Upper and lower bounds for first order expressibility

AbstractWe study first order expressibility as a measure of complexity. We introduce the new class Var&Sz[v(n),z(n)] of languages expressible by a uniform sequence of sentences with v(n) variables and size O[z(n)]. When v(n) is constant our uniformity condition is syntactical and thus the following characterizations of P and PSPACE come entirely from logic. NSPACE|log n|⊆⋃k=1,2,…Var&Sz|k, log(n)|⊆DSPACE|log2(n)|,P=⋃k=1,2,…Var&Sz|k, nk|,PSPACE=⋃k=1,2,…Var&Sz|k, 2nk|. The above means, for example, that the properties expressible with constantly many variables in polynomial size sentences are just the polynomial time recognizable properties. These results hold for languages with an ordering relation, e.g., for graphs the vertices are numbered. We introduce an “alternating pebbling game” to prove lower bounds on the number of variables and size needed to express properties without the ordering. We show, for example, that k variables are needed to express Clique(k), suggesting that this problem requires DTIME[nk]

DSPACE(n)=<SUP>2</SUP> NSPACE(n): a degree theoretic characterization

It is shown that the following are equivalent. 1. DSPACE(n)=NSPACE(n). 2. There is a nontrivial &#8804; 1-NLm-degree that coincides with &#8804;1-Lm-degree. 3. For every class C closed under log-lin reductions, the &#8804;1-NLm-complete degree of C coincides with the &#8804;1-Lm-complete degree of C

Streaming Algorithms Via Reductions

In the streaming algorithms model of computation we must process data in order and without enough memory to remember the entire input. We study reductions between problems in the streaming model with an eye to using reductions as an algorithm design technique. Our contributions include: * Linear Transformation reductions, which compose with existing linear sketch techniques. We use these for small-space algorithms for numeric measurements of distance-from-periodicity, finding the period of a numeric stream, and detecting cyclic shifts. * The first streaming graph algorithms in the sliding window\u27 model, where we must consider only the most recent L elements for some fixed threshold L. We develop basic algorithms for connectivity and unweighted maximum matching, then develop a variety of other algorithms via reductions to these problems. * A new reduction from maximum weighted matching to maximum unweighted matching. This reduction immediately yields improved approximation guarantees for maximum weighted matching in the semistreaming, sliding window, and MapReduce models, and extends to the more general problem of finding maximum independent sets in p-systems. * Algorithms in a stream-of-samples model which exhibit clear sample vs. space tradeoffs. These algorithms are also inspired by examining reductions. We provide algorithms for calculating F_k frequency moments and graph connectivity

Order-Related Problems Parameterized by Width

In the main body of this thesis, we study two different order theoretic problems. The first problem, called Completion of an Ordering, asks to extend a given finite partial order to a complete linear order while respecting some weight constraints. The second problem is an order reconfiguration problem under width constraints. While the Completion of an Ordering problem is NP-complete, we show that it lies in FPT when parameterized by the interval width of ρ. This ordering problem can be used to model several ordering problems stemming from diverse application areas, such as graph drawing, computational social choice, and computer memory management. Each application yields a special partial order ρ. We also relate the interval width of ρ to parameterizations for these problems that have been studied earlier in the context of these applications, sometimes improving on parameterized algorithms that have been developed for these parameterizations before. This approach also gives some practical sub-exponential time algorithms for ordering problems. In our second main result, we combine our parameterized approach with the paradigm of solution diversity. The idea of solution diversity is that instead of aiming at the development of algorithms that output a single optimal solution, the goal is to investigate algorithms that output a small set of sufficiently good solutions that are sufficiently diverse from one another. In this way, the user has the opportunity to choose the solution that is most appropriate to the context at hand. It also displays the richness of the solution space. There, we show that the considered diversity version of the Completion of an Ordering problem is fixed-parameter tractable with respect to natural paramaters that capture the notion of diversity and the notion of sufficiently good solutions. We apply this algorithm in the study of the Kemeny Rank Aggregation class of problems, a well-studied class of problems lying in the intersection of order theory and social choice theory. Up to this point, we have been looking at problems where the goal is to find an optimal solution or a diverse set of good solutions. In the last part, we shift our focus from finding solutions to studying the solution space of a problem. There we consider the following order reconfiguration problem: Given a graph G together with linear orders τ and τ ′ of the vertices of G, can one transform τ into τ ′ by a sequence of swaps of adjacent elements in such a way that at each time step the resulting linear order has cutwidth (pathwidth) at most w? We show that this problem always has an affirmative answer when the input linear orders τ and τ ′ have cutwidth (pathwidth) at most w/2. Using this result, we establish a connection between two apparently unrelated problems: the reachability problem for two-letter string rewriting systems and the graph isomorphism problem for graphs of bounded cutwidth. This opens an avenue for the study of the famous graph isomorphism problem using techniques from term rewriting theory. In addition to the main part of this work, we present results on two unrelated problems, namely on the Steiner Tree problem and on the Intersection Non-emptiness problem from automata theory.Doktorgradsavhandlin

Linear and Exact Extended Formulations

Matematicko-fyzikální fakult

One-Way Log-Tape Reductions

One-way log-tape (1-L) reductions are mappings defined by log-tape Turing machines whose read head on the input can only move to the right. The 1-L reductions provide a more refined tool for studying the feasible complexity classes than the P-time [2,7] or log-tape [4] reductions. Although the 1-L computations are provably weaker than the feasible classes L, NL, P and NP, the known complete sets for those classes are complete under 1-L reductions. However, using known techniques of counting arguments and recursion theory we show that certain log-tape reductions cannot be 1-L and we construct sets that are complete under log-tape reductions but not under 1-L reductions