A Parameterized Strongly Polynomial Algorithm for Block Structured Integer Programs

Abstract

The theory of n-fold integer programming has been recently emerging as an important tool in parameterized complexity. The input to an n-fold integer program (IP) consists of parameter A, dimension n, and numerical data of binary encoding length L. It was known for some time that such programs can be solved in polynomial time using O(n g(A) L) arithmetic operations where g is an exponential function of the parameter. In 2013 it was shown that it can be solved in fixed-parameter tractable time using O(f (A)n 3 L) arithmetic operations for a single-exponential function f . This, and a faster algorithm for a special case of combinatorial n-fold IP, have led to several very recent breakthroughs in the parameterized complexity of scheduling, stringology, and computational social choice. In 2015 it was shown that it can be solved in strongly polynomial time using O(n g(A) ) arithmetic operations.Here we establish a result which subsumes all three of the above results by showing that n-fold IP can be solved in strongly polynomial fixed-parameter tractable time using O(f (A)n 3 ) arithmetic operations. In fact, our results are much more general, briefly outlined as follows.There is a strongly polynomial algorithm for integer linear programming (ILP) whenever a so-called Graver-best oracle is realizable for it. Graver-best oracles for the large classes of multi-stage stochastic and tree-fold ILPs can be realized in fixed-parameter tractable time. Together with the previous oracle algorithm, this newly shows two large classes of ILP to be strongly polynomial; in contrast, only few classes of ILP were previously known to be strongly polynomial.We show that ILP is fixed-parameter tractable parameterized by the largest coefficient A ∞ and the primal or dual treedepth of A, and that this parameterization cannot be relaxed, signifying substantial progress in understanding the parameterized complexity of ILP.

New Bounds on Augmenting Steps of Block-structured Integer Programs

“…Therefore, even augmenting via a different set of steps may have to deal with steps that are unbounded by O FPT (1). Combining Theorem 1 with the original idea of Hemmecke et al [14] and a strongly polynomial framework of Koutecký et al [25], we obtain the currently fastest algorithm for 4-block n-fold IP: Theorem 3. 4-block n-fold IP can be solved in time O FPT (n O(s D t B ) ).…”

“…Later, this FPT algorithm inspired a better algorithm for a special case of combinatorial n-fold IP developed by Knop et al [23], who also apply it to problems in stringology and graph algorithms. Finally, this algorithm was lifted to the general n-fold IP by Koutecký et al [25] and Eisenbrand et al [7].…”

“…Ganian and Ordyniak [10] studied the structural parameters primal treedepth and treewidth, and later Ganian et al [11] studied dual and incidence treedepth and treewidth. Koutecký et al [25] discovered that tree-fold and multi-stage stochastic IPs are essentially equivalent to IPs with small dual and primal treedepth, settling the parameterized complexity with respect to these parameters. The work of Koutecký et al [25] subsumes essentially all current knowledge about the solvability of IP in variable dimension with the exception of totally unimodular constraint matrices and two related classes [2,3], with the main remaining open problem being the complexity of 4-block n-fold IP and, more generally, IP with respect to incidence treedepth.…”

“…Historically, all tractable classes of IP were discovered by proving new norm bounds and subsequently designing a dynamic program around them, with the former typically being much harder than the latter. Moreover, recent runtime improvements have followed from improving existing bounds [7,25], and the most challenging questions in the field are tightly connected to norm bounds. Our focus here is the currently least understood class of IPs, 4-block n-fold IP:…”

“…, s D ,t D , ∆)n O (1) (i.e., an FPT algorithm; see below) remains unknown. From another perspective, Koutecký et al [25] has recently resolved the complexity of IP with respect to the structural parameters primal and dual treedepth td P and td D , respectively, by showing that IPs with small td P and td D are efficiently solveable. IPs with small incidence treedepth td I subsume both of the aforementioned classes as well as 4block n-fold IP, and 4-block n-fold IP remains the simplest open case with respect to td I .…”