[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86253-en":3,"doc-seo-86253-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},86253,687197207919,"Theodora","https://ap-avatar.wpscdn.com/avatar/a000253d6f5f7c60be?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779446848396160552",8,"Research & Report","Tropical Circuits with Scalar Multiplication Gates","Study of tropical circuits augmented with scalar multiplication gates that use max, plus, and multiplication by a positive constant. Exponential size lower bounds are proved for computing maximum weight directed spanning trees and maximum weight bipartite perfect matchings. The results yield an exponential separation between monotone and non-monotone maxout neural networks, generalizing ReLU. Consequently, models enforcing convexity constraints, such as ICNNs, may require exponentially larger size than unrestricted networks to express the same functions.","arXiv :2607 . 1 1540v 1 [ cs .CC] 13 Jul 2026  \nTropical Circuits with Scalar Multiplication Gates  \nChristoph Hertrich 1 and Moritz Stargalla 1  \n1 University of Technology Nuremberg  \n[christoph.hertrich@utn.de](christoph.hertrich@utn.de), [moritz.stargalla@utn.de](moritz.stargalla@utn.de)  \nJuly 14, 2026  \nAbstract  \nWe study tropical circuits with scalar multiplication gates, that is, algebraic circuits whose gates implement max, +, or multiplication with a positive constant. For such circuits, we prove exponential size lower bounds for computing maximum weight directed spanning trees and maximum weight bipartite perfect matchings. As a corollary, we obtain an exponential size separation between monotone and non-monotone maxout neural networks, which generalize the popularly used ReLU neural networks. One conclusion from this is that neural network models with enforced convexity constraints, such as input-convex neural networks (ICNNs), sometimes need to be exponentially larger than their unrestricted counterparts in order to express the same functions.  \n1 Introduction  \nTropical circuits [Juk23], also known as max-plus circuits, are a variant of classical arithmetic circuits that use maximum and addition gates instead of addition and multiplication gates. Besides fundamental interest in the power of different models of computation and their dependence on theset of allowed operations, a primary motivation to study tropical circuits is to prove lower boundson pure dynamic programs. By definition, a pure dynamic program consists of a predefined sequence of max (or min) and plus operations only. One example is the Bellman-Ford algorithm. Since every pure dynamic program can be written as a tropical circuit, lower bounds on the latter imply lower bounds on the former. That way, it has for example been shown that every pure dynamic program for the minimum spanning tree problem needs exponentially many iterations [JS19] .  \nRecently, tropical circuits received increased attention due to their close connection to neural networks. Variants of tropical circuits have been used in order to prove size upper bounds for neural networks with rectified linear unit (ReLU) activations [HS25; HKL26] . Such neural networks can be defined as a circuit in which each node (neuron) computes an affine function of the outputs of its predecessors composed with the ReLU function x 7→ max{0, x} . It is straightforward to verify that a ReLU network can exactly simulate every tropical circuit. However, ReLU networks are strictly more powerful than tropical circuits, as they can, for instance, solve the minimum spanning tree problem in polynomial size [FGK16; HS25] . The main reason for this distinction seems to be the ability of neural networks to implement subtraction via negative weights: the construction at hand basically implements a (max, + , −)-circuit.  \nHowever, besides subtraction, there is a second feature that seemingly makes neural networks more powerful than tropical circuits: namely scalar multiplication with arbitrary real constants. This opens up the question of how much additional power this feature provides alone, without allowing subtraction. To study this question, we propose to augment the model of tropical circuits by scalar multiplication gates with positive constants, calling the resulting model scalar tropical circuits (STCs) .  \nEvery STC computes a continuous piecewise linear (CPWL) function of the form c 7→ max x∈P c⊤ x for a polytope P, which is the support function fP of the polytope P.  \n1.1 Our Contributions  \nIn the following we first detail our contribution in the context of tropical circuit theory and afterwards discuss implications, particularly in the context of neural networks.  \nLower Bounds on the Size of STCs. The size of a regular tropical circuit is the number of max and plus gates. We also measure the size of an STC Φ, denoted as size(Φ), as the number of max and plus gates, not counting scalar multiplication","cbCaigJPrrhlOqAW","https://ap.wps.com/l/cbCaigJPrrhlOqAW","pdf",461335,4,1,23,"English","en",105,"# Abstract\n# Introduction\n## Our Contributions","[{\"question\":\"What are scalar tropical circuits (STCs)?\",\"answer\":\"STCs are tropical circuits extended with scalar multiplication gates using positive real constants, in addition to max and plus operations.\"},{\"question\":\"Which problems receive exponential size lower bounds?\",\"answer\":\"The paper proves exponential size lower bounds for computing maximum weight directed spanning trees and maximum weight bipartite perfect matchings.\"},{\"question\":\"How do the results relate to neural networks such as ReLU and maxout?\",\"answer\":\"The work derives an exponential separation between monotone and non-monotone maxout neural networks, and notes that these networks generalize ReLU networks.\"}]",1784209835,58,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"tropical-circuits-with-scalar-multiplication-gates","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/tropical-circuits-with-scalar-multiplication-gates/86253/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-26","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":50},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What are scalar tropical circuits (STCs)?","Question",{"text":75,"@type":76},"STCs are tropical circuits extended with scalar multiplication gates using positive real constants, in addition to max and plus operations.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Which problems receive exponential size lower bounds?",{"text":80,"@type":76},"The paper proves exponential size lower bounds for computing maximum weight directed spanning trees and maximum weight bipartite perfect matchings.",{"name":82,"@type":73,"acceptedAnswer":83},"How do the results relate to neural networks such as ReLU and maxout?",{"text":84,"@type":76},"The work derives an exponential separation between monotone and non-monotone maxout neural networks, and notes that these networks generalize ReLU networks.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]