18 research outputs found
The Complexity of Power-Index Comparison
We study the complexity of the following problem: Given two weighted voting
games G' and G'' that each contain a player p, in which of these games is p's
power index value higher? We study this problem with respect to both the
Shapley-Shubik power index [SS54] and the Banzhaf power index [Ban65,DS79]. Our
main result is that for both of these power indices the problem is complete for
probabilistic polynomial time (i.e., is PP-complete). We apply our results to
partially resolve some recently proposed problems regarding the complexity of
weighted voting games. We also study the complexity of the raw Shapley-Shubik
power index. Deng and Papadimitriou [DP94] showed that the raw Shapley-Shubik
power index is #P-metric-complete. We strengthen this by showing that the raw
Shapley-Shubik power index is many-one complete for #P. And our strengthening
cannot possibly be further improved to parsimonious completeness, since we
observe that, in contrast with the raw Banzhaf power index, the raw
Shapley-Shubik power index is not #P-parsimonious-complete.Comment: 12 page
Path Puzzles: Discrete Tomography with a Path Constraint is Hard
We prove that path puzzles with complete row and column information--or
equivalently, 2D orthogonal discrete tomography with Hamiltonicity
constraint--are strongly NP-complete, ASP-complete, and #P-complete. Along the
way, we newly establish ASP-completeness and #P-completeness for 3-Dimensional
Matching and Numerical 3-Dimensional Matching.Comment: 16 pages, 8 figures. Revised proof of Theorem 2.4. 2-page abstract
appeared in Abstracts from the 20th Japan Conference on Discrete and
Computational Geometry, Graphs, and Games (JCDCGGG 2017
Reductions for Frequency-Based Data Mining Problems
Studying the computational complexity of problems is one of the - if not the
- fundamental questions in computer science. Yet, surprisingly little is known
about the computational complexity of many central problems in data mining. In
this paper we study frequency-based problems and propose a new type of
reduction that allows us to compare the complexities of the maximal frequent
pattern mining problems in different domains (e.g. graphs or sequences). Our
results extend those of Kimelfeld and Kolaitis [ACM TODS, 2014] to a broader
range of data mining problems. Our results show that, by allowing constraints
in the pattern space, the complexities of many maximal frequent pattern mining
problems collapse. These problems include maximal frequent subgraphs in
labelled graphs, maximal frequent itemsets, and maximal frequent subsequences
with no repetitions. In addition to theoretical interest, our results might
yield more efficient algorithms for the studied problems.Comment: This is an extended version of a paper of the same title to appear in
the Proceedings of the 17th IEEE International Conference on Data Mining
(ICDM'17
Possible Winners in Noisy Elections
We consider the problem of predicting winners in elections, for the case
where we are given complete knowledge about all possible candidates, all
possible voters (together with their preferences), but where it is uncertain
either which candidates exactly register for the election or which voters cast
their votes. Under reasonable assumptions, our problems reduce to counting
variants of election control problems. We either give polynomial-time
algorithms or prove #P-completeness results for counting variants of control by
adding/deleting candidates/voters for Plurality, k-Approval, Approval,
Condorcet, and Maximin voting rules. We consider both the general case, where
voters' preferences are unrestricted, and the case where voters' preferences
are single-peaked.Comment: 34 page
Planar 3-SAT with a Clause/Variable Cycle
In the Planar 3-SAT problem, we are given a 3-SAT formula together with its
incidence graph, which is planar, and are asked whether this formula is
satisfiable. Since Lichtenstein's proof that this problem is NP-complete, it
has been used as a starting point for a large number of reductions. In the
course of this research, different restrictions on the incidence graph of the
formula have been devised, for which the problem also remains hard.
In this paper, we investigate the restriction in which we require that the
incidence graph can be augmented by the edges of a Hamiltonian cycle that first
passes through all variables and then through all clauses, in a way that the
resulting graph is still planar. We show that the problem of deciding
satisfiability of a 3-SAT formula remains NP-complete even if the incidence
graph is restricted in that way and the Hamiltonian cycle is given. This
complements previous results demanding cycles only through either the variables
or clauses.
The problem remains hard for monotone formulas, as well as for instances with
exactly three distinct variables per clause. In the course of this
investigation, we show that monotone instances of Planar 3-SAT with exactly
three distinct variables per clause are always satisfiable, thus settling the
question by Darmann, D\"ocker, and Dorn on the complexity of this problem
variant in a surprising way.Comment: Implementing style of DMTCS journa
The Surprising Power of Graph Neural Networks with Random Node Initialization
Graph neural networks (GNNs) are effective models for representation learning
on relational data. However, standard GNNs are limited in their expressive
power, as they cannot distinguish graphs beyond the capability of the
Weisfeiler-Leman graph isomorphism heuristic. In order to break this
expressiveness barrier, GNNs have been enhanced with random node initialization
(RNI), where the idea is to train and run the models with randomized initial
node features. In this work, we analyze the expressive power of GNNs with RNI,
and prove that these models are universal, a first such result for GNNs not
relying on computationally demanding higher-order properties. This universality
result holds even with partially randomized initial node features, and
preserves the invariance properties of GNNs in expectation. We then empirically
analyze the effect of RNI on GNNs, based on carefully constructed datasets. Our
empirical findings support the superior performance of GNNs with RNI over
standard GNNs.Comment: Proceedings of the Thirtieth International Joint Conference on
Artificial Intelligence (IJCAI-21). Code and data available at
http://www.github.com/ralphabb/GNN-RN
Who witnesses The Witness? Finding witnesses in The Witness is hard and sometimes impossible
We analyze the computational complexity of the many types of
pencil-and-paper-style puzzles featured in the 2016 puzzle video game The
Witness. In all puzzles, the goal is to draw a simple path in a rectangular
grid graph from a start vertex to a destination vertex. The different puzzle
types place different constraints on the path: preventing some edges from being
visited (broken edges); forcing some edges or vertices to be visited
(hexagons); forcing some cells to have certain numbers of incident path edges
(triangles); or forcing the regions formed by the path to be partially
monochromatic (squares), have exactly two special cells (stars), or be singly
covered by given shapes (polyominoes) and/or negatively counting shapes
(antipolyominoes). We show that any one of these clue types (except the first)
is enough to make path finding NP-complete ("witnesses exist but are hard to
find"), even for rectangular boards. Furthermore, we show that a final clue
type (antibody), which necessarily "cancels" the effect of another clue in the
same region, makes path finding -complete ("witnesses do not exist"),
even with a single antibody (combined with many anti/polyominoes), and the
problem gets no harder with many antibodies. On the positive side, we give a
polynomial-time algorithm for monomino clues, by reducing to hexagon clues on
the boundary of the puzzle, even in the presence of broken edges, and solving
"subset Hamiltonian path" for terminals on the boundary of an embedded planar
graph in polynomial time.Comment: 72 pages, 59 figures. Revised proof of Lemma 3.5. A short version of
this paper appeared at the 9th International Conference on Fun with
Algorithms (FUN 2018