Huge tables and multicommodity flows are fixed-parameter tractable via unimodular integer Carathéodory

Onn, Shmuel

doi:10.1016/j.jcss.2016.07.004

Cited by 4 publications

(5 citation statements)

References 17 publications

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“…, n} : x i = 0} denote the support of x. The main purpose of this paper is to establish lower and and upper bounds on the minimal size of support of an optimal solution to Problem (1) which are polynomial in m and the largest binary encoding length of an entry of A. Polynomial support bounds for integer programming [5,1] have been successfully used in many areas such as in logic and complexity, see [14,12] in the design of efficient polynomial-time approximation schemes [10,11], in fixed parameter tractability [13,16] and they were an ingredient in the solution of cutting stock with a fixed number of item types [8]. These previous bounds however were tailored for the integer feasibility problem only and thus depend on the largest encoding length of a component of the objective function vector if applied to the optimization problem (1).…”

Section: Introductionmentioning

confidence: 99%

The Support of Integer Optimal Solutions

Aliev¹,

Loera²,

Eisenbrand³

et al. 2018

SIAM J. Optim.

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The support of a vector is the number of nonzero-components. We show that given an integral m × n matrix A, the integer linear optimization problem max c T x : Ax = b, x ≥ 0, x ∈ Z n has an optimal solution whose support is bounded by 2m log(2 √ m A ∞ ), where A ∞ is the largest absolute value of an entry of A. Compared to previous bounds, the one presented here is independent on the objective function. We furthermore provide a nearly matching asymptotic lower bound on the support of optimal solutions.

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Section: Introductionmentioning

confidence: 99%

The Support of Integer Optimal Solutions

Aliev¹,

Loera²,

Eisenbrand³

et al. 2018

SIAM J. Optim.

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“…[4,8,20,25]. In particular, it was also used by Goemans and Rothvoß [23] and Onn [47] who are primary inspiration for our work.…”

Section: Related Workmentioning

confidence: 99%

“…Goemans and Rothvoß implicitely deal with the Monoid Decomposition problem [23]. Onn [47] considered the Monoid Decomposition problem in the case when the monoid is defined by a totally unimodular matrix. Recently, Jansen et al [35] considered a different extension of the configuration IP notion to multiple "levels" of configurations, that is, where placements of items are called modules and placements of modules are called configurations.…”

Section: Related Workmentioning

confidence: 99%

Multitype Integer Monoid Optimization and Applications

Knop,

Koutecký,

Levin

et al. 2019

Preprint

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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.

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“…[1,14,22,27]. Jansen and Solis-Oba use a mixed ConfLP to give a parameterized OPT + 1 algorithm for bin packing [24]; Onn [36] gave a weaker form of Theorem 1 which only applies to the setting where E i 1 = I and E i 2 is totally unimodular, for all i. Jansen et al [25] extend the ConfIP to multiple "levels" of configurations. An extended version [31] of this paper shows how to model many scheduling problems as high multiplicity N -fold IPs, so that an application of Theorem 1 yields new parameterized algorithms for these problems.…”

Section: Related Workmentioning

confidence: 99%

High-multiplicity N-fold IP via configuration LP

et al. 2022

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N-fold integer programs (IPs) form an important class of block-structured IPs for which increasingly fast algorithms have recently been developed and successfully applied. We study high-multiplicityN-fold IPs, which encode IPs succinctly by presenting a description of each block type and a vector of block multiplicities. Our goal is to design algorithms which solve N-fold IPs in time polynomial in the size of the succinct encoding, which may be significantly smaller than the size of the explicit (non-succinct) instance. We present the first fixed-parameter algorithm for high-multiplicity N-fold IPs, which even works for convex objectives. Our key contribution is a novel proximity theorem which relates fractional and integer optima of the Configuration LP, a fundamental notion by Gilmore and Gomory [Oper. Res., 1961] which we generalize. Our algorithm for N-fold IP is faster than previous algorithms whenever the number of blocks is much larger than the number of block types, such as in N-fold IP models for various scheduling problems.

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Huge tables and multicommodity flows are fixed-parameter tractable via unimodular integer Carathéodory

Cited by 4 publications

References 17 publications

The Support of Integer Optimal Solutions

The Support of Integer Optimal Solutions

Multitype Integer Monoid Optimization and Applications

High-multiplicity N-fold IP via configuration LP

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