
Faster Algorithms for Integer Programs with Block Structure
We consider integer programming problems { c^T x : A x = b, l ≤ x ≤ u, x...
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Evaluating and Tuning nfold Integer Programming
In recent years, algorithmic breakthroughs in stringology, computational...
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Determining rRobustness of Arbitrary Digraphs Using ZeroOne Linear Integer Programming
There has been an increase in the use of resilient control algorithms ba...
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A Parameterized Strongly Polynomial Algorithm for Block Structured Integer Programs
The theory of nfold integer programming has been recently emerging as a...
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Most IPs with bounded determinants can be solved in polynomial time
In 1983 Lenstra showed that an integer program (IP) is fixed parameter t...
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On the Integrality Gap of Binary Integer Programs with Gaussian Data
For a binary integer program (IP) max c^𝖳 x, Ax ≤ b, x ∈{0,1}^n, where A...
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NearLinear Time Algorithm for nfold ILPs via Color Coding
We study an important case of ILPs {c^Tx Ax = b, l ≤ x ≤ u, x ∈Z^n t...
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On the Graver basis of blockstructured integer programming
We consider the 4block nfold integer programming (IP), in which the constraint matrix consists of n copies of small matrices A, B, D and one copy of C in a specific block structure. We prove that, the ℓ_∞norm of the Graver basis elements of 4block nfold IP is upper bounded by O_FPT(n^s_c) where s_c is the number of rows of matrix C and O_FPT hides a multiplicative factor that is only dependent on the parameters of the small matrices A,B,C,D (i.e., the number of rows and columns, and the largest absolute value among the entries). This improves upon the existing upper bound of O_FPT(n^2^s_c). We provide a matching lower bounded of Ω(n^s_c), which even holds for an arbitrary nonzero integral element in the kernel space. We then consider a special case of 4block nfold in which C is a zero matrix (called 3block nfold IP). We show that, surprisingly, 3block nfold IP admits a Hilbert basis whose ℓ_∞norm is bounded by O_FPT(1), despite the fact that the ℓ_∞norm of its Graver basis elements is still Ω(n). Finally, we provide upper bounds on the ℓ_∞norm of Graver basis elements for 3block nfold IP. Based on these upper bounds, we establish algorithms for 3block nfold IP and provide improved algorithms for 4block nfold IP.
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