Lower Bounds for the Parameterized Complexity of Minimum Fill-in and Other Completion Problems

Answer Set Programming (ASP) is a paradigm and problem modeling/solving toolkit for KR that is often invoked. There are plenty of results dedicated to studying the hardness of (fragments of) ASP. So far, these studies resulted in characterizations in terms of computational complexity as well as in fine-grained insights presented in form of dichotomy-style results, lower bounds when translating to other formalisms like propositional satisfiability (SAT), and even detailed parameterized complexity landscapes. A quite generic and prominent parameter in parameterized complexity originating from graph theory is the so-called treewidth, which in a sense captures structural density of a program. Recently, there was an increase in the number of treewidth-based solvers related to SAT. While there exist several translations from (normal) ASP to SAT, yet there is no reduction preserving treewidth or at least being aware of the treewidth increase. This paper deals with a novel reduction from normal ASP to SAT that is aware of the treewidth, and guarantees that a slight increase of treewidth is indeed sufficient. Then, we also present a new result establishing that when considering treewidth, already the fragment of normal ASP is slightly harder than SAT (under reasonable assumptions in computational complexity). This also confirms that our reduction probably cannot be significantly improved and that the slight increase of treewidth is unavoidable.

Get full-text (via PubEx)

The Parameterized Complexity of Connected Fair Division

Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence ◽

10.24963/ijcai.2021/20 ◽

2021 ◽

Author(s):

Argyrios Deligkas ◽

Eduard Eiben ◽

Robert Ganian ◽

Thekla Hamm ◽

Sebastian Ordyniak

Keyword(s):

Lower Bounds ◽

Parameterized Complexity ◽

Fundamental Problem ◽

Fair Division ◽

Division Problem ◽

Additional Parameter ◽

Fixed Parameter Tractability ◽

Connected Subgraph ◽

Fixed Parameter ◽

New Algorithms

We study the Connected Fair Division problem (CFD), which generalizes the fundamental problem of fairly allocating resources to agents by requiring that the items allocated to each agent form a connected subgraph in a provided item graph G. We expand on previous results by providing a comprehensive complexity-theoretic understanding of CFD based on several new algorithms and lower bounds while taking into account several well-established notions of fairness: proportionality, envy-freeness, EF1 and EFX. In particular, we show that to achieve tractability, one needs to restrict both the agents and the item graph in a meaningful way. We design (XP)-algorithms for the problem parameterized by (1) clique-width of G plus the number of agents and (2) treewidth of G plus the number of agent types, along with corresponding lower bounds. Finally, we show that to achieve fixed-parameter tractability, one needs to not only use a more restrictive parameterization of G, but also include the maximum item valuation as an additional parameter.

Get full-text (via PubEx)

Solving Integer Quadratic Programming via Explicit and Structural Restrictions

Proceedings of the AAAI Conference on Artificial Intelligence ◽

10.1609/aaai.v33i01.33011477 ◽

2019 ◽

Vol 33 ◽

pp. 1477-1484

Author(s):

Eduard Eiben ◽

Robert Ganian ◽

Dusan Knop ◽

Sebastian Ordyniak

Keyword(s):

Quadratic Programming ◽

Lower Bounds ◽

Parameterized Complexity ◽

Integer Quadratic Programming ◽

New Algorithms ◽

Variable Interactions

We study the parameterized complexity of Integer Quadratic Programming under two kinds of restrictions: explicit restrictions on the domain or coefficients, and structural restrictions on variable interactions. We argue that both kinds of restrictions are necessary to achieve tractability for Integer Quadratic Programming, and obtain four new algorithms for the problem that are tuned to possible explicit restrictions of instances that we may wish to solve. The presented algorithms are exact, deterministic, and complemented by appropriate lower bounds.

Get full-text (via PubEx)

Parameterized Complexity of Envy-Free Resource Allocation in Social Networks

Proceedings of the AAAI Conference on Artificial Intelligence ◽

10.1609/aaai.v34i05.6201 ◽

2020 ◽

Vol 34 (05) ◽

pp. 7135-7142

Author(s):

Eduard Eiben ◽

Robert Ganian ◽

Thekla Hamm ◽

Sebastian Ordyniak

Keyword(s):

Social Networks ◽

Resource Allocation ◽

Social Network ◽

Lower Bounds ◽

Parameterized Complexity ◽

Classical Problem ◽

Full Information ◽

Bounded Treewidth ◽

Basic Model ◽

The Social

We consider the classical problem of allocating resources among agents in an envy-free (and, where applicable, proportional) way. Recently, the basic model was enriched by introducing the concept of a social network which allows to capture situations where agents might not have full information about the allocation of all resources. We initiate the study of the parameterized complexity of these resource allocation problems by considering natural parameters which capture structural properties of the network and similarities between agents and items. In particular, we show that even very general fragments of the considered problems become tractable as long as the social network has bounded treewidth or bounded clique-width. We complement our results with matching lower bounds which show that our algorithms cannot be substantially improved.

Get full-text (via PubEx)