3,938 research outputs found
Fair Knapsack
We study the following multiagent variant of the knapsack problem. We are
given a set of items, a set of voters, and a value of the budget; each item is
endowed with a cost and each voter assigns to each item a certain value. The
goal is to select a subset of items with the total cost not exceeding the
budget, in a way that is consistent with the voters' preferences. Since the
preferences of the voters over the items can vary significantly, we need a way
of aggregating these preferences, in order to select the socially best valid
knapsack. We study three approaches to aggregating voters' preferences, which
are motivated by the literature on multiwinner elections and fair allocation.
This way we introduce the concepts of individually best, diverse, and fair
knapsack. We study the computational complexity (including parameterized
complexity, and complexity under restricted domains) of the aforementioned
multiagent variants of knapsack.Comment: Extended abstract will appear in Proc. of 33rd AAAI 201
Dynamic programming for graphs on surfaces
We provide a framework for the design and analysis of dynamic
programming algorithms for surface-embedded graphs on n vertices
and branchwidth at most k. Our technique applies to general families
of problems where standard dynamic programming runs in 2O(k·log k).
Our approach combines tools from topological graph theory and
analytic combinatorics.Postprint (updated version
Parameterized Algorithmics for Computational Social Choice: Nine Research Challenges
Computational Social Choice is an interdisciplinary research area involving
Economics, Political Science, and Social Science on the one side, and
Mathematics and Computer Science (including Artificial Intelligence and
Multiagent Systems) on the other side. Typical computational problems studied
in this field include the vulnerability of voting procedures against attacks,
or preference aggregation in multi-agent systems. Parameterized Algorithmics is
a subfield of Theoretical Computer Science seeking to exploit meaningful
problem-specific parameters in order to identify tractable special cases of in
general computationally hard problems. In this paper, we propose nine of our
favorite research challenges concerning the parameterized complexity of
problems appearing in this context
Dynamic Programming for Graphs on Surfaces
We provide a framework for the design and analysis of dynamic programming
algorithms for surface-embedded graphs on n vertices and branchwidth at most k.
Our technique applies to general families of problems where standard dynamic
programming runs in 2^{O(k log k)} n steps. Our approach combines tools from
topological graph theory and analytic combinatorics. In particular, we
introduce a new type of branch decomposition called "surface cut
decomposition", generalizing sphere cut decompositions of planar graphs
introduced by Seymour and Thomas, which has nice combinatorial properties.
Namely, the number of partial solutions that can be arranged on a surface cut
decomposition can be upper-bounded by the number of non-crossing partitions on
surfaces with boundary. It follows that partial solutions can be represented by
a single-exponential (in the branchwidth k) number of configurations. This
proves that, when applied on surface cut decompositions, dynamic programming
runs in 2^{O(k)} n steps. That way, we considerably extend the class of
problems that can be solved in running times with a single-exponential
dependence on branchwidth and unify/improve most previous results in this
direction.Comment: 28 pages, 3 figure
Structural Rounding: Approximation Algorithms for Graphs Near an Algorithmically Tractable Class
We develop a framework for generalizing approximation algorithms from the structural graph algorithm literature so that they apply to graphs somewhat close to that class (a scenario we expect is common when working with real-world networks) while still guaranteeing approximation ratios. The idea is to edit a given graph via vertex- or edge-deletions to put the graph into an algorithmically tractable class, apply known approximation algorithms for that class, and then lift the solution to apply to the original graph. We give a general characterization of when an optimization problem is amenable to this approach, and show that it includes many well-studied graph problems, such as Independent Set, Vertex Cover, Feedback Vertex Set, Minimum Maximal Matching, Chromatic Number, (l-)Dominating Set, Edge (l-)Dominating Set, and Connected Dominating Set.
To enable this framework, we develop new editing algorithms that find the approximately-fewest edits required to bring a given graph into one of a few important graph classes (in some cases these are bicriteria algorithms which simultaneously approximate both the number of editing operations and the target parameter of the family). For bounded degeneracy, we obtain an O(r log{n})-approximation and a bicriteria (4,4)-approximation which also extends to a smoother bicriteria trade-off. For bounded treewidth, we obtain a bicriteria (O(log^{1.5} n), O(sqrt{log w}))-approximation, and for bounded pathwidth, we obtain a bicriteria (O(log^{1.5} n), O(sqrt{log w} * log n))-approximation. For treedepth 2 (related to bounded expansion), we obtain a 4-approximation. We also prove complementary hardness-of-approximation results assuming P != NP: in particular, these problems are all log-factor inapproximable, except the last which is not approximable below some constant factor 2 (assuming UGC)
On Directed Feedback Vertex Set parameterized by treewidth
We study the Directed Feedback Vertex Set problem parameterized by the
treewidth of the input graph. We prove that unless the Exponential Time
Hypothesis fails, the problem cannot be solved in time on general directed graphs, where is the treewidth of
the underlying undirected graph. This is matched by a dynamic programming
algorithm with running time .
On the other hand, we show that if the input digraph is planar, then the
running time can be improved to .Comment: 20
Combinatorial Voter Control in Elections
Voter control problems model situations such as an external agent trying to
affect the result of an election by adding voters, for example by convincing
some voters to vote who would otherwise not attend the election. Traditionally,
voters are added one at a time, with the goal of making a distinguished
alternative win by adding a minimum number of voters. In this paper, we
initiate the study of combinatorial variants of control by adding voters: In
our setting, when we choose to add a voter~, we also have to add a whole
bundle of voters associated with . We study the computational
complexity of this problem for two of the most basic voting rules, namely the
Plurality rule and the Condorcet rule.Comment: An extended abstract appears in MFCS 201
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