{"paper":{"title":"The approximability of MAX CSP with fixed-value constraints","arxiv_id":"cs/0602075","primary_cat":"cs.CC","submitted_at":"2006-02-21T14:13:37+00:00","version_compiled":"v1"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13-20260917 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v1","sentences":[{"i":0,"p":1.0,"id":"s0","own":false,"role":"background","text":"In the maximum constraint satisfaction problem (MAX CSP), one is given a finite collection of (possibly weighted) constraints on overlapping sets of variables, and the goal is to assign values from a given finite domain to the variables so as to maximize the number (or the total weight, for the weighted case) of satisfied constraints.","hedge":{"p":0.98,"label":"asserted"},"own_p":0.04,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.1,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":1,"p":1.0,"id":"s1","own":false,"role":"background","text":"This problem is NP-hard in general, and, therefore, it is natural to study how restricting the allowed types of constraints affects the approximability of the problem.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.04,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":2,"p":0.87,"id":"s2","own":true,"role":"result","text":"In this paper, we show that any MAX CSP problem with a finite set of allowed constraint types, which includes all fixed-value constraints (i.e., constraints of the form x=a), is either solvable exactly in polynomial-time or else is APX-complete, even if the number of occurrences of variables in instances are bounded.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.95,"quantitative":false,"quantitative_p":0.38,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.87,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.13,"interpretation":0.0}},{"i":3,"p":0.84,"id":"s3","own":true,"role":"contribution","text":"Moreover, we present a simple description of all polynomial-time solvable cases of our problem.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.06,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.16,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.84,"interpretation":0.0}},{"i":4,"p":0.9,"id":"s4","own":true,"role":"method","text":"This description relies on the well-known algebraic combinatorial property of 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