Algorithms to Approximate Column-Sparse Packing Problems

Column-sparse packing problems arise in several contexts in both deterministic and stochastic discrete optimization. We present two unifying ideas, (non-uniform) attenuation and multiple-chance algorithms, to obtain improved approximation algorithms for some well-known families of such problems. As three main examples, we attain the integrality gap, up to lower-order terms, for known LP relaxations for k-column sparse packing integer programs (Bansal et al., Theory of Computing, 2012) and stochastic k-set packing (Bansal et al., Algorithmica, 2012), and go "half the remaining distance" to optimal for a major integrality-gap conjecture of Furedi, Kahn and Seymour on hypergraph matching (Combinatorica, 1993).Comment: Extended abstract appeared in SODA 2018. Full version in ACM Transactions of Algorithm

Improved Deterministic Distributed Matching via Rounding

We present improved deterministic distributed algorithms for a number of well-studied matching problems, which are simpler, faster, more accurate, and/or more general than their known counterparts. The common denominator of these results is a deterministic distributed rounding method for certain linear programs, which is the first such rounding method, to our knowledge. A sampling of our end results is as follows. - An O(log^2 Delta log n)-round deterministic distributed algorithm for computing a maximal matching, in n-node graphs with maximum degree Delta. This is the first improvement in about 20 years over the celebrated O(log^4 n)-round algorithm of Hanckowiak, Karonski, and Panconesi [SODA\u2798, PODC\u2799]. - A deterministic distributed algorithm for computing a (2+epsilon)-approximation of maximum matching in O(log^2 Delta log(1/epsilon) + log^* n) rounds. This is exponentially faster than the classic O(Delta + log^* n)-round 2-approximation of Panconesi and Rizzi [DIST\u2701]. With some modifications, the algorithm can also find an epsilon-maximal matching which leaves only an epsilon-fraction of the edges on unmatched nodes. - An O(log^2 Delta log(1/epsilon) + log^* n)-round deterministic distributed algorithm for computing a (2+epsilon)-approximation of a maximum weighted matching, and also for the more general problem of maximum weighted b-matching. These improve over the O(log^4 n log_(1+epsilon) W)-round (6+epsilon)-approximation algorithm of Panconesi and Sozio [DIST\u2710], where W denotes the maximum normalized weight. - A deterministic local computation algorithm for a (2+epsilon)-approximation of maximum matching with 2^O(log^2 Delta) log^* n queries. This improves almost exponentially over the previous deterministic constant approximations which have query-complexity of 2^Omega(Delta log Delta) log^* n

Analyzing Massive Graphs in the Semi-streaming Model

Massive graphs arise in a many scenarios, for example, traffic data analysis in large networks, large scale scientific experiments, and clustering of large data sets. The semi-streaming model was proposed for processing massive graphs. In the semi-streaming model, we have a random accessible memory which is near-linear in the number of vertices. The input graph (or equivalently, edges in the graph) is presented as a sequential list of edges (insertion-only model) or edge insertions and deletions (dynamic model). The list is read-only but we may make multiple passes over the list. There has been a few results in the insertion-only model such as computing distance spanners and approximating the maximum matching. In this thesis, we present some algorithms and techniques for (i) solving more complex problems in the semi-streaming model, (for example, problems in the dynamic model) and (ii) having better solutions for the problems which have been studied (for example, the maximum matching problem). In course of both of these, we develop new techniques with broad applications and explore the rich trade-offs between the complexity of models (insertion-only streams vs. dynamic streams), the number of passes, space, accuracy, and running time. 1. We initiate the study of dynamic graph streams. We start with basic problems such as the connectivity problem and computing the minimum spanning tree. These problems are trivial in the insertion-only model. However, they require non-trivial (and multiple passes for computing the exact minimum spanning tree) algorithms in the dynamic model. 2. Second, we present a graph sparsification algorithm in the semi-streaming model. A graph sparsification is a sparse graph that approximately preserves all the cut values of a graph. Such a graph acts as an oracle for solving cut-related problems, for example, the minimum cut problem and the multicut problem. Our algorithm produce a graph sparsification with high probability in one pass. 3. Third, we use the primal-dual algorithms to develop the semi-streaming algorithms. The primal-dual algorithms have been widely accepted as a framework for solving linear programs and semidefinite programs faster. In contrast, we apply the method for reducing space and number of passes in addition to reducing the running time. We also present some examples that arise in applications and show how to apply the techniques: the multicut problem, the correlation clustering problem, and the maximum matching problem. As a consequence, we also develop near-linear time algorithms for the $b$-matching problems which were not known before

Approximability of Sparse Integer Programs

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 ε>0, if P≠NP this ratio cannot be improved to k−1−ε, and under the unique games conjecture this ratio cannot be improved to k−ε. 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 (2k 2+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 A ij is small compared to b i. Finally, we demonstrate a 17/16-inapproximability for covering integer programs with at most two nonzeroes per colum

Approximability of Sparse Integer Programs

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 eps>0, if P != NP this ratio cannot be improved to k-1-eps, and under the unique games conjecture this ratio cannot be improved to k-eps. 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 (2k^2+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 A_{ij} is small compared to b_i. Finally, we demonstrate a 17/16-inapproximability for covering integer programs with at most two nonzeroes per column.Comment: Version submitted to Algorithmica special issue on ESA 2009. Previous conference version: http://dx.doi.org/10.1007/978-3-642-04128-0_

Greedy D-Approximation Algorithm for Covering with Arbitrary Constraints and Submodular Cost

This paper describes a simple greedy D-approximation algorithm for any covering problem whose objective function is submodular and non-decreasing, and whose feasible region can be expressed as the intersection of arbitrary (closed upwards) covering constraints, each of which constrains at most D variables of the problem. (A simple example is Vertex Cover, with D = 2.) The algorithm generalizes previous approximation algorithms for fundamental covering problems and online paging and caching problems

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum

"Rotterdam econometrics": publications of the econometric institute 1956-2005

This paper contains a list of all publications over the period 1956-2005, as reported in the Rotterdam Econometric Institute Reprint series during 1957-2005.