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Multitype integer monoid optimization and applications
Citation Link: https://doi.org/10.15480/882.4138
Publikationstyp
Preprint
Date Issued
2019-09-16
Sprache
English
Institut
TORE-DOI
TORE-URI
Configuration integer programs (IP) have been key in the design of algorithms for NP-hard
high-multiplicity problems since the pioneering work of Gilmore and Gomory [Oper. Res., 1961].
Configuration IPs have one variable for each possible configuration, which describes a placement
of items into a location, and whose value corresponds to the number of locations with that
placement. In high multiplicity problems items come in types, and are represented succinctly
by a vector of multiplicities; solving the configuration IP then amounts to deciding whether the
input vector of multiplicities of items of each type can be decomposed into a given number of
configurations.
We make this typically implicit notion explicit by observing that the set of all input vectors
which can be decomposed into configurations forms a monoid of configurations, and the problem
corresponding to solving the configuration IP is the Monoid Decomposition problem. Then,
motivated by applications, we enrich this problem in two ways. First, in certain problems each
configuration additionally has an objective value, and the problem becomes an optimization
problem of finding a “best” decomposition under the given objective. Second, there are often
different types of configurations derived from different types of locations. The resulting problem
is then to optimize over decompositions of the input multiplicity vector into configurations of
several types, and we call it Multitype Integer Monoid Optimization, or simply MIMO.
We develop fast exact (exponential-time) algorithms for various MIMO with few or many
location types and with various objectives. Our algorithms build on a novel proximity theorem
which connects the solutions of a certain configuration IP to those of its continuous relaxation.
We then cast several fundamental scheduling and bin packing problems as MIMOs, and thereby
obtain new or substantially faster algorithms for them.
We complement our positive algorithmic results by hardness results which show that, under
common complexity assumptions, the algorithms cannot be extended into more relaxed regimes.
high-multiplicity problems since the pioneering work of Gilmore and Gomory [Oper. Res., 1961].
Configuration IPs have one variable for each possible configuration, which describes a placement
of items into a location, and whose value corresponds to the number of locations with that
placement. In high multiplicity problems items come in types, and are represented succinctly
by a vector of multiplicities; solving the configuration IP then amounts to deciding whether the
input vector of multiplicities of items of each type can be decomposed into a given number of
configurations.
We make this typically implicit notion explicit by observing that the set of all input vectors
which can be decomposed into configurations forms a monoid of configurations, and the problem
corresponding to solving the configuration IP is the Monoid Decomposition problem. Then,
motivated by applications, we enrich this problem in two ways. First, in certain problems each
configuration additionally has an objective value, and the problem becomes an optimization
problem of finding a “best” decomposition under the given objective. Second, there are often
different types of configurations derived from different types of locations. The resulting problem
is then to optimize over decompositions of the input multiplicity vector into configurations of
several types, and we call it Multitype Integer Monoid Optimization, or simply MIMO.
We develop fast exact (exponential-time) algorithms for various MIMO with few or many
location types and with various objectives. Our algorithms build on a novel proximity theorem
which connects the solutions of a certain configuration IP to those of its continuous relaxation.
We then cast several fundamental scheduling and bin packing problems as MIMOs, and thereby
obtain new or substantially faster algorithms for them.
We complement our positive algorithmic results by hardness results which show that, under
common complexity assumptions, the algorithms cannot be extended into more relaxed regimes.
Subjects
integer programming
configuration IP
proximity theorems
scheduling
DDC Class
510: Mathematik
More Funding Information
Deutsche Forschungsgemeinschaft (DFG),
OP VVV MEYS Research Center for Informatics,
Israel Science Foundation,
Charles University,
GA ČR,
German-Israeli Foundation for Scientific Research and Development (GIF),
Dresner Chair,
MaMu, NI 369/19,
CZ.02.1.01/0.0/0.0/16 019/0000765,
308/18,
UNCE/SCI/004,
7-0914-2S,
I-1366- 407.6/2016,
MN 59/4-1
OP VVV MEYS Research Center for Informatics,
Israel Science Foundation,
Charles University,
GA ČR,
German-Israeli Foundation for Scientific Research and Development (GIF),
Dresner Chair,
MaMu, NI 369/19,
CZ.02.1.01/0.0/0.0/16 019/0000765,
308/18,
UNCE/SCI/004,
7-0914-2S,
I-1366- 407.6/2016,
MN 59/4-1
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