Evaluating and Tuning n -fold Integer Programming

Abstract

In recent years, algorithmic breakthroughs in stringology, computational social choice, scheduling, etc., were achieved by applying the theory of so-called n-fold integer programming. An n-fold integer program (IP) has a highly uniform block structured constraint matrix. Hemmecke, Onn, and Romanchuk [Math. Programming, 2013] showed an algorithm with runtime ∆ O(rst+r 2 s) n 3 , where ∆ is the largest coefficient, r, s, and t are dimensions of blocks of the constraint matrix and n is the total dimension of the IP; thus, an algorithm efficient if the blocks are of small size and with small coefficients. The algorithm works by iteratively improving a feasible solution with augmenting steps, and n-fold IPs have the special property that augmenting steps are guaranteed to exist in a not-too-large neighborhood. However, this algorithm has never been implemented and evaluated.We have implemented the algorithm and learned the following along the way. The original algorithm is practically unusable, but we discover a series of improvements which make its evaluation possible. Crucially, we observe that a certain constant in the algorithm can be treated as a tuning parameter, which yields an efficient heuristic (essentially searching in a smaller-than-guaranteed neighborhood). Furthermore, the algorithm uses an overly expensive strategy to find a "best" step, while finding only an "approximately best" step is much cheaper, yet sufficient for quick convergence. Using this insight, we improve the asymptotic dependence on n from n 3 to n 2 log n.Finally, we tested the behavior of the algorithm with various values of the tuning parameter and different strategies of finding improving steps. First, we show that decreasing the tuning parameter initially leads to an increased number of iterations needed for convergence and eventually to getting stuck in local optima, as expected. However, surprisingly small values of the parameter already exhibit good behavior while significantly lowering the time the algorithm spends per single iteration. Second, our new strategy for finding "approximately best" steps wildly outperforms the original construction. ACM Subject ClassificationTheory of computation → Parameterized complexity and exact algorithms; Mathematics of computing → Solvers; Theory of computation → Discrete optimization Keywords and phrases n-fold integer programming, integer programming, analysis of algorithms, primal heuristic, local search Supplement Material https://github.com/katealtmanova/nfoldexperiment

Integer Programming and Incidence Treedepth

“…In other words, maxdeg C (A) denotes the maximum number of nonzeros in a row of A and maxdeg V (A) denotes the maximum number of nonzeros in a column of A. Now, we get that ILP can be solved in time f (maxdeg C (A), maxdeg V (A), td I (A))L O (1) , where f is some computable function and L is the length of the encoding of the given ILP thanks to Lenstra's algorithm [19]. The above observation can in fact be strengthened-namely, if the arity of all the constraints or the number of occurences of all the variables in the given SIP is bounded, then we obtain a bound on either primal or dual treedepth.…”

“…In other words, maxdeg C (A) denotes the maximum number of nonzeros in a row of A and maxdeg V (A) denotes the maximum number of nonzeros in a column of A. Now, we get that ILP can be solved in time f (maxdeg C (A), maxdeg V (A), td I (A))L O (1) , where f is some computable function and L is the length of the encoding of the given ILP thanks to Lenstra's algorithm [19].…”

“…Their algorithm has been subsequently improved [18,9]. Altmanová et al [1] implemented the algorithm of Hemmecke et al [14] and improved the polynomial factor (achieving the same running time as Eisenbrand et al [9]) the above algorithms (from cubic dependence to n 2 log n). The best running time of an algorithm solving n-fold IP is due to Jansen et al [16] and runs in nearly linear time in terms of n.…”

“…It follows that if we bound either maxdeg V (A) or maxdeg C (A), that is, formally set maxdeg(A) = min {maxdeg V (A), maxdeg C (A)}, then we can use the results of Koutecký et al [18] to solve the linear IP with such a solution set in time f (maxdeg(A), A ∞ ) • n O (1) • L. Consequently, the use of high-degree constraints and variables in the proof of Theorem 1 is unavoidable.…”

Near-Linear Time Algorithm for $n$-Fold ILPs via Color Coding

“…The corresponding columns in A will be called blocks. Lately, n-fold ILPs received great attention [2,7,12,14,16] and were studied intensively due to two reasons. Firstly, many optimization problems are expressible as n-fold ILPs [5,10,12,14].…”