测试，该启发式方法能缩小最优性间隙且性能优于基准方法。
Resilience and robustness are important properties in the reliability and attack-tolerance analysis of networks. In recent decades, various qualitative and heuristic-based quantitative approaches have made significant contributions in addressing network resilience and robustness. However, the lack of exact methods such as mixed-integer programming (MIP) models is sensible in the literature. In this paper, we contribute to the literature on the network resilience and robustness for targeted and random attacks and propose a MIP model considering graph-theoretical aspects of networks. The propose…
Integer Maximization over $\ell_p$ Balls: Hardness and Exact Algorithms
本文研究在原点中心 $\ell_p$ 球的整数点上最大化线性函数的问题，证明其 NP 完全性，并给出两个伪多项式精确算法及与最近向量问题的联系，同时探讨维度相关方法。
In this work we propose an improvement of the $x$-space algorithm developed for solving a class of min--max bilevel optimization problems (Tang Y., Richard J.P.P., Smith J.C. (2016), A class of algorithms for mixed-integer bilevel min--max optimization. Journal of Global Optimization, 66(2), 225--262). In this setting, the leader of the upper level problem aims at restricting the follower's decisions by minimizing an objective function, which the follower intends to maximize in the lower level problem by making decisions still available to her. The $x$-space algorithm solves upper and lower bo…
PyEPO: A PyTorch-based End-to-End Predict-then-Optimize Library for Linear and Integer Programming
PyEPO是一种基于PyTorch的端到端预测-然后优化库，用于线性和整数规划，提供用户友好的接口和先进的算法，通过实验比较不同方法并讨论未来研究的启示。
Branch-and-Cut for Mixed-Integer Nash Equilibrium Problems
本文提出了一种用于混合整数纳什均衡问题的分支-切割算法，通过双层优化技术将问题转化为可计算形式，并证明在特定条件下算法可有限终止。
Deep Learning for Two-Stage Robust Integer Optimization
针对整数变量下两阶段鲁棒优化（2RO）难以求解大规模算例的问题，提出结合定制神经网络的列与约束生成算法Neur2RO，可快速生成高质量解，显著优于现有方法。
From approximate to exact integer programming
本文提出两种基于近似整数规划求解精确整数规划的方法，分别在给定模余数时达到 $2^{O(n)}$ 复杂度，并改进多项式范围背包和子集和问题至 $(\log n)^{O(n)}$。
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