The binary network flow problem is logspace complete for P

AbstractIt is shown that the problem of whether the maximum flow in a given network exceeds a given natural number is logspace many-one complete for P if the edge capacities are presented in binary (even if the problem is restricted to acyclic graphs). This improves a result by Goldschlager et al. (1982) that this problem is logspace Turing complete for P

Stream Reasoning in Temporal Datalog

In recent years, there has been an increasing interest in extending traditional stream processing engines with logical, rule-based, reasoning capabilities. This poses significant theoretical and practical challenges since rules can derive new information and propagate it both towards past and future time points; as a result, streamed query answers can depend on data that has not yet been received, as well as on data that arrived far in the past. Stream reasoning algorithms, however, must be able to stream out query answers as soon as possible, and can only keep a limited number of previous input facts in memory. In this paper, we propose novel reasoning problems to deal with these challenges, and study their computational properties on Datalog extended with a temporal sort and the successor function (a core rule-based language for stream reasoning applications)

The Parallelism Tradeoff: Limitations of Log-Precision Transformers

Despite their omnipresence in modern NLP, characterizing the computational power of transformer neural nets remains an interesting open question. We prove that transformers whose arithmetic precision is logarithmic in the number of input tokens (and whose feedforward nets are computable using space linear in their input) can be simulated by constant-depth logspace-uniform threshold circuits. This provides insight on the power of transformers using known results in complexity theory. For example, if $\mathsf L \neq \mathsf P$ (i.e., not all poly-time problems can be solved using logarithmic space), then transformers cannot even accurately solve linear equalities or check membership in an arbitrary context-free grammar with empty productions. Our result intuitively emerges from the transformer architecture's high parallelizability. We thus speculatively introduce the idea of a fundamental parallelism tradeoff: any model architecture as parallelizable as the transformer will obey limitations similar to it. Since parallelism is key to training models at massive scale, this suggests a potential inherent weakness of the scaling paradigm.Comment: Accepted at TACL. Formerly entitled "Log-Precision Transformers are Constant-Depth Threshold Circuits". Updated with minor corrections in Section 2 (Implications) on March 6, 2023. Update with minor edits to the proof of Lemma 3 on April 26, 202

The Window Validity Problem in Rule-Based Stream Reasoning

Rule-based temporal query languages provide the expressive power and flexibility required to capture in a natural way complex analysis tasks over streaming data. Stream processing applications, however, typically require near real-time response using limited resources. In particular, it becomes essential that the underpinning query language has favourable computational properties and that stream processing algorithms are able to keep only a small number of previously received facts in memory at any point in time without sacrificing correctness. In this paper, we propose a recursive fragment of temporal Datalog with tractable data complexity and study the properties of a generic stream reasoning algorithm for this fragment. We focus on the window validity problem as a way to minimise the number of time points for which the stream reasoning algorithm needs to keep data in memory at any point in time

Randomized Search of Graphs in Log Space and Probabilistic Computation

Reingold has shown that L = SL, that s-t connectivity in a poly-mixing digraph is complete for promise-RL, and that s-t connectivity for a poly-mixing out-regular digraph with known stationary distribution is in L. Several properties that bound the mixing times of random walks on digraphs have been identified, including the digraph conductance and the digraph spectral expansion. However, rapidly mixing digraphs can still have exponential cover time, thus it is important to specifically identify structural properties of digraphs that effect cover times. We examine the complexity of random walks on a basic parameterized family of unbalanced digraphs called Strong Chains (which model weakly symmetric logspace computations), and a special family of Strong Chains called Harps. We show that the worst case hitting times of Strong Chain families vary smoothly with the number of asymmetric vertices and identify the necessary condition for non-polynomial cover time. This analysis also yields bounds on the cover times of general digraphs. Next we relate random walks on graphs to the random walks that arise in Monte Carlo methods applied to optimization problems. We introduce the notion of the asymmetric states of Markov chains and use this definition to obtain some results about Markov chains. We also obtain some results on the mixing times for Markov Chain Monte Carlo Methods. Finally, we consider the question of whether a single long random walk or many short walks is a better strategy for exploration. These are walks which reset to the start after a fixed number of steps. We exhibit digraph families for which a few short walks are far superior to a single long walk. We introduce an iterative deepening random search. We use this strategy estimate the cover time for poly-mixing subgraphs. Finally we discuss complexity theoretic implications and future work

Derandomizing Isolation in Space-Bounded Settings

We study the possibility of deterministic and randomness-efficient isolation in space-bounded models of computation: Can one efficiently reduce instances of computational problems to equivalent instances that have at most one solution? We present results for the NL-complete problem of reachability on digraphs, and for the LogCFL-complete problem of certifying acceptance on shallow semi-unbounded circuits. A common approach employs small weight assignments that make the solution of minimum weight unique. The Isolation Lemma and other known procedures use Omega(n) random bits to generate weights of individual bitlength O(log(n)). We develop a derandomized version for both settings that uses O(log(n)^{3/2}) random bits and produces weights of bitlength O(log(n)^{3/2}) in logarithmic space. The construction allows us to show that every language in NL can be accepted by a nondeterministic machine that runs in polynomial time and O(log(n)^{3/2}) space, and has at most one accepting computation path on every input. Similarly, every language in LogCFL can be accepted by a nondeterministic machine equipped with a stack that does not count towards the space bound, that runs in polynomial time and O(log(n)^{3/2}) space, and has at most one accepting computation path on every input. We also show that the existence of somewhat more restricted isolations for reachability on digraphs implies that NL can be decided in logspace with polynomial advice. A similar result holds for certifying acceptance on shallow semi-unbounded circuits and LogCFL

On the parameterized complexity of computing tree-partitions

We study the parameterized complexity of computing the tree-partition-width, a graph parameter equivalent to treewidth on graphs of bounded maximum degree. On one hand, we can obtain approximations of the tree-partition-width efficiently: we show that there is an algorithm that, given an $n$-vertex graph $G$ and an integer $k$, constructs a tree-partition of width $O(k^7)$ for $G$ or reports that $G$ has tree-partition width more than $k$, in time $k^{O(1)}n^2$. We can improve on the approximation factor or the dependence on $n$ by sacrificing the dependence on $k$. On the other hand, we show the problem of computing tree-partition-width exactly is XALP-complete, which implies that it is $W[t]$-hard for all $t$. We deduce XALP-completeness of the problem of computing the domino treewidth. Finally, we adapt some known results on the parameter tree-partition-width and the topological minor relation, and use them to compare tree-partition-width to tree-cut width

On the Parameterized Complexity of Computing Tree-Partitions

We study the parameterized complexity of computing the tree-partition-width, a graph parameter equivalent to treewidth on graphs of bounded maximum degree. On one hand, we can obtain approximations of the tree-partition-width efficiently: we show that there is an algorithm that, given an $n$-vertex graph $G$ and an integer $k$, constructs a tree-partition of width $O(k^7)$ for $G$ or reports that $G$ has tree-partition width more than $k$, in time $k^{O(1)}n^2$. We can improve on the approximation factor or the dependence on $n$ by sacrificing the dependence on $k$. On the other hand, we show the problem of computing tree-partition-width exactly is XALP-complete, which implies that it is $W[t]$-hard for all $t$. We deduce XALP-completeness of the problem of computing the domino treewidth. Finally, we adapt some known results on the parameter tree-partition-width and the topological minor relation, and use them to compare tree-partition-width to tree-cut width