Knapsack cover inequalities for a particular covering problem
$begingroup$
The Knapsack cover inequalities for a constraint $ A_i x geq b$ where $x_{j}in{0,1}$ are:
$$sum_{ j notin S} tilde{ a}_j x_j geq b_i −sum_{ j in S } 1 $$
with $tilde{ a}_j = min {a_j , b_i − sum_{ j in S } 1 }$.
Given a covering polyhedron $P$ given by $ { x : Ax geq b, 0 leq x leq 1 } $ it is known that the integrality gap of $P'$ (given by constraints in $P$ and additional knapsack cover inequalites) and the integral hull of $P$ is at most 2.
Suppose our covering polyhedron is given, that $xin{0,1}^n$ with $n$ even and large, and that the rows of A are all vectors with $n/2$ ones. How would one go about finding a knapsack cover inequality that cuts the point 0.1 * $mathbb{1}$ (with $mathbb{1}$ the all ones vector)? I understand one can do this with iterative Chvátal cuts, but I'm wondering how one would use knapsack cover inequalities to cut such a point.
combinatorics linear-programming integer-programming
edited Dec 28 '18 at 17:20
LinAlg
10.1k1521
asked Dec 14 '18 at 5:08
Hao SunHao Sun
155214
$endgroup$
add a comment |
$begingroup$
The Knapsack cover inequalities for a constraint $ A_i x geq b$ where $x_{j}in{0,1}$ are:
$$sum_{ j notin S} tilde{ a}_j x_j geq b_i −sum_{ j in S } 1 $$
with $tilde{ a}_j = min {a_j , b_i − sum_{ j in S } 1 }$.
Given a covering polyhedron $P$ given by $ { x : Ax geq b, 0 leq x leq 1 } $ it is known that the integrality gap of $P'$ (given by constraints in $P$ and additional knapsack cover inequalites) and the integral hull of $P$ is at most 2.
Suppose our covering polyhedron is given, that $xin{0,1}^n$ with $n$ even and large, and that the rows of A are all vectors with $n/2$ ones. How would one go about finding a knapsack cover inequality that cuts the point 0.1 * $mathbb{1}$ (with $mathbb{1}$ the all ones vector)? I understand one can do this with iterative Chvátal cuts, but I'm wondering how one would use knapsack cover inequalities to cut such a point.
combinatorics linear-programming integer-programming
edited Dec 28 '18 at 17:20
LinAlg
10.1k1521
asked Dec 14 '18 at 5:08
Hao SunHao Sun
155214
$endgroup$
add a comment |
$begingroup$
The Knapsack cover inequalities for a constraint $ A_i x geq b$ where $x_{j}in{0,1}$ are:
$$sum_{ j notin S} tilde{ a}_j x_j geq b_i −sum_{ j in S } 1 $$
with $tilde{ a}_j = min {a_j , b_i − sum_{ j in S } 1 }$.
Given a covering polyhedron $P$ given by $ { x : Ax geq b, 0 leq x leq 1 } $ it is known that the integrality gap of $P'$ (given by constraints in $P$ and additional knapsack cover inequalites) and the integral hull of $P$ is at most 2.
Suppose our covering polyhedron is given, that $xin{0,1}^n$ with $n$ even and large, and that the rows of A are all vectors with $n/2$ ones. How would one go about finding a knapsack cover inequality that cuts the point 0.1 * $mathbb{1}$ (with $mathbb{1}$ the all ones vector)? I understand one can do this with iterative Chvátal cuts, but I'm wondering how one would use knapsack cover inequalities to cut such a point.
combinatorics linear-programming integer-programming
edited Dec 28 '18 at 17:20
LinAlg
10.1k1521
asked Dec 14 '18 at 5:08
Hao SunHao Sun
155214
$endgroup$
The Knapsack cover inequalities for a constraint $ A_i x geq b$ where $x_{j}in{0,1}$ are:
$$sum_{ j notin S} tilde{ a}_j x_j geq b_i −sum_{ j in S } 1 $$
with $tilde{ a}_j = min {a_j , b_i − sum_{ j in S } 1 }$.
Given a covering polyhedron $P$ given by $ { x : Ax geq b, 0 leq x leq 1 } $ it is known that the integrality gap of $P'$ (given by constraints in $P$ and additional knapsack cover inequalites) and the integral hull of $P$ is at most 2.
Suppose our covering polyhedron is given, that $xin{0,1}^n$ with $n$ even and large, and that the rows of A are all vectors with $n/2$ ones. How would one go about finding a knapsack cover inequality that cuts the point 0.1 * $mathbb{1}$ (with $mathbb{1}$ the all ones vector)? I understand one can do this with iterative Chvátal cuts, but I'm wondering how one would use knapsack cover inequalities to cut such a point.
combinatorics linear-programming integer-programming
combinatorics linear-programming integer-programming
edited Dec 28 '18 at 17:20
LinAlg
10.1k1521
asked Dec 14 '18 at 5:08
Hao SunHao Sun
155214
edited Dec 28 '18 at 17:20
LinAlg
10.1k1521
asked Dec 14 '18 at 5:08
Hao SunHao Sun
155214
edited Dec 28 '18 at 17:20
LinAlg
10.1k1521
edited Dec 28 '18 at 17:20
LinAlg
10.1k1521
edited Dec 28 '18 at 17:20
LinAlg
10.1k1521
10.1k1521
asked Dec 14 '18 at 5:08
Hao SunHao Sun
155214
asked Dec 14 '18 at 5:08
Hao SunHao Sun
155214
asked Dec 14 '18 at 5:08
Hao SunHao Sun
155214
155214
add a comment |
add a comment |
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