[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81971-en":3,"doc-seo-81971-105":30,"detail-sidebar-cat-0-en-105":92},{"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},81971,1099514068035,"Ezra","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","On the Approximability of Parameterized Minimum Monotone Satisfying Assignment","Parameterized Minimum Monotone Satisfying Assignment (k-MMSA) studies monotone Boolean circuits that admit a satisfying assignment of Hamming weight at most k, with MMSA levels defined by bounded alternations between AND and OR gates. While polynomial-time approximability is well studied, parameterized approximability is less understood: k-MMSA2 corresponds to k-SetCover and k-MMSA4 to k-MinLabel. This work gives an FPT approximation for k-MMSA3 and gap-preserving reductions relating k-MMSA3 to k-MMSA2, plus hardness separations for k-MMSA4.","arXiv :2607 .06852v 1 [ cs .CC] 7 Jul 2026  \nOn the Approximability of Parameterized Minimum Monotone  \nSatisfying Assignment  \nVenkatesan Guruswami∗ Bingkai Lin† Xuandi Ren‡ Xin Zheng§  \nAbstract  \nThe parameterized Minimum Monotone Satisfying Assignment (k-MMSA) problem asks whether a monotone Boolean circuit admits a satisfying assignment of Hamming weight at most k. The MMSA hierarchy is defined by allowing a bounded number of alternations between AND and OR gates in the circuit. While the polynomial-time approximability of the MMSA hierarchy has been studied extensively, much less is known in the parameterized setting. In particular, k-MMSA2 is the well-known k-SetCover problem, whose parameterized inapproximability lies in the polylog (n) regime. In contrast, k-MMSA4 captures k-MinLabel, for which known lower bounds give poly (n) inapproximability. Sandwiched by k-MMSA2 and k-MMSA4 , the inapproximability of k-MMSA3 remained comparatively unexplored.  \nIn this paper, we give an FPT-time O(2k log n)-approximation algorithm for k-MMSA3 , suggesting that in the fixed-parameter regime, the third level of MMSA remains surprisingly close to the second level. Complementing this algorithm, we also give an FPT-time gap-preserving reduction from k-MMSA3 to k-MMSA2 . Thus, stronger inapproximability for k-MMSA3 would imply new hardness for k-MMSA2 , potentially offering a route around the current barriers for the latter problem.  \nRevisiting Marx’s reduction from k-MMSAt to gap k-MMSAt+2, we also show that k-MMSA4 admits no no(1)-factor FPT approximation unless W[2]=FPT, and no nO(1/k)-factor approximation running in no (k) time under ETH. These results separate the parameterized approximability behavior of the third and fourth levels and clarify where stronger inapproximability enters the k-MMSA hierarchy.  \n∗ Simons Institute for the Theory of Computing, and Departments of EECS & Mathematics, UC Berkeley. Email: [venkatg@berkeley.edu](venkatg@berkeley.edu. Research)[. Research](venkatg@berkeley.edu. Research) supported in part by NSF grant CCF-2211972, a DARPA grant under Contract No. HR0011262E031, and a Simons Investigator award.  \n†State Key Laboratory of Novel Software Technology, Nanjing University. Email: [lin@nju.edu.cn](lin@nju.edu.cn).  \n‡Department of EECS, UC Berkeley. Email: [xuandi_ren@berkeley.edu](xuandi_ren@berkeley.edu. Supported)[. Supported](xuandi_ren@berkeley.edu. Supported) in part by NSF grant CCF- 2228287 and V.G.’s Simons Investigator award.  \n§State Key Laboratory of Novel Software Technology, Nanjing University. Email: [xinzheng@smail.nju.edu.cn](xinzheng@smail.nju.edu.cn).  \n1 Introduction  \nGiven a Boolean circuit C consisting of only ∧ and ∨ gates on n input bits, the Minimum Monotone Satisfying Assignment (MMSA) problem asks to find a satisfying assignment of C with minimum Hamming weight.  \nMMSA was first introduced by Alekhnovich, Buss, Moran and Pitassi [ABMP01] . In a later work, Dinur and Safra [DS04] defined a hierarchy of MMSA problems by restricting the number of alternations of the monotone circuit 1. Specifically, MMSAt refers to the problem where the alternation starts with an ∧, and has at most t levels. This hierarchy turns out to be equivalent to an AND/OR scheduling hierarchy suggested by Goldwasser and Motwani [GM97] .  \nLooking into this hierarchy, MMSA 1 can be solved easily by simply outputting the width of the ∧ gate. MMSA2 is equivalent to the classical Set Cover problem, by identifying each set with a variable and each element to cover with an inner disjunction. Thus, MMSA2 inherits the O(log n)-approximation [Joh74 , Lov75 , Chv79] and (1−o(1))ln n NP-inapproximability [DS14] of Set Cover. In [DS04], Dinur and Safra showed a 2log 1−o(1) n inapproximability for MMSA3 via a reduction from PCP, and placed the Label-Cover problem2 between MMSA3 and MMSA4 . Recently, Chlamtáč, Makarychev, and Vakilian [CMV23] revisited the polynomial-time approximability of this hierarchy. Thappeyr","cbCaikRk3yuGBCZ9","https://ap.wps.com/l/cbCaikRk3yuGBCZ9","pdf",610594,5,1,14,"English","en",105,"# Introduction\n## MMSA and its hierarchy\n## Connections to Set Cover and Label-Cover\n## Parameterized complexity and approximation results","[{\"question\":\"What does the k-MMSA problem ask for, and how are MMSA levels defined?\",\"answer\":\"k-MMSA asks whether a monotone Boolean circuit has a satisfying assignment whose Hamming weight is at most k. MMSA levels are defined by restricting the number of alternations between AND and OR gates in the circuit.\"},{\"question\":\"Why are k-MMSA2 and k-MMSA4 of special interest?\",\"answer\":\"k-MMSA2 is equivalent to the well-known k-SetCover problem, inheriting strong parameterized inapproximability. k-MMSA4 captures k-MinLabel, for which known lower bounds imply poly(n) inapproximability.\"},{\"question\":\"What are the paper’s main contributions regarding k-MMSA3 and its relation to other levels?\",\"answer\":\"The paper provides an FPT-time approximation algorithm for k-MMSA3 with factor O(2^k log n). It also gives an FPT-time gap-preserving reduction from k-MMSA3 to k-MMSA2, so stronger inapproximability for k-MMSA3 would entail new hardness for k-MMSA2.\"}]",1784177353,35,{"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":87,"head_meta":89,"extra_data":91,"updated_unix":28},"on-the-approximability-of-parameterized-minimum-monotone-satisfying-assignment","",{"@graph":36,"@context":86},[37,54,69],{"@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":53},"https://docshare.wps.com/document/on-the-approximability-of-parameterized-minimum-monotone-satisfying-assignment/81971/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-29","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What does the k-MMSA problem ask for, and how are MMSA levels defined?","Question",{"text":76,"@type":77},"k-MMSA asks whether a monotone Boolean circuit has a satisfying assignment whose Hamming weight is at most k. MMSA levels are defined by restricting the number of alternations between AND and OR gates in the circuit.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"Why are k-MMSA2 and k-MMSA4 of special interest?",{"text":81,"@type":77},"k-MMSA2 is equivalent to the well-known k-SetCover problem, inheriting strong parameterized inapproximability. k-MMSA4 captures k-MinLabel, for which known lower bounds imply poly(n) inapproximability.",{"name":83,"@type":74,"acceptedAnswer":84},"What are the paper’s main contributions regarding k-MMSA3 and its relation to other levels?",{"text":85,"@type":77},"The paper provides an FPT-time approximation algorithm for k-MMSA3 with factor O(2^k log n). It also gives an FPT-time gap-preserving reduction from k-MMSA3 to k-MMSA2, so stronger inapproximability for k-MMSA3 would entail new hardness for k-MMSA2.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},"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":20,"slug":138},19,"General","general"]