Combinatorial n-fold integer programming and applications

Abstract

Many fundamental NP-hard problems can be formulated as integer linear programs (ILPs). A famous algorithm by Lenstra solves ILPs in time that is exponential only in the dimension of the program, and polynomial in the size of the ILP. That algorithm became a ubiquitous tool in the design of fixedparameter algorithms for NP-hard problems, where one wishes to isolate the hardness of a problem by some parameter. However, in many cases using Lenstra's algorithm has two drawbacks: First, the run time of the resulting algorithms is often doubly-exponential in the parameter, and second, an ILP formulation in small dimension cannot easily express problems involving many different costs.Inspired by the work of Hemmecke, Onn and Romanchuk [Math. Prog. 2013], we develop a singleexponential algorithm for so-called combinatorial n-fold integer programs, which are remarkably similar to prior ILP formulations for various problems, but unlike them, also allow variable dimension. We then apply our algorithm to a few representative problems like Closest String, Swap Bribery, Weighted Set Multicover, and obtain exponential speedups in the dependence on the respective parameters, the input size, or both.Unlike Lenstra's algorithm, which is essentially a bounded search tree algorithm, our result uses the technique of augmenting steps. At its heart is a deep result stating that in combinatorial n-fold IPs, existence of an augmenting step implies existence of a "local" augmenting step, which can be found using dynamic programming. Our results provide an important insight into many problems by showing that they exhibit this phenomenon, and highlights the importance of augmentation techniques.1 Kannan's algorithm in parameterized complexity: they modeled Closest String with k input strings as an ILP of dimension k O(k) , and thereby concluded with the first fixed-parameter algorithm for Closest String. This success led Niedermeier [Nie04] to propose in his book:[...] It remains to investigate further examples besides Closest String where the described ILP approach turns out to be applicable. More generally, it would be interesting to discover more connections between fixed-parameter algorithms and (integer) linear programming.Since then, many more applications of Lenstra's and Kannan's algorithm for parameterized problems have been proposed. However, essentially all of them [BFN + 15, DS12, FLM + 08, HR15, Lam12, MW15] share a common trait with the algorithm for Closest String: they have a doubly-exponential run time dependence on the parameter. Moreover, it is difficult to find ILP formulations with small dimension for problems whose input contains many objects with varying cost functions, such as in Swap Bribery [BCF + 14, Challenge #2].

The Support of Integer Optimal Solutions

“…, 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).…”

Parameterized resiliency problems

“…Many of the prototypical uses of Lenstra's algorithm lead to FPT algorithms which have a double-exponential (i.e., 2 2 k O (1) ) dependency on the parameter, such as the algorithms for Closest String [15] or Swap Bribery [8]. Very recently, Knop et al [22] showed that many of these ILP formulations have a particular format which is solvable exponentially faster than by Lenstra's algorithm, thus bringing down the dependency on the parameter down to single-exponential. This leads us to wonder what is the true complexity of, e.g., Resiliency Closest String?…”

Mixed integer programming with convex/concave constraints: Fixed-parameter tractability and applications to multicovering and voting

“…In spite of this restrictive structure, n-fold integer programming found a number of applications, e.g., in scheduling [34] and voting [38]. See also the works of Dvorák et al [15] and Knop et al [35] for very recent generalizations and extensions of the this technique. It is quite non-obvious how the technique of n-fold integer programming compares to ours.…”