{"id":324296,"date":"2016-11-18T15:11:39","date_gmt":"2016-11-18T23:11:39","guid":{"rendered":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/?post_type=msr-research-item&#038;p=324296"},"modified":"2018-10-16T20:38:46","modified_gmt":"2018-10-17T03:38:46","slug":"testing-coverage-functions","status":"publish","type":"msr-research-item","link":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/publication\/testing-coverage-functions\/","title":{"rendered":"Testing Coverage Functions"},"content":{"rendered":"<p class=\"Para\">A <em class=\"EmphasisTypeItalic \">coverage function<\/em> <em class=\"EmphasisTypeItalic \">f<\/em> over a ground set [<em class=\"EmphasisTypeItalic \">m<\/em>] is associated with a universe <em class=\"EmphasisTypeItalic \">U<\/em> of weighted elements and <em class=\"EmphasisTypeItalic \">m<\/em> sets <em class=\"EmphasisTypeItalic \">A<\/em> <sub>1<\/sub>,\u2026,<em class=\"EmphasisTypeItalic \">A<\/em> <sub><em class=\"EmphasisTypeItalic \">m<\/em> <\/sub>\u2009\u2286\u2009<em class=\"EmphasisTypeItalic \">U<\/em>, and for any <em class=\"EmphasisTypeItalic \">T<\/em>\u2009\u2286\u2009[<em class=\"EmphasisTypeItalic \">m<\/em>], <em class=\"EmphasisTypeItalic \">f<\/em>(<em class=\"EmphasisTypeItalic \">T<\/em>) is defined as the total weight of the elements in the union \u222a\u2009<sub> <em class=\"EmphasisTypeItalic \">j<\/em>\u2009\u2208\u2009<em class=\"EmphasisTypeItalic \">T<\/em> <\/sub><em class=\"EmphasisTypeItalic \">A<\/em> <sub><em class=\"EmphasisTypeItalic \">j<\/em> <\/sub>. Coverage functions are an important special case of submodular functions, and arise in many applications, for instance as a class of utility functions of agents in combinatorial auctions.<\/p>\n<p class=\"Para\">Set functions such as coverage functions often lack succinct representations, and in algorithmic applications, an access to a value oracle is assumed. In this paper, we ask whether one can test if a given oracle is that of a coverage function or not. We demonstrate an algorithm which makes <em class=\"EmphasisTypeItalic \">O<\/em>(<em class=\"EmphasisTypeItalic \">m<\/em>|<em class=\"EmphasisTypeItalic \">U<\/em>|) queries to an oracle of a coverage function and completely reconstructs it. This gives a polytime tester for <em class=\"EmphasisTypeItalic \">succinct<\/em> coverage functions for which |<em class=\"EmphasisTypeItalic \">U<\/em>| is polynomially bounded in <em class=\"EmphasisTypeItalic \">m<\/em>. In contrast, we demonstrate a set function which is \u201cfar\u201d from coverage, but requires <span id=\"IEq1\" class=\"InlineEquation\"><span id=\"MathJax-Element-1-Frame\" class=\"MathJax\" style=\"border: 0px; font-style: normal; font-variant: inherit; font-weight: normal; font-stretch: inherit; font-size: 13px; line-height: normal; font-family: inherit; margin: 0px; padding: 0px; vertical-align: baseline; outline: 0px; display: inline; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;\" tabindex=\"0\" data-mathml=\"<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mn>2<\/mn><mrow class=\"MJX-TeXAtom-ORD\"><mrow class=\"MJX-TeXAtom-ORD\"><mover><mi mathvariant=\"normal\">&#x0398;<\/mi><mo stretchy=\"false\">&#x007E;<\/mo><\/mover><\/mrow><mo stretchy=\"false\">(<\/mo><mi>m<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/msup><\/math>\"><span id=\"MathJax-Span-1\" class=\"math\"><span id=\"MathJax-Span-2\" class=\"mrow\"><span id=\"MathJax-Span-3\" class=\"msubsup\"><span id=\"MathJax-Span-4\" class=\"mn\">2<\/span><span id=\"MathJax-Span-5\" class=\"texatom\"><span id=\"MathJax-Span-6\" class=\"mrow\"><span id=\"MathJax-Span-7\" class=\"texatom\"><span id=\"MathJax-Span-8\" class=\"mrow\"><span id=\"MathJax-Span-9\" class=\"munderover\"><span id=\"MathJax-Span-10\" class=\"mi\">\u0398<\/span><span id=\"MathJax-Span-11\" class=\"mo\">~<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-12\" class=\"mo\">(<\/span><span id=\"MathJax-Span-13\" class=\"mi\">m<\/span><span id=\"MathJax-Span-14\" class=\"mo\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00a0queries to distinguish it from the class of coverage functions.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A coverage function f over a ground set [m] is associated with a universe U of weighted elements and m sets A 1,\u2026,A m \u2009\u2286\u2009U, and for any T\u2009\u2286\u2009[m], f(T) is defined as the total weight of the elements in the union \u222a\u2009 j\u2009\u2208\u2009T A j . Coverage functions are an important special case of [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"Springer Berlin Heidelberg","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"39th International Colloquium, ICALP 2012, Warwick, UK","msr_editors":"","msr_how_published":"","msr_isbn":"978-3-642-31593-0","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"170-181","msr_page_range_start":"170","msr_page_range_end":"181","msr_series":"","msr_volume":"7391","msr_copyright":"","msr_conference_name":"39th International Colloquium, ICALP 2012, Warwick, 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