{"paper":{"title":"Hitting Axis-Parallel Segments with Weighted Points","license":"http://creativecommons.org/licenses/by/4.0/","headline":"An LP-rounding algorithm achieves a randomized (1 + 2/e)-approximation for hitting weighted axis-parallel segments, breaking the factor-2 barrier.","cross_cats":["cs.DS"],"primary_cat":"cs.CG","authors_text":"Jatin Yadav, Rajiv Raman, Siddhartha Sarkar","submitted_at":"2026-05-14T07:38:53Z","abstract_excerpt":"We study a geometric hitting-set problem in which the input consists of a set $P$ of weighted points and a family $S=H\\cup V$ of axis-parallel segments in the plane. The goal is to select a minimum-weight subset of $P$ that hits every segment in $S$. Even restricted geometric hitting-set problems are known to be computationally hard, and for axis-parallel segments the standard decomposition into horizontal and vertical sub-instances yields only a simple factor-$2$ approximation.\n  We present an LP-rounding algorithm that breaks the factor-2 barrier. For the weighted problem, we obtain a random"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We present an LP-rounding algorithm that breaks the factor-2 barrier. For the weighted problem, we obtain a randomized (1+2/e)-approximation by combining systematic rounding on horizontal lines with an exact repair step on residual vertical sub-instances.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The LP relaxation admits an efficient solution whose fractional optimum is within a constant factor of the integral optimum, and that the rounding analysis holds without additional assumptions on point or segment positions beyond axis-parallelism.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"An LP-rounding algorithm yields a randomized (1 + 2/e)-approximation for weighted hitting set of axis-parallel segments, with a (1 + 1/(e-1)) bound in the unweighted case and 1 + 1/e when one orientation consists of lines.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"An LP-rounding algorithm achieves a randomized (1 + 2/e)-approximation for hitting weighted axis-parallel segments, breaking the factor-2 barrier.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"c31ff41486780d14e61ae45da56ea8860db0f8ea942348c6edb38b60943860c4"},"source":{"id":"2605.14499","kind":"arxiv","version":1},"verdict":{"id":"e5f08f6d-769e-407b-bbd9-cc3bc666878e","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T01:10:53.806777Z","strongest_claim":"We present an LP-rounding algorithm that breaks the factor-2 barrier. For the weighted problem, we obtain a randomized (1+2/e)-approximation by combining systematic rounding on horizontal lines with an exact repair step on residual vertical sub-instances.","one_line_summary":"An LP-rounding algorithm yields a randomized (1 + 2/e)-approximation for weighted hitting set of axis-parallel segments, with a (1 + 1/(e-1)) bound in the unweighted case and 1 + 1/e when one orientation consists of lines.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The LP relaxation admits an efficient solution whose fractional optimum is within a constant factor of the integral optimum, and that the rounding analysis holds without additional assumptions on point or segment positions beyond axis-parallelism.","pith_extraction_headline":"An LP-rounding algorithm achieves a randomized (1 + 2/e)-approximation for hitting weighted axis-parallel segments, breaking the factor-2 barrier."},"references":{"count":34,"sample":[{"doi":"","year":2019,"title":"Approximationschemesforindependent set and sparse subsets of polygons.Journal of the ACM, 66(4):29:1–29:40, 2019","work_id":"8b2999d8-49e2-4601-9e26-22ea57dc1177","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2012,"title":"A non-linear lower bound for planar epsilon-nets.Discret","work_id":"20aecde9-b3dd-4ed4-ab34-20b83222306b","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2010,"title":"Small-sizeϵ-nets for axis-parallel rectangles and boxes.SIAM J","work_id":"a59230f4-3ca2-4e4e-a022-8ecfd1ebaa06","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2013,"title":"Katz, Gila Morgenstern, and Yelena Yuditsky","work_id":"c3afed10-afbc-4cf1-b508-a02ae7220bd0","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"On the number of points in general position in the plane","work_id":"2db04dd3-3eac-4738-a324-652cf1cdb30b","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":34,"snapshot_sha256":"23ef8044d10a4bf6980b7cd5e290253318e2b863e5089860bdc20e4d45b55b9e","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}