Paul Hunter

DAG-Width and parity games

Tree-width is a well-known metric on undirected graphs that measures how tree-like a graph is and gives a notion of graph decomposition that proves useful in algorithm development. Tree-width is characterised by a game known as the cops-and-robber game where a number of cops chase a robber on the graph. We consider the natural adaptation of this game to directed graphs and show that monotone strategies in the game yield a measure with an associated notion of graph decomposition that can be seen to describe how close a directed graph is to a directed acyclic graph (DAG). This promises to be useful in developing algorithms on directed graphs. In particular, we show that the problem of determining the winner of a parity game is solvable in polynomial time on graphs of bounded DAG-width. We also consider the relationship between DAG-width and other measures such as entanglement and directed tree-width. One consequence we obtain is that certain NP-complete problems such as Hamiltonicity and disjoint paths are polynomial-time computable on graphs of bounded DAG-width.

Digraph measures: Kelly decompositions, games, and orderings

We consider various well-known, equivalent complexity measures for graphs such as elimination orderings, κ-trees and cops and robber games and study their natural translations to digraphs. We show that on digraphs all these measures are also equivalent and induce a natural connectivity measure. We introduce a decomposition for digraphs and an associated width, Kelly-width, which is equivalent to the aforementioned measure. We demonstrate its usefulness by exhibiting a number of potential applications including polynomial-time algorithms for NP-complete problems on graphs of bounded Kelly-width, and complexity analysis of asymmetric matrix factorization. Finally, we compare the new width to other known decompositions of digraphs.

The dag-width of directed graphs

Tree-width is a well-known metric on undirected graphs that measures how tree-like a graph is and gives a notion of graph decomposition that proves useful in algorithm design. Tree-width can be characterised by a graph searching game where a number of cops attempt to capture a robber. We consider the natural adaptation of this game to directed graphs and show that monotone strategies in the game yield a measure, called dag-width, that can be seen to describe how close a directed graph is to a directed acyclic graph (dag). We also provide an associated decomposition and show how it is useful for developing algorithms on directed graphs. In particular, we show that the problem of determining the winner of a parity game is solvable in polynomial time on graphs of bounded dag-width. We also consider the relationship between dag-width and other connectivity measures such as directed tree-width and path-width. A consequence we obtain is that certain NP-complete problems such as Hamiltonicity and disjoint paths are polynomial-time computable on graphs of bounded dag-width.

Expressive Completeness for Metric Temporal Logic

Metric Temporal Logic (MTL) is a generalisation of Linear Temporal Logic in which the Until and Since modalities are annotated with intervals that express metric constraints. Hirshfeld and Rabinovich have shown that over the reals, firstorder logic with binary order relation <; and unary function +1 is strictly more expressive than MTL with integer constants. Indeed they prove that no temporal logic whose modalities are definable by formulas of bounded quantifier depth can be expressively complete for FO(<;, +1). In this paper we show that if we allow unary functions +q, q ∈ Q, in first-order logic and correspondingly allow rational constants in MTL, then the two logics have the same expressive power. This gives the first generalisation of Kamps theorem on the expressive completeness of LTL for FO(<;) to the quantitative setting. The proof of this result involves a generalisation of Gabbays notion of separation to the metric setting.

Reachability in Succinct One-Counter Games

We consider two-player games with reachability objectives played on transition systems of succinct one-counter machines, that is, machines where the counter is incremented or decremented by a value given in binary. We show that the winner-determination problem is EXPSPACE-complete regardless of whether transitions are guarded by constraints on the counter or if the counter is restricted to non-negative values.

Non-Zero Sum Games for Reactive Synthesis

In this invited contribution, we summarize new solution concepts useful for the synthesis of reactive systems that we have introduced in several recent publications. These solution concepts are developed in the context of non-zero sum games played on graphs. They are part of the contributions obtained in the inVEST project funded by the European Research Council.

LIFO-search: A min-max theorem and a searching game for cycle-rank and tree-depth

We introduce a variant of the classic node search game called LIFO-search where searchers are assigned different numbers. The additional rule is that a searcher can be removed only if no searchers of lower rank are in the graph at that moment. We show that all common variations of the game require the same number of searchers. We then introduce the notion of (directed) shelters in (di)graphs and prove a min-max theorem implying their equivalence to the cycle-rank/tree-depth parameter in (di)graphs. As (directed) shelters provide escape strategies for the fugitive, this implies that the LIFO-search game is monotone and that the LIFO-search parameter is equivalent to the one of cycle-rank/tree-depth in (di)graphs.

Mean-Payoff Games with Partial-Observation

Mean-payoff games are important quantitative models for open reactive systems. They have been widely studied as games of perfect information. In this paper we investigate the algorithmic properties of several subclasses of mean-payoff games where the players have asymmetric information about the state of the game. These games are in general undecidable and not determined according to the classical definition. We show that such games are determined under a more general notion of winning strategy. We also consider mean-payoff games where the winner can be determined by the winner of a finite cycle-forming game. This yields several decidable classes of mean-payoff games of asymmetric information that require only finite-memory strategies, including a generalization of perfect information games where positional strategies are sufficient. We give an exponential time algorithm for determining the winner of the latter.

When is Metric Temporal Logic expressively complete

A seminal result of Kamp is that over the reals Linear Temporal Logic (LTL) has the same expressive power as first-order logic with binary order relation < and monadic predicates. A key question is whether there exists an analogue of Kamps theorem for Metric Temporal Logic (MTL) -- a generalization of LTL in which the Until and Since modalities are annotated with intervals that express metric constraints. Hirshfeld and Rabinovich gave a negative answer, showing that first-order logic with binary order relation < and unary function +1 is strictly more expressive than MTL with integer constants. However, a recent result of Hunter, Ouaknine and Worrell shows that with rational timing constants, MTL has the same expressive power as first-order logic, giving a positive answer. In this paper we generalize these results by giving a precise characterization of those sets of constants for which MTL and first-order logic have the same expressive power. We also show that full first-order expressiveness can be recovered with the addition of counting modalities, strongly supporting the assertion of Hirshfeld and Rabinovich that Q2MLO is one of the most expressive decidable fragments of FO(<,+1).

LIFO-search on digraphs: a searching game for cycle-rank

We consider the extension of the last-in-first-out graph searching game of Giannopoulou and Thilikos to digraphs. We show that all common variations of the game require the same number of searchers, and the minimal number of searchers required is one more than the cyclerank of the digraph. We also obtain a tight duality theorem, giving a precise min-max characterization of obstructions for cycle-rank.