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2013•arXiv (Cornell University)Open access

On Polynomial Kernels for Integer Linear Programs: Covering, Packing and\n Feasibility

Stefan Kratsch

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Abstract

We study the existence of polynomial kernels for the problem of deciding\nfeasibility of integer linear programs (ILPs), and for finding good solutions\nfor covering and packing ILPs. Our main results are as follows: First, we show\nthat the ILP Feasibility problem admits no polynomial kernelization when\nparameterized by both the number of variables and the number of constraints,\nunless NP \\subseteq coNP/poly. This extends to the restricted cases of bounded\nvariable degree and bounded number of variables per constraint, and to covering\nand packing ILPs. Second, we give a polynomial kernelization for the Cover ILP\nproblem, asking for a solution to Ax >= b with c^Tx <= k, parameterized by k,\nwhen A is row-sparse; this generalizes a known polynomial kernelization for the\nspecial case with 0/1-variables and coefficients (d-Hitting Set).\n

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We study the existence of polynomial kernels for the problem of deciding\nfeasibility of integer linear programs (ILPs), and for finding good solutions\nfor covering and packing ILPs. Our main results are as follows: First, we show\nthat the ILP Feasibility problem admits no polynomial kernelization when\nparameterized by both the number of variables and the number of constraints,\nunless NP \\subseteq coNP/poly. This extends to the restricted cases of bounded\nvariable degree and bounded number of variables per constraint, and to covering\nand packing ILPs. Second, we give a polynomial kernelization for the Cover ILP\nproblem, asking for a solution to Ax >= b with c^Tx <= k, parameterized by k,\nwhen A is row-sparse; this generalizes a known polynomial kernelization for the\nspecial case with 0/1-variables and coefficients (d-Hitting Set).\n

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We study the existence of polynomial kernels for the problem of deciding\nfeasibility of integer linear programs (ILPs), and for finding good solutions\nfor covering and packing ILPs. Our main results are as follows: First, we show\nthat the ILP Feasibility problem admits no polynomial kernelization when\nparameterized by both the number of variables and the number of constraints,\nunless NP \\subseteq coNP/poly. This extends to the restricted cases of bounded\nvariable degree and bounded number of variables per constraint, and to covering\nand packing ILPs. Second, we give a polynomial kernelization for the Cover ILP\nproblem, asking for a solution to Ax >= b with c^Tx <= k, parameterized by k,\nwhen A is row-sparse; this generalizes a known polynomial kernelization for the\nspecial case with 0/1-variables and coefficients (d-Hitting Set).\n

Key concepts: Kernelization, Parameterized complexity, Packing problems, Bounded function, Mathematics, Combinatorics, Polynomial, Integer (computer science)

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