3,999 research outputs found

    Cross-intersecting sub-families of hereditary families

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    Families A1,A2,...,Ak\mathcal{A}_1, \mathcal{A}_2, ..., \mathcal{A}_k of sets are said to be \emph{cross-intersecting} if for any ii and jj in {1,2,...,k}\{1, 2, ..., k\} with iji \neq j, any set in Ai\mathcal{A}_i intersects any set in Aj\mathcal{A}_j. For a finite set XX, let 2X2^X denote the \emph{power set of XX} (the family of all subsets of XX). A family H\mathcal{H} is said to be \emph{hereditary} if all subsets of any set in H\mathcal{H} are in H\mathcal{H}; so H\mathcal{H} is hereditary if and only if it is a union of power sets. We conjecture that for any non-empty hereditary sub-family H{}\mathcal{H} \neq \{\emptyset\} of 2X2^X and any kX+1k \geq |X|+1, both the sum and product of sizes of kk cross-intersecting sub-families A1,A2,...,Ak\mathcal{A}_1, \mathcal{A}_2, ..., \mathcal{A}_k (not necessarily distinct or non-empty) of H\mathcal{H} are maxima if A1=A2=...=Ak=S\mathcal{A}_1 = \mathcal{A}_2 = ... = \mathcal{A}_k = \mathcal{S} for some largest \emph{star S\mathcal{S} of H\mathcal{H}} (a sub-family of H\mathcal{H} whose sets have a common element). We prove this for the case when H\mathcal{H} is \emph{compressed with respect to an element xx of XX}, and for this purpose we establish new properties of the usual \emph{compression operation}. For the product, we actually conjecture that the configuration A1=A2=...=Ak=S\mathcal{A}_1 = \mathcal{A}_2 = ... = \mathcal{A}_k = \mathcal{S} is optimal for any hereditary H\mathcal{H} and any k2k \geq 2, and we prove this for a special case too.Comment: 13 page

    Cross-intersecting non-empty uniform subfamilies of hereditary families

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    A set AA tt-intersects a set BB if AA and BB have at least tt common elements. A set of sets is called a family. Two families A\mathcal{A} and B\mathcal{B} are cross-tt-intersecting if each set in A\mathcal{A} tt-intersects each set in B\mathcal{B}. A family H\mathcal{H} is hereditary if for each set AA in H\mathcal{H}, all the subsets of AA are in H\mathcal{H}. The rrth level of H\mathcal{H}, denoted by H(r)\mathcal{H}^{(r)}, is the family of rr-element sets in H\mathcal{H}. A set BB in H\mathcal{H} is a base of H\mathcal{H} if for each set AA in H\mathcal{H}, BB is not a proper subset of AA. Let μ(H)\mu(\mathcal{H}) denote the size of a smallest base of H\mathcal{H}. We show that for any integers tt, rr, and ss with 1trs1 \leq t \leq r \leq s, there exists an integer c(r,s,t)c(r,s,t) such that the following holds for any hereditary family H\mathcal{H} with μ(H)c(r,s,t)\mu(\mathcal{H}) \geq c(r,s,t). If A\mathcal{A} is a non-empty subfamily of H(r)\mathcal{H}^{(r)}, B\mathcal{B} is a non-empty subfamily of H(s)\mathcal{H}^{(s)}, A\mathcal{A} and B\mathcal{B} are cross-tt-intersecting, and A+B|\mathcal{A}| + |\mathcal{B}| is maximum under the given conditions, then for some set II in H\mathcal{H} with tIrt \leq |I| \leq r, either A={AH(r) ⁣:IA}\mathcal{A} = \{A \in \mathcal{H}^{(r)} \colon I \subseteq A\} and B={BH(s) ⁣:BIt}\mathcal{B} = \{B \in \mathcal{H}^{(s)} \colon |B \cap I| \geq t\}, or r=sr = s, t<It < |I|, A={AH(r) ⁣:AIt}\mathcal{A} = \{A \in \mathcal{H}^{(r)} \colon |A \cap I| \geq t\}, and B={BH(s) ⁣:IB}\mathcal{B} = \{B \in \mathcal{H}^{(s)} \colon I \subseteq B\}. This was conjectured by the author for t=1t=1 and generalizes well-known results for the case where H\mathcal{H} is a power set.Comment: 15 pages. arXiv admin note: text overlap with arXiv:1805.0524

    Cross-intersecting integer sequences

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    We call (a1,,an)(a_1, \dots, a_n) an \emph{rr-partial sequence} if exactly rr of its entries are positive integers and the rest are all zero. For c=(c1,,cn){\bf c} = (c_1, \dots, c_n) with 1c1cn1 \leq c_1 \leq \dots \leq c_n, let Sc(r)S_{\bf c}^{(r)} be the set of rr-partial sequences (a1,,an)(a_1, \dots, a_n) with 0aici0 \leq a_i \leq c_i for each ii in {1,,n}\{1, \dots, n\}, and let Sc(r)(1)S_{\bf c}^{(r)}(1) be the set of members of Sc(r)S_{\bf c}^{(r)} which have a1=1a_1 = 1. We say that (a1,,an)(a_1, \dots, a_n) \emph{meets} (b1,,bm)(b_1, \dots, b_m) if ai=bi0a_i = b_i \neq 0 for some ii. Two sets AA and BB of sequences are said to be \emph{cross-intersecting} if each sequence in AA meets each sequence in BB. Let d=(d1,,dm){\bf d} = (d_1, \dots, d_m) with 1d1dm1 \leq d_1 \leq \dots \leq d_m. Let ASc(r)A \subseteq S_{\bf c}^{(r)} and BSd(s)B \subseteq S_{\bf d}^{(s)} such that AA and BB are cross-intersecting. We show that ABSc(r)(1)Sd(s)(1)|A||B| \leq |S_{\bf c}^{(r)}(1)||S_{\bf d}^{(s)}(1)| if either c13c_1 \geq 3 and d13d_1 \geq 3 or c=d{\bf c} = {\bf d} and r=s=nr = s = n. We also determine the cases of equality. We obtain this by proving a general cross-intersection theorem for \emph{weighted} sets. The bound generalises to one for k2k \geq 2 cross-intersecting sets.Comment: 20 pages, submitted for publication, presentation improve

    Symmetric Submodular Function Minimization Under Hereditary Family Constraints

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    We present an efficient algorithm to find non-empty minimizers of a symmetric submodular function over any family of sets closed under inclusion. This for example includes families defined by a cardinality constraint, a knapsack constraint, a matroid independence constraint, or any combination of such constraints. Our algorithm make O(n3)O(n^3) oracle calls to the submodular function where nn is the cardinality of the ground set. In contrast, the problem of minimizing a general submodular function under a cardinality constraint is known to be inapproximable within o(n/logn)o(\sqrt{n/\log n}) (Svitkina and Fleischer [2008]). The algorithm is similar to an algorithm of Nagamochi and Ibaraki [1998] to find all nontrivial inclusionwise minimal minimizers of a symmetric submodular function over a set of cardinality nn using O(n3)O(n^3) oracle calls. Their procedure in turn is based on Queyranne's algorithm [1998] to minimize a symmetric submodularComment: 13 pages, Submitted to SODA 201

    Strongly intersecting integer partitions

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    We call a sum a1+a2+• • •+ak a partition of n of length k if a1, a2, . . . , ak and n are positive integers such that a1 ≤ a2 ≤ • • • ≤ ak and n = a1 + a2 + • • • + ak. For i = 1, 2, . . . , k, we call ai the ith part of the sum a1 + a2 + • • • + ak. Let Pn,k be the set of all partitions of n of length k. We say that two partitions a1+a2+• • •+ak and b1+b2+• • •+bk strongly intersect if ai = bi for some i. We call a subset A of Pn,k strongly intersecting if every two partitions in A strongly intersect. Let Pn,k(1) be the set of all partitions in Pn,k whose first part is 1. We prove that if 2 ≤ k ≤ n, then Pn,k(1) is a largest strongly intersecting subset of Pn,k, and uniquely so if and only if k ≥ 4 or k = 3 ≤ n ̸∈ {6, 7, 8} or k = 2 ≤ n ≤ 3.peer-reviewe
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