5 research outputs found
Slim Tree-Cut Width
Tree-cut width is a parameter that has been introduced as an attempt to obtain an analogue of treewidth for edge cuts. Unfortunately, in spite of its desirable structural properties, it turned out that tree-cut width falls short as an edge-cut based alternative to treewidth in algorithmic aspects. This has led to the very recent introduction of a simple edge-based parameter called edge-cut width [WG 2022], which has precisely the algorithmic applications one would expect from an analogue of treewidth for edge cuts, but does not have the desired structural properties.
In this paper, we study a variant of tree-cut width obtained by changing the threshold for so-called thin nodes in tree-cut decompositions from 2 to 1. We show that this "slim tree-cut width" satisfies all the requirements of an edge-cut based analogue of treewidth, both structural and algorithmic, while being less restrictive than edge-cut width. Our results also include an alternative characterization of slim tree-cut width via an easy-to-use spanning-tree decomposition akin to the one used for edge-cut width, a characterization of slim tree-cut width in terms of forbidden immersions as well as an approximation algorithm for computing the parameter
The Complexity of Fair Division of Indivisible Items with Externalities
We study the computational complexity of fairly allocating a set of
indivisible items under externalities. In this recently-proposed setting, in
addition to the utility the agent gets from their bundle, they also receive
utility from items allocated to other agents. We focus on the extended
definitions of envy-freeness up to one item (EF1) and of envy-freeness up to
any item (EFX), and we provide the landscape of their complexity for several
different scenarios. We prove that it is NP-complete to decide whether there
exists an EFX allocation, even when there are only three agents, or even when
there are only six different values for the items. We complement these negative
results by showing that when both the number of agents and the number of
different values for items are bounded by a parameter the problem becomes
fixed-parameter tractable. Furthermore, we prove that two-valued and
binary-valued instances are equivalent and that EFX and EF1 allocations
coincide for this class of instances. Finally, motivated from real-life
scenarios, we focus on a class of structured valuation functions, which we term
agent/item-correlated. We prove their equivalence to the ``standard'' setting
without externalities. Therefore, all previous results for EF1 and EFX apply
immediately for these valuations
A Structural Complexity Analysis of Synchronous Dynamical Systems
Synchronous dynamical systems are well-established models that have been used to capture a range of phenomena in networks, including opinion diffusion, spread of disease and product adoption. We study the three most notable problems in synchronous dynamical systems: whether the system will transition to a target configuration from a starting configuration, whether the system will reach convergence from a starting configuration, and whether the system is guaranteed to converge from every possible starting configuration. While all three problems were known to be intractable in the classical sense, we initiate the study of their exact boundaries of tractability from the perspective of structural parameters of the network by making use of the more fine-grained parameterized complexity paradigm. As our first result, we consider treewidth - as the most prominent and ubiquitous structural parameter - and show that all three problems remain intractable even on instances of constant treewidth. We complement this negative finding with fixed-parameter algorithms for the former two problems parameterized by treedepth, a well-studied restriction of treewidth. While it is possible to rule out a similar algorithm for convergence guarantee under treedepth, we conclude with a fixed-parameter algorithm for this last problem when parameterized by treedepth and the maximum in-degree
The Fine-Grained Complexity of Graph Homomorphism Parameterized by Clique-Width
The generic homomorphism problem, which asks whether an input graph G admits a homomorphism into a fixed target graph H, has been widely studied in the literature. In this article, we provide a fine-grained complexity classification of the running time of the homomorphism problem with respect to the clique-width of G (denoted cw) for virtually all choices of H under the Strong Exponential Time Hypothesis. In particular, we identify a property of H called the signature number s(H) and show that for each H, the homomorphism problem can be solved in time O^*(s(H)^cw). Crucially, we then show that this algorithm can be used to obtain essentially tight upper bounds. Specifically, we provide a reduction that yields matching lower bounds for each H that is either a projective core or a graph admitting a factorization with additional properties - allowing us to cover all possible target graphs under long-standing conjectures