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<button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.18321">arXiv:2411.18321</a> <span> [<a href="https://arxiv.org/pdf/2411.18321">pdf</a>, <a href="https://arxiv.org/format/2411.18321">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Artificial Intelligence">cs.AI</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Software">cs.MS</span> </div> </div> <p class="title is-5 mathjax"> Learning optimal objective values for MILP </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Scavuzzo%2C+L">Lara Scavuzzo</a>, <a href="/search/cs?searchtype=author&query=Aardal%2C+K">Karen Aardal</a>, <a href="/search/cs?searchtype=author&query=Yorke-Smith%2C+N">Neil Yorke-Smith</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.18321v1-abstract-short" style="display: inline;"> Modern Mixed Integer Linear Programming (MILP) solvers use the Branch-and-Bound algorithm together with a plethora of auxiliary components that speed up the search. In recent years, there has been an explosive development in the use of machine learning for enhancing and supporting these algorithmic components. Within this line, we propose a methodology for predicting the optimal objective value, o… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.18321v1-abstract-full').style.display = 'inline'; document.getElementById('2411.18321v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.18321v1-abstract-full" style="display: none;"> Modern Mixed Integer Linear Programming (MILP) solvers use the Branch-and-Bound algorithm together with a plethora of auxiliary components that speed up the search. In recent years, there has been an explosive development in the use of machine learning for enhancing and supporting these algorithmic components. Within this line, we propose a methodology for predicting the optimal objective value, or, equivalently, predicting if the current incumbent is optimal. For this task, we introduce a predictor based on a graph neural network (GNN) architecture, together with a set of dynamic features. Experimental results on diverse benchmarks demonstrate the efficacy of our approach, achieving high accuracy in the prediction task and outperforming existing methods. These findings suggest new opportunities for integrating ML-driven predictions into MILP solvers, enabling smarter decision-making and improved performance. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.18321v1-abstract-full').style.display = 'none'; document.getElementById('2411.18321v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2402.05501">arXiv:2402.05501</a> <span> [<a href="https://arxiv.org/pdf/2402.05501">pdf</a>, <a href="https://arxiv.org/format/2402.05501">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> </div> </div> <p class="title is-5 mathjax"> Machine Learning Augmented Branch and Bound for Mixed Integer Linear Programming </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Scavuzzo%2C+L">Lara Scavuzzo</a>, <a href="/search/cs?searchtype=author&query=Aardal%2C+K">Karen Aardal</a>, <a href="/search/cs?searchtype=author&query=Lodi%2C+A">Andrea Lodi</a>, <a href="/search/cs?searchtype=author&query=Yorke-Smith%2C+N">Neil Yorke-Smith</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2402.05501v1-abstract-short" style="display: inline;"> Mixed Integer Linear Programming (MILP) is a pillar of mathematical optimization that offers a powerful modeling language for a wide range of applications. During the past decades, enormous algorithmic progress has been made in solving MILPs, and many commercial and academic software packages exist. Nevertheless, the availability of data, both from problem instances and from solvers, and the desir… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.05501v1-abstract-full').style.display = 'inline'; document.getElementById('2402.05501v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.05501v1-abstract-full" style="display: none;"> Mixed Integer Linear Programming (MILP) is a pillar of mathematical optimization that offers a powerful modeling language for a wide range of applications. During the past decades, enormous algorithmic progress has been made in solving MILPs, and many commercial and academic software packages exist. Nevertheless, the availability of data, both from problem instances and from solvers, and the desire to solve new problems and larger (real-life) instances, trigger the need for continuing algorithmic development. MILP solvers use branch and bound as their main component. In recent years, there has been an explosive development in the use of machine learning algorithms for enhancing all main tasks involved in the branch-and-bound algorithm, such as primal heuristics, branching, cutting planes, node selection and solver configuration decisions. This paper presents a survey of such approaches, addressing the vision of integration of machine learning and mathematical optimization as complementary technologies, and how this integration can benefit MILP solving. In particular, we give detailed attention to machine learning algorithms that automatically optimize some metric of branch-and-bound efficiency. We also address how to represent MILPs in the context of applying learning algorithms, MILP benchmarks and software. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.05501v1-abstract-full').style.display = 'none'; document.getElementById('2402.05501v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2207.06533">arXiv:2207.06533</a> <span> [<a href="https://arxiv.org/pdf/2207.06533">pdf</a>, <a href="https://arxiv.org/format/2207.06533">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Physics">quant-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Performance">cs.PF</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1038/s41534-023-00713-9">10.1038/s41534-023-00713-9 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Optimal entanglement distribution policies in homogeneous repeater chains with cutoffs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=I%C3%B1esta%2C+%C3%81+G">脕lvaro G. I帽esta</a>, <a href="/search/cs?searchtype=author&query=Vardoyan%2C+G">Gayane Vardoyan</a>, <a href="/search/cs?searchtype=author&query=Scavuzzo%2C+L">Lara Scavuzzo</a>, <a href="/search/cs?searchtype=author&query=Wehner%2C+S">Stephanie Wehner</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2207.06533v3-abstract-short" style="display: inline;"> We study the limits of bipartite entanglement distribution using a chain of quantum repeaters that have quantum memories. To generate end-to-end entanglement, each node can attempt the generation of an entangled link with a neighbor, or perform an entanglement swapping measurement. A maximum storage time, known as cutoff, is enforced on the memories to ensure high-quality entanglement. Nodes follo… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2207.06533v3-abstract-full').style.display = 'inline'; document.getElementById('2207.06533v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2207.06533v3-abstract-full" style="display: none;"> We study the limits of bipartite entanglement distribution using a chain of quantum repeaters that have quantum memories. To generate end-to-end entanglement, each node can attempt the generation of an entangled link with a neighbor, or perform an entanglement swapping measurement. A maximum storage time, known as cutoff, is enforced on the memories to ensure high-quality entanglement. Nodes follow a policy that determines when to perform each operation. Global-knowledge policies take into account all the information about the entanglement already produced. Here, we find global-knowledge policies that minimize the expected time to produce end-to-end entanglement. Our methods are based on Markov decision processes and value and policy iteration. We compare optimal policies to a policy in which nodes only use local information. We find that the advantage in expected delivery time provided by an optimal global-knowledge policy increases with increasing number of nodes and decreasing probability of successful swapping. Our work sheds light on how to distribute entangled pairs in large quantum networks using a chain of intermediate repeaters with cutoffs. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2207.06533v3-abstract-full').style.display = 'none'; document.getElementById('2207.06533v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 April, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 13 July, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">9 pages, 8 figures, 15 pages appendix with 10 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> npj Quantum Inf 9, 46 (2023) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2205.11107">arXiv:2205.11107</a> <span> [<a href="https://arxiv.org/pdf/2205.11107">pdf</a>, <a href="https://arxiv.org/format/2205.11107">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Learning to branch with Tree MDPs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Scavuzzo%2C+L">Lara Scavuzzo</a>, <a href="/search/cs?searchtype=author&query=Chen%2C+F+Y">Feng Yang Chen</a>, <a href="/search/cs?searchtype=author&query=Ch%C3%A9telat%2C+D">Didier Ch茅telat</a>, <a href="/search/cs?searchtype=author&query=Gasse%2C+M">Maxime Gasse</a>, <a href="/search/cs?searchtype=author&query=Lodi%2C+A">Andrea Lodi</a>, <a href="/search/cs?searchtype=author&query=Yorke-Smith%2C+N">Neil Yorke-Smith</a>, <a href="/search/cs?searchtype=author&query=Aardal%2C+K">Karen Aardal</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2205.11107v3-abstract-short" style="display: inline;"> State-of-the-art Mixed Integer Linear Program (MILP) solvers combine systematic tree search with a plethora of hard-coded heuristics, such as the branching rule. The idea of learning branching rules from data has received increasing attention recently, and promising results have been obtained by learning fast approximations of the strong branching expert. In this work, we instead propose to learn… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2205.11107v3-abstract-full').style.display = 'inline'; document.getElementById('2205.11107v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2205.11107v3-abstract-full" style="display: none;"> State-of-the-art Mixed Integer Linear Program (MILP) solvers combine systematic tree search with a plethora of hard-coded heuristics, such as the branching rule. The idea of learning branching rules from data has received increasing attention recently, and promising results have been obtained by learning fast approximations of the strong branching expert. In this work, we instead propose to learn branching rules from scratch via Reinforcement Learning (RL). We revisit the work of Etheve et al. (2020) and propose tree Markov Decision Processes, or tree MDPs, a generalization of temporal MDPs that provides a more suitable framework for learning to branch. We derive a tree policy gradient theorem, which exhibits a better credit assignment compared to its temporal counterpart. We demonstrate through computational experiments that tree MDPs improve the learning convergence, and offer a promising framework for tackling the learning-to-branch problem in MILPs. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2205.11107v3-abstract-full').style.display = 'none'; document.getElementById('2205.11107v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 October, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 23 May, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">10 pages, 2 figures, plus supplementary material</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2203.02433">arXiv:2203.02433</a> <span> [<a href="https://arxiv.org/pdf/2203.02433">pdf</a>, <a href="https://arxiv.org/ps/2203.02433">ps</a>, <a href="https://arxiv.org/format/2203.02433">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Neural and Evolutionary Computing">cs.NE</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> The Machine Learning for Combinatorial Optimization Competition (ML4CO): Results and Insights </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Gasse%2C+M">Maxime Gasse</a>, <a href="/search/cs?searchtype=author&query=Cappart%2C+Q">Quentin Cappart</a>, <a href="/search/cs?searchtype=author&query=Charfreitag%2C+J">Jonas Charfreitag</a>, <a href="/search/cs?searchtype=author&query=Charlin%2C+L">Laurent Charlin</a>, <a href="/search/cs?searchtype=author&query=Ch%C3%A9telat%2C+D">Didier Ch茅telat</a>, <a href="/search/cs?searchtype=author&query=Chmiela%2C+A">Antonia Chmiela</a>, <a href="/search/cs?searchtype=author&query=Dumouchelle%2C+J">Justin Dumouchelle</a>, <a href="/search/cs?searchtype=author&query=Gleixner%2C+A">Ambros Gleixner</a>, <a href="/search/cs?searchtype=author&query=Kazachkov%2C+A+M">Aleksandr M. Kazachkov</a>, <a href="/search/cs?searchtype=author&query=Khalil%2C+E">Elias Khalil</a>, <a href="/search/cs?searchtype=author&query=Lichocki%2C+P">Pawel Lichocki</a>, <a href="/search/cs?searchtype=author&query=Lodi%2C+A">Andrea Lodi</a>, <a href="/search/cs?searchtype=author&query=Lubin%2C+M">Miles Lubin</a>, <a href="/search/cs?searchtype=author&query=Maddison%2C+C+J">Chris J. Maddison</a>, <a href="/search/cs?searchtype=author&query=Morris%2C+C">Christopher Morris</a>, <a href="/search/cs?searchtype=author&query=Papageorgiou%2C+D+J">Dimitri J. Papageorgiou</a>, <a href="/search/cs?searchtype=author&query=Parjadis%2C+A">Augustin Parjadis</a>, <a href="/search/cs?searchtype=author&query=Pokutta%2C+S">Sebastian Pokutta</a>, <a href="/search/cs?searchtype=author&query=Prouvost%2C+A">Antoine Prouvost</a>, <a href="/search/cs?searchtype=author&query=Scavuzzo%2C+L">Lara Scavuzzo</a>, <a href="/search/cs?searchtype=author&query=Zarpellon%2C+G">Giulia Zarpellon</a>, <a href="/search/cs?searchtype=author&query=Yang%2C+L">Linxin Yang</a>, <a href="/search/cs?searchtype=author&query=Lai%2C+S">Sha Lai</a>, <a href="/search/cs?searchtype=author&query=Wang%2C+A">Akang Wang</a>, <a href="/search/cs?searchtype=author&query=Luo%2C+X">Xiaodong Luo</a> , et al. (16 additional authors not shown) </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2203.02433v2-abstract-short" style="display: inline;"> Combinatorial optimization is a well-established area in operations research and computer science. Until recently, its methods have focused on solving problem instances in isolation, ignoring that they often stem from related data distributions in practice. However, recent years have seen a surge of interest in using machine learning as a new approach for solving combinatorial problems, either dir… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2203.02433v2-abstract-full').style.display = 'inline'; document.getElementById('2203.02433v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2203.02433v2-abstract-full" style="display: none;"> Combinatorial optimization is a well-established area in operations research and computer science. Until recently, its methods have focused on solving problem instances in isolation, ignoring that they often stem from related data distributions in practice. However, recent years have seen a surge of interest in using machine learning as a new approach for solving combinatorial problems, either directly as solvers or by enhancing exact solvers. Based on this context, the ML4CO aims at improving state-of-the-art combinatorial optimization solvers by replacing key heuristic components. The competition featured three challenging tasks: finding the best feasible solution, producing the tightest optimality certificate, and giving an appropriate solver configuration. Three realistic datasets were considered: balanced item placement, workload apportionment, and maritime inventory routing. This last dataset was kept anonymous for the contestants. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2203.02433v2-abstract-full').style.display = 'none'; document.getElementById('2203.02433v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 March, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 4 March, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Neurips 2021 competition. arXiv admin note: text overlap with arXiv:2112.12251 by other authors</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2011.06069">arXiv:2011.06069</a> <span> [<a href="https://arxiv.org/pdf/2011.06069">pdf</a>, <a href="https://arxiv.org/format/2011.06069">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Ecole: A Gym-like Library for Machine Learning in Combinatorial Optimization Solvers </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Prouvost%2C+A">Antoine Prouvost</a>, <a href="/search/cs?searchtype=author&query=Dumouchelle%2C+J">Justin Dumouchelle</a>, <a href="/search/cs?searchtype=author&query=Scavuzzo%2C+L">Lara Scavuzzo</a>, <a href="/search/cs?searchtype=author&query=Gasse%2C+M">Maxime Gasse</a>, <a href="/search/cs?searchtype=author&query=Ch%C3%A9telat%2C+D">Didier Ch茅telat</a>, <a href="/search/cs?searchtype=author&query=Lodi%2C+A">Andrea Lodi</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2011.06069v2-abstract-short" style="display: inline;"> We present Ecole, a new library to simplify machine learning research for combinatorial optimization. Ecole exposes several key decision tasks arising in general-purpose combinatorial optimization solvers as control problems over Markov decision processes. Its interface mimics the popular OpenAI Gym library and is both extensible and intuitive to use. We aim at making this library a standardized p… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2011.06069v2-abstract-full').style.display = 'inline'; document.getElementById('2011.06069v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2011.06069v2-abstract-full" style="display: none;"> We present Ecole, a new library to simplify machine learning research for combinatorial optimization. Ecole exposes several key decision tasks arising in general-purpose combinatorial optimization solvers as control problems over Markov decision processes. Its interface mimics the popular OpenAI Gym library and is both extensible and intuitive to use. We aim at making this library a standardized platform that will lower the bar of entry and accelerate innovation in the field. Documentation and code can be found at https://www.ecole.ai. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2011.06069v2-abstract-full').style.display = 'none'; document.getElementById('2011.06069v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 November, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 11 November, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Published at the 1st Workshop on Learning Meets Combinatorial Algorithms @ NeurIPS 2020, Vancouver, Canada</span> </p> </li> </ol> <div class="is-hidden-tablet">  <span class="help" style="display: inline-block;"><a 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