Streaming Approximation Resistance of Every Ordering CSP

An ordering constraint satisfaction problem (OCSP) is given by a positive integer $k$ and a constraint predicate $\Pi$ mapping permutations on $\{1,\ldots,k\}$ to $\{0,1\}$. Given an instance of OCSP$(\Pi)$ on $n$ variables and $m$ constraints, the goal is to find an ordering of the $n$ variables that maximizes the number of constraints that are satisfied, where a constraint specifies a sequence of $k$ distinct variables and the constraint is satisfied by an ordering on the $n$ variables if the ordering induced on the $k$ variables in the constraint satisfies $\Pi$. OCSPs capture natural problems including "Maximum acyclic subgraph (MAS)" and "Betweenness". In this work we consider the task of approximating the maximum number of satisfiable constraints in the (single-pass) streaming setting, where an instance is presented as a stream of constraints. We show that for every $\Pi$, OCSP$(\Pi)$ is approximation-resistant to $o(n)$-space streaming algorithms. This space bound is tight up to polylogarithmic factors. In the case of MAS our result shows that for every $\epsilon>0$, MAS is not $1/2+\epsilon$-approximable in $o(n)$ space. The previous best inapproximability result only ruled out a $3/4$-approximation in $o(\sqrt n)$ space. Our results build on recent works of Chou, Golovnev, Sudan, Velingker, and Velusamy who show tight, linear-space inapproximability results for a broad class of (non-ordering) constraint satisfaction problems over arbitrary (finite) alphabets. We design a family of appropriate CSPs (one for every $q$) from any given OCSP, and apply their work to this family of CSPs. We show that the hard instances from this earlier work have a particular "small-set expansion" property. By exploiting this combinatorial property, in combination with the hardness results of the resulting families of CSPs, we give optimal inapproximability results for all OCSPs.Comment: 23 pages, 1 figure. Replaces earlier version with $o(\sqrt{n})$ lower bound, using new bounds from arXiv:2106.13078. To appear in APPROX'2

Adaptive Near-Optimal Rank Tensor Approximation for High-Dimensional Operator Equations

We consider a framework for the construction of iterative schemes for operator equations that combine low-rank approximation in tensor formats and adaptive approximation in a basis. Under fairly general assumptions, we obtain a rigorous convergence analysis, where all parameters required for the execution of the methods depend only on the underlying infinite-dimensional problem, but not on a concrete discretization. Under certain assumptions on the rates for the involved low-rank approximations and basis expansions, we can also give bounds on the computational complexity of the iteration as a function of the prescribed target error. Our theoretical findings are illustrated and supported by computational experiments. These demonstrate that problems in very high dimensions can be treated with controlled solution accuracy.Comment: 51 page

Reordering Rows for Better Compression: Beyond the Lexicographic Order

Sorting database tables before compressing them improves the compression rate. Can we do better than the lexicographical order? For minimizing the number of runs in a run-length encoding compression scheme, the best approaches to row-ordering are derived from traveling salesman heuristics, although there is a significant trade-off between running time and compression. A new heuristic, Multiple Lists, which is a variant on Nearest Neighbor that trades off compression for a major running-time speedup, is a good option for very large tables. However, for some compression schemes, it is more important to generate long runs rather than few runs. For this case, another novel heuristic, Vortex, is promising. We find that we can improve run-length encoding up to a factor of 3 whereas we can improve prefix coding by up to 80%: these gains are on top of the gains due to lexicographically sorting the table. We prove that the new row reordering is optimal (within 10%) at minimizing the runs of identical values within columns, in a few cases.Comment: to appear in ACM TOD

Computational Performance Evaluation of Two Integer Linear Programming Models for the Minimum Common String Partition Problem

In the minimum common string partition (MCSP) problem two related input strings are given. "Related" refers to the property that both strings consist of the same set of letters appearing the same number of times in each of the two strings. The MCSP seeks a minimum cardinality partitioning of one string into non-overlapping substrings that is also a valid partitioning for the second string. This problem has applications in bioinformatics e.g. in analyzing related DNA or protein sequences. For strings with lengths less than about 1000 letters, a previously published integer linear programming (ILP) formulation yields, when solved with a state-of-the-art solver such as CPLEX, satisfactory results. In this work, we propose a new, alternative ILP model that is compared to the former one. While a polyhedral study shows the linear programming relaxations of the two models to be equally strong, a comprehensive experimental comparison using real-world as well as artificially created benchmark instances indicates substantial computational advantages of the new formulation.Comment: arXiv admin note: text overlap with arXiv:1405.5646 This paper version replaces the one submitted on January 10, 2015, due to detected error in the calculation of the variables involved in the ILP model