Approximability of Sparse Integer Programs

arXiv:0904.0859v5 [cs.DS] 9 Feb 2010 Approximability of Sparse Integer Programs David Pritchard Deeparnab Chakrabarty October 23, 2018 Abstract The main focus of this paper is a pair of new approximation algorithms for certain integer programs. First, for covering integer programs {min cx : Ax ≥b, 0 ≤x ≤d} where A has at most k nonzeroes per row, we give a k-approximation algorithm. (We assume A, b, c, d are nonnegative.) For any k ≥2 and ǫ > 0, if P ̸= NP this ratio cannot be improved to k −1 −ǫ, and under the unique games conjecture this ratio cannot be improved to k −ǫ. One key idea is to replace individual constraints by others that have better rounding properties but the same nonnegative integral solutions; another critical ingredient is knapsack-cover inequalities. Second, for packing integer programs {max cx : Ax ≤b, 0 ≤x ≤d} where A has at most k nonzeroes per column, we give a (2k2 + 2)-approximation algorithm. Our approach builds on the iterated LP relaxation framework. In addition, we obtain improved approximations for the second problem when k = 2, and for both problems when every Aij is small compared to bi. Finally, we demonstrate a 17/16-inapproximability for covering integer programs with at most two nonzeroes per column. 1 Introduction We investigate the following problem: what is the best possible approximation ratio for integer programs where the constraint matrix is sparse? To put this in context we recall a famous result of Lenstra [29]: integer programs with a constant number of variables or a constant number of constraints can be solved in polynomial time. Our investigations analogously ask what is possible if each constraint involves at most k variables, or if each variable appears in at most k constraints. Rather than consider all integer programs, we consider only packing and covering problems. Such pro- grams have only positive quantities in their parameters. One reason for this is that every integer program can be rewritten (possibly with additional variables) in such a way that each constraint contains at most 3 variables and each variable appears in at most 3 constraints, if both positive and negative coefficients are allowed. Aside from this, packing programs and covering programs capture a substantial number of combinatorial optimization problems and are interesting in their own right. A covering (resp. packing) integer program, shorthanded as CIP (resp. PIP) henceforth, is an integer program of the form {min cx : Ax ≥b, 0 ≤x ≤d} (resp. {max cx : Ax ≤b, 0 ≤x ≤d}) with A, b, c, d nonnegative and rational. Note that CIPs are sometimes called multiset multicover when A and b are integral. We call constraints x ≤d multiplicity constraints (also known as capacity constraints). We allow for entries of d to be infinite, and without loss of generality, all finite entries of d are integral. An integer program with constraint matrix A is k-row-sparse, or k-RS, if each row of A has at most k entries; we define k-column- sparse (k-CS) similarly. As a rule of thumb we ignore the case k = 1, since such problems trivially admit fully polynomial-time approximation schemes (FPTAS’s) or poly-time algorithms. The symbol 0 denotes the all-zero vector, and similarly 1 denotes the all-ones vector. For covering problems an α-approximation algorithm returns a feasible solution with objective value at most α times optimal; for packing, the algorithm returns a feasible solution with objective value is at least 1/α times optimal. We use n to denote the number of variables and m the number of constraints (i.e. the number of columns and rows of A, respectively). Throughout the paper, A will be used as a matrix. We let Aj denote the jth column of A, and let ai denote the ith row of A. 1 1.1 k-Row-Sparse Covering IPs The special case of 2-RS CIP where A, b, c, d are 0-1 is the same as Min Vertex Cover, which is APX-hard. More generally, 0-1 k-RS CIP is the same as k-Bounded Hypergraph Min Vertex Cover (a.k.a. Set Cover with maximum frequency k) which is not approximable to k −1 −ǫ for any fixed ǫ > 0 unless P=NP [8] (k −ǫ under the unique games conjecture [22]). This special case is known to admit a matching positive result: set cover with maximum frequency k can be k-approximated by direct rounding of the naive LP [15] or local ratio/primal-dual methods [2]. The following results are known for other special cases of k-RS CIP with multiplicity constraints: Hochbaum [12] gave a k-approximation in the special case that A is 0-1; Hochbaum et al. [17] and Bar- Yehuda & Rawitz [3] gave pseudopolynomial 2-approximation algorithms for the case that k = 2 and d is finite. For the special case d = 1, Carr et al. [5, §2.6] gave a k-approximation, and Fujito & Yabuta [9] gave a primal-dual k-approximation. Moreover [5, 9] claim a k-approximation for general d, however, the papers do not give a proof and we do not see a straightforward method of extending their techniques to the general d case. Our first main result, given

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