3,999 research outputs found
Cross-intersecting sub-families of hereditary families
Families of sets are said
to be \emph{cross-intersecting} if for any and in
with , any set in intersects any set in
. For a finite set , let denote the \emph{power set of
} (the family of all subsets of ). A family is said to be
\emph{hereditary} if all subsets of any set in are in
; so is hereditary if and only if it is a union of
power sets. We conjecture that for any non-empty hereditary sub-family
of and any , both the sum
and product of sizes of cross-intersecting sub-families (not necessarily distinct or non-empty) of
are maxima if for some largest \emph{star of
} (a sub-family of whose sets have a common
element). We prove this for the case when is \emph{compressed
with respect to an element of }, and for this purpose we establish new
properties of the usual \emph{compression operation}. For the product, we
actually conjecture that the configuration is optimal for any hereditary and
any , and we prove this for a special case too.Comment: 13 page
Cross-intersecting non-empty uniform subfamilies of hereditary families
A set -intersects a set if and have at least common
elements. A set of sets is called a family. Two families and
are cross--intersecting if each set in
-intersects each set in . A family is hereditary
if for each set in , all the subsets of are in
. The th level of , denoted by
, is the family of -element sets in . A set
in is a base of if for each set in
, is not a proper subset of . Let denote
the size of a smallest base of . We show that for any integers
, , and with , there exists an integer
such that the following holds for any hereditary family
with . If is a
non-empty subfamily of , is a non-empty
subfamily of , and are
cross--intersecting, and is maximum under
the given conditions, then for some set in with , either and ,
or , , , and . This was conjectured by the author for and generalizes well-known
results for the case where is a power set.Comment: 15 pages. arXiv admin note: text overlap with arXiv:1805.0524
Cross-intersecting integer sequences
We call an \emph{-partial sequence} if exactly of
its entries are positive integers and the rest are all zero. For with , let
be the set of -partial sequences with for each in , and let be the set
of members of which have . We say that \emph{meets} if for some . Two
sets and of sequences are said to be \emph{cross-intersecting} if each
sequence in meets each sequence in . Let
with . Let and such that and are cross-intersecting. We
show that if either and or and . 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 cross-intersecting sets.Comment: 20 pages, submitted for publication, presentation improve
Symmetric Submodular Function Minimization Under Hereditary Family Constraints
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 oracle calls to the submodular
function where 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 (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 using 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
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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