{"title":"On Approximability of Satisfiable k-CSPs: V","authors":"Amey Bhangale, Subhash Khot, Dor Minzer","doi":"arxiv-2408.15377","DOIUrl":"https://doi.org/arxiv-2408.15377","url":null,"abstract":"We propose a framework of algorithm vs. hardness for all Max-CSPs and\u0000demonstrate it for a large class of predicates. This framework extends the work\u0000of Raghavendra [STOC, 2008], who showed a similar result for almost satisfiable\u0000Max-CSPs. Our framework is based on a new hybrid approximation algorithm, which uses a\u0000combination of the Gaussian elimination technique (i.e., solving a system of\u0000linear equations over an Abelian group) and the semidefinite programming\u0000relaxation. We complement our algorithm with a matching dictator vs.\u0000quasirandom test that has perfect completeness. The analysis of our dictator vs. quasirandom test is based on a novel\u0000invariance principle, which we call the mixed invariance principle. Our mixed\u0000invariance principle is an extension of the invariance principle of Mossel,\u0000O'Donnell and Oleszkiewicz [Annals of Mathematics, 2010] which plays a crucial\u0000role in Raghavendra's work. The mixed invariance principle allows one to relate\u00003-wise correlations over discrete probability spaces with expectations over\u0000spaces that are a mixture of Guassian spaces and Abelian groups, and may be of\u0000independent interest.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185875","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Lossy Catalytic Computation","authors":"Chetan Gupta, Rahul Jain, Vimal Raj Sharma, Raghunath Tewari","doi":"arxiv-2408.14670","DOIUrl":"https://doi.org/arxiv-2408.14670","url":null,"abstract":"A catalytic Turing machine is a variant of a Turing machine in which there\u0000exists an auxiliary tape in addition to the input tape and the work tape. This\u0000auxiliary tape is initially filled with arbitrary content. The machine can read\u0000and write on the auxiliary tape, but it is constrained to restore its initial\u0000content when it halts. Studying such a model and finding its powers and\u0000limitations has practical applications. In this paper, we study catalytic Turing machines with O(log n)-sized work\u0000tape and polynomial-sized auxiliary tape that are allowed to lose at most\u0000constant many bits of the auxiliary tape when they halt. We show that such\u0000catalytic Turing machines can only decide the same set of languages as standard\u0000catalytic Turing machines with the same size work and auxiliary tape.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185877","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"CSPs with Few Alien Constraints","authors":"Peter Jonsson, Victor Lagerkvist, George Osipov","doi":"arxiv-2408.12909","DOIUrl":"https://doi.org/arxiv-2408.12909","url":null,"abstract":"The constraint satisfaction problem asks to decide if a set of constraints\u0000over a relational structure $mathcal{A}$ is satisfiable (CSP$(mathcal{A})$).\u0000We consider CSP$(mathcal{A} cup mathcal{B})$ where $mathcal{A}$ is a\u0000structure and $mathcal{B}$ is an alien structure, and analyse its\u0000(parameterized) complexity when at most $k$ alien constraints are allowed. We\u0000establish connections and obtain transferable complexity results to several\u0000well-studied problems that previously escaped classification attempts. Our\u0000novel approach, utilizing logical and algebraic methods, yields an FPT versus\u0000pNP dichotomy for arbitrary finite structures and sharper dichotomies for\u0000Boolean structures and first-order reducts of $(mathbb{N},=)$ (equality CSPs),\u0000together with many partial results for general $omega$-categorical structures.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185878","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the computational power of $C$-random strings","authors":"Alexey Milovanov","doi":"arxiv-2409.04448","DOIUrl":"https://doi.org/arxiv-2409.04448","url":null,"abstract":"Denote by $H$ the Halting problem. Let $R_U: = { x | C_U(x) ge |x|}$,\u0000where $C_U(x)$ is the plain Kolmogorov complexity of $x$ under a universal\u0000decompressor $U$. We prove that there exists a universal $U$ such that $H in\u0000P^{R_U}$, solving the problem posted by Eric Allender.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185887","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Rida Ait El Manssour, Nikhil Balaji, Klara Nosan, Mahsa Shirmohammadi, James Worrell
{"title":"A parametric version of the Hilbert Nullstellensatz","authors":"Rida Ait El Manssour, Nikhil Balaji, Klara Nosan, Mahsa Shirmohammadi, James Worrell","doi":"arxiv-2408.13027","DOIUrl":"https://doi.org/arxiv-2408.13027","url":null,"abstract":"Hilbert's Nullstellensatz is a fundamental result in algebraic geometry that\u0000gives a necessary and sufficient condition for a finite collection of\u0000multivariate polynomials to have a common zero in an algebraically closed\u0000field. Associated with this result, there is the computational problem HN of\u0000determining whether a system of polynomials with coefficients in the field of\u0000rational numbers has a common zero over the field of algebraic numbers. In an influential paper, Koiran showed that HN can be determined in the\u0000polynomial hierarchy assuming the Generalised Riemann Hypothesis (GRH). More\u0000precisely, he showed that HN lies in the complexity class AM under GRH. In a\u0000later work he generalised this result by showing that the problem DIM, which\u0000asks to determine the dimension of the set of solutions of a given polynomial\u0000system, also lies in AM subject to GRH. In this paper we study the solvability of polynomial equations over arbitrary\u0000algebraically closed fields of characteristic zero. Up to isomorphism, every\u0000such field is the algebraic closure of a field of rational functions. We thus\u0000formulate a parametric version of HN, called HNP, in which the input is a\u0000system of polynomials with coefficients in a function field\u0000$mathbb{Q}(mathbf{x})$ and the task is to determine whether the polynomials\u0000have a common zero in the algebraic closure\u0000$overline{mathbb{Q}(mathbf{x})}$. We observe that Koiran's proof that DIM lies in AM can be interpreted as a\u0000randomised polynomial-time reduction of DIM to HNP, followed by an argument\u0000that HNP lies in AM. Our main contribution is a self-contained proof that HNP\u0000lies in AM that follows the same basic idea as Koiran's argument -- namely\u0000random instantiation of the parameters -- but whose justification is purely\u0000algebraic, relying on a parametric version of Hilbert's Nullstellensatz, and\u0000avoiding recourse to semi-algebraic geometry.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185876","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Domination in Diameter-Two Graphs and the 2-Club Cluster Vertex Deletion Parameter","authors":"Faisal N. Abu-Khzam, Lucas Isenmann","doi":"arxiv-2408.08418","DOIUrl":"https://doi.org/arxiv-2408.08418","url":null,"abstract":"The s-club cluster vertex deletion number of a graph, or sccvd, is the\u0000minimum number of vertices whose deletion results in a disjoint union of\u0000s-clubs, or graphs whose diameter is bounded above by s. We launch a study of\u0000several domination problems on diameter-two graphs, or 2-clubs, and study their\u0000parameterized complexity with respect to the 2ccvd number as main parameter. We\u0000further propose to explore the class of problems that become solvable in\u0000sub-exponential time when the running time is independent of some input\u0000parameter. Hardness of problems for this class depends on the Exponential-Time\u0000Hypothesis. We give examples of problems that are in the proposed class and\u0000problems that are hard for it.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142224303","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Oracle without Optimal Proof Systems outside Nondeterministic Subexponential Time","authors":"Fabian Egidy, Christian Glaßer","doi":"arxiv-2408.07408","DOIUrl":"https://doi.org/arxiv-2408.07408","url":null,"abstract":"We study the existence of optimal proof systems for sets outside of\u0000$mathrm{NP}$. Currently, no set $L notin mathrm{NP}$ is known that has\u0000optimal proof systems. Our main result shows that this is not surprising,\u0000because we can rule out relativizable proofs of optimality for all sets outside\u0000$mathrm{NTIME}(t)$ where $t$ is slightly superpolynomial. We construct an\u0000oracle $O$, such that for any set $L subseteq Sigma^*$ at least one of the\u0000following two properties holds: $L$ does not have optimal proof systems\u0000relative to $O$. $L in mathrm{UTIME}^O(2^{2(log\u0000n)^{8+4log(log(log(n)))}})$. The runtime bound is slightly superpolynomial.\u0000So there is no relativizable proof showing that a complex set has optimal proof\u0000systems. Hence, searching for non-trivial optimal proof systems with\u0000relativizable methods can only be successful (if at all) in a narrow range\u0000above $mathrm{NP}$.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185879","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Giuseppe Mazzotta, Francesco Ricca, Mirek Truszczynski
{"title":"Quantifying over Optimum Answer Sets","authors":"Giuseppe Mazzotta, Francesco Ricca, Mirek Truszczynski","doi":"arxiv-2408.07697","DOIUrl":"https://doi.org/arxiv-2408.07697","url":null,"abstract":"Answer Set Programming with Quantifiers (ASP(Q)) has been introduced to\u0000provide a natural extension of ASP modeling to problems in the polynomial\u0000hierarchy (PH). However, ASP(Q) lacks a method for encoding in an elegant and\u0000compact way problems requiring a polynomial number of calls to an oracle in\u0000$Sigma_n^p$ (that is, problems in $Delta_{n+1}^p$). Such problems include, in\u0000particular, optimization problems. In this paper we propose an extension of\u0000ASP(Q), in which component programs may contain weak constraints. Weak\u0000constraints can be used both for expressing local optimization within\u0000quantified component programs and for modeling global optimization criteria. We\u0000showcase the modeling capabilities of the new formalism through various\u0000application scenarios. Further, we study its computational properties obtaining\u0000complexity results and unveiling non-obvious characteristics of ASP(Q) programs\u0000with weak constraints.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185880","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the NP-Hardness Approximation Curve for Max-2Lin(2)","authors":"Björn Martinsson","doi":"arxiv-2408.04832","DOIUrl":"https://doi.org/arxiv-2408.04832","url":null,"abstract":"In the maxtlint{} problem you are given a system of equations on the form\u0000$x_i + x_j equiv b pmod{2}$, and your objective is to find an assignment that\u0000satisfies as many equations as possible. Let $c in [0.5, 1]$ denote the\u0000maximum fraction of satisfiable equations. In this paper we construct a curve\u0000$s (c)$ such that it is NPhard{} to find a solution satisfying at least a\u0000fraction $s$ of equations. This curve either matches or improves all of the\u0000previously known inapproximability NP-hardness results for maxtlint{}. In\u0000particular, we show that if $c geqslant 0.9232$ then $frac{1 - s (c)}{1 - c}\u0000> 1.48969$, which improves the NP-hardness inapproximability constant for the\u0000min deletion version of maxtlint{}. Our work complements the work of O'Donnell\u0000and Wu that studied the same question assuming the Unique Games Conjecture. Similar to earlier inapproximability results for maxtlint{}, we use a gadget\u0000reduction from the $(2^k - 1)$-ary Hadamard predicate. Previous works used $k$\u0000ranging from $2$ to $4$. Our main result is a procedure for taking a gadget for\u0000some fixed $k$, and use it as a building block to construct better and better\u0000gadgets as $k$ tends to infinity. Our method can be used to boost the result of\u0000both smaller gadgets created by hand $(k = 3)$ or larger gadgets constructed\u0000using a computer $(k = 4)$.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141941786","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Léo Saulières, Martin C. Cooper, Florence Dupin de Saint Cyr
{"title":"Backward explanations via redefinition of predicates","authors":"Léo Saulières, Martin C. Cooper, Florence Dupin de Saint Cyr","doi":"arxiv-2408.02606","DOIUrl":"https://doi.org/arxiv-2408.02606","url":null,"abstract":"History eXplanation based on Predicates (HXP), studies the behavior of a\u0000Reinforcement Learning (RL) agent in a sequence of agent's interactions with\u0000the environment (a history), through the prism of an arbitrary predicate. To\u0000this end, an action importance score is computed for each action in the\u0000history. The explanation consists in displaying the most important actions to\u0000the user. As the calculation of an action's importance is #W[1]-hard, it is\u0000necessary for long histories to approximate the scores, at the expense of their\u0000quality. We therefore propose a new HXP method, called Backward-HXP, to provide\u0000explanations for these histories without having to approximate scores.\u0000Experiments show the ability of B-HXP to summarise long histories.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141941787","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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