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Tai256c is the largest unsolved quadratic assignment problem (QAP) instance in QAPLIB. It is known that QAP tai256c can be converted into a 256 dimensional binary quadratic optimization problem (BQOP) with a single cardinality constraint which requires the sum of the binary variables to be 92. As the BQOP is much simpler than the original QAP, the conversion increases the possibility to solve the QAP. Solving exactly the BQOP, however, is still very difficult. Indeed, a 1.48% gap remains between the best known upper bound (UB) and lower bound (LB) of the unknown optimal value. This paper shows that the BQOP admits a nontrivial symmetry, a property that makes the BQOP very hard to solve. The symmetry induces equivalent subproblems in branch and bound (BB) methods. To effectively improve the LB, we propose an efficient BB method that incorporates a doubly nonnegative relaxation, the standard orbit branching and a technique to prune equivalent subproblems. With this BB method, a new LB with 1.25% gap is successfully obtained, and computing an LB with 1.0% gap is shown to be still quite difficult.
The optimization of periodic timetables is an indispensable planning task in public transport. Although the periodic event scheduling problem (PESP) provides an elegant mathematical formulation of the periodic timetabling problem that led to many insights for primal heuristics, it is notoriously hard to solve to optimality. One reason is that for the standard mixed-integer linear programming formulations, linear programming relaxations are weak and the integer variables are of pure technical nature and in general do not correlate with the objective value. While the first problem has been addressed by developing several families of cutting planes, we focus on the second aspect. We discuss integral forward cycle bases as a concept to compute improved dual bounds for PESP instances. To this end, we develop the theory of forward cycle bases on general digraphs. Specifically for the application of timetabling, we devise a generic procedure to construct line-based event-activity networks, and give a simple recipe for an integral forward cycle basis on such networks. Finally, we analyze the 16 railway instances of the benchmark library PESPlib, match them to the line-based structure and use forward cycle bases to compute better dual bounds for 14 out of the 16 instances.
Consolidation of commodities and coordination of vehicle routes are fundamental features of supply chain management problems. While locations for consolidation and coordination are typically known a priori, in adaptive transportation networks this is not the case. The identification of such consolidation locations forms part of the decision making process. Supply chain management problems integrating the designation of consolidation locations with the coordination of long haul and local vehicle routing is not only challenging to solve, but also very difficult to formulate mathematically. In this paper, the first mathematical model integrating location clustering with long haul and local vehicle routing is proposed. This mathematical formulation is used to develop algorithms to find high quality solutions. A novel parallel framework is developed that combines exact and heuristic methods to improve the search for high quality solutions and provide valid bounds. The results demonstrate that using exact methods to guide heuristic search is an effective approach to find high quality solutions for difficult supply chain management problems.
大規模二次割当問題への挑戦
(2022)
Sampling rare events in metastable dynamical systems is often a computationally expensive task and one needs to resort to enhanced sampling methods such as importance sampling. Since we can formulate the problem of finding optimal importance sampling controls as a stochastic optimization problem, this then brings additional numerical challenges and the convergence of corresponding algorithms might as well suffer from metastabilty. In this article we address this issue by combining systematic control approaches with the heuristic adaptive metadynamics method. Crucially, we approximate the importance sampling control by a neural network, which makes the algorithm in principle feasible for high dimensional applications. We can numerically demonstrate in relevant metastable problems that our algorithm is more effective than previous attempts and that only the combination of the two approaches leads to a satisfying convergence and therefore to an efficient sampling in certain metastable settings.
For cryptanalysis in lattice-based schemes, the performance evaluation of lattice basis reduction using high-performance computers is becoming increasingly important for the determination of the security level. We propose a distributed and asynchronous parallel reduction algorithm based on randomization and DeepBKZ, which is an improved variant of the block Korkine-Zolotarev (BKZ) reduction algorithm. Randomized copies of a lattice basis are distributed to up to 103,680 cores and independently reduced in parallel, while some basis vectors are shared asynchronously among all processes via MPI. There is a trade-off between randomization and information sharing; if a substantial amount of information is shared, all processes will work on the same problem, thereby diminishing the benefit of parallelization. To monitor this balance between randomness and sharing, we propose a metric to quantify the variety of lattice bases. We empirically find an optimal parameter of sharing for high-dimensional lattices. We demonstrate the efficacy of our proposed parallel algorithm and implementation with respect to both performance and scalability through our experiments.
Lattice problems are a class of optimization problems that are notably hard. There are no classical or quantum algorithms known to solve these problems efficiently. Their hardness has made lattices a major cryptographic primitive for post-quantum cryptography. Several different approaches have been used for lattice problems with different computational profiles; some suffer from super-exponential time, and others require exponential space. This motivated us to develop a novel lattice problem solver, CMAP-LAP, based on the clever coordination of different algorithms that run massively in parallel. With our flexible framework, heterogeneous modules run asynchronously in parallel on a large-scale distributed system while exchanging information, which drastically boosts the overall performance. We also implement full checkpoint-and-restart functionality, which is vital to high-dimensional lattice problems. Through numerical experiments with up to 103,680 cores, we evaluated the performance and stability of our system and demonstrated its high capability for future massive-scale experiments.
We report our progress on the project for solving larger scale quadratic assignment problems (QAPs). Our main approach to solve large scale NP-hard combinatorial optimization problems such as QAPs is a parallel branch-and-bound method efficiently implemented on a powerful computer system using the Ubiquity Generator(UG) framework that can utilize more than 100,000 cores. Lower bounding procedures incorporated in the branch-and-bound method play a crucial role in solving the problems. For a strong lower bounding procedure, we employ the Lagrangian doubly nonnegative (DNN) relaxation and the Newton-bracketing method developed by the authors’ group. In this report, we describe some basic tools used in the project including the lower bounding procedure and branching rules, and present some preliminary numerical results.
Our next target problem is QAPs with dimension at least 50, as we have succeeded to solve tai30a and sko42 from QAPLIB for the first time.
Mixed-integer programming (MIP) problem is arguably among the hardest classes of optimization problems. This paper describes how we solved 21 previously unsolved MIP instances from the MIPLIB benchmark sets. To achieve these results we used an enhanced version of ParaSCIP, setting a new record for the largest scale MIP computation: up to 80,000 cores in parallel on the Titan supercomputer. In this paper, we describe the basic parallelization mechanism of ParaSCIP, improvements of the dynamic load balancing and novel techniques to exploit the power of parallelization for MIP solving. We give a detailed overview of computing times and statistics for solving open MIPLIB instances.
Public transportation networks are typically operated with a periodic timetable. The Periodic Event Scheduling Problem (PESP) is the standard mathematical modelling tool for periodic timetabling. Since PESP can be solved in linear time on trees, it is a natural question to ask whether there are polynomial-time algorithms for input networks of bounded treewidth. We show that deciding the feasibility of a PESP instance is NP-hard even when the treewidth is 2, the branchwidth is 2, or the carvingwidth is 3. Analogous results hold for the optimization of reduced PESP instances, where the feasibility problem is trivial. To complete the picture, we present two pseudo-polynomial-time dynamic programming algorithms solving PESP on input networks with bounded tree- or branchwidth. We further analyze the parameterized complexity of PESP with bounded cyclomatic number, diameter, or vertex cover number. For event-activity networks with a special -- but standard -- structure, we give explicit and sharp bounds on the branchwidth in terms of the maximum degree and the carvingwidth of an underlying line network. Finally, we investigate several parameters on the smallest instance of the benchmarking library PESPlib.