1,052 research outputs found
Hardness of Games and Graph Sampling
The work presented in this document is divided into two parts. The �rst part presents the hardness of games and
the second part presents Graph sampling. Non-deterministic constraint logic[1] is used to prove the hardness of
games. The games which are considered in this work is Reversi (2 player bounded game), Peg Solitaire (single
player bounded game), Badland (single player bounded game). It also contains a theoretical study of peg
solitaire on special graph classes. Reversi is proved to be PSPACE-Complete using Bounded 2CL, Peg Solitaire
is proved to be NP-Complete using Bounded NCL. Badland is proved to be NP-Complete by a reduction from
3-SAT. The objective of study of peg solitaire of special graph classes is to �nd the maximum number of marbles
we can remove from a fully �lled board, if the player is given the privilege to remove a marble from any cell
initially, then following the rules after the initial move.
The second part of the work is dedicated to graph sampling. Given a graph G, we try to sample a represen-
tative subgraph Gs which is similar to the original graph G. The properties that are being studied are Degree
Distribution, Clustering Coefficient, Average Shortest Path Length, Largest Connected Component Size. To
measure the similarity between the original graph and sample we use the metrics Kolmogorov - Smirnov test
and Kullback - Leibler divergence test. Tightly Induced Edge Sampling performs well on general graphs but
it's performance decreases when the graph is a tree. Overall TIBFS and KARGER produces a sample which
closely matches the distribution of original graphs.
On Polynomial Kernelization of H-free Edge Deletion
For a set H of graphs, the H-free Edge Deletion problem is to decide whether there exist at most k edges in the input graph, for some k∈N, whose deletion results in a graph without an induced copy of any of the graphs in H . The problem is known to be fixed-parameter tractable if H is of finite cardinality. In this paper, we present a polynomial kernel for this problem for any fixed finite set H of connected graphs for the case where the input graphs are of bounded degree. We use a single kernelization rule which deletes vertices ‘far away’ from the induced copies of every H∈H in the input graph. With a slightly modified kernelization rule, we obtain polynomial kernels for H-free Edge Deletion under the following three settings
An FPT algorithm for Matching Cut and d-cut
Given a positive integer , the -CUT problem is to decide if an
undirected graph has a non trivial bipartition of such
that every vertex in (resp. ) has at most neighbors in (resp.
). When , this is the MATCHING CUT problem. Gomes and Sau, in IPEC
2019, gave the first fixed parameter tractable algorithm for -CUT, when
parameterized by maximum number of the crossing edges in the cut (i.e. the size
of edge cut). However, their paper doesn't provide an explicit bound on the
running time, as it indirectly relies on a MSOL formulation and Courcelle's
Theorem. Motivated by this, we design and present an FPT algorithm for the
MATCHING CUT (and more generally for -CUT) for general graphs with running
time where is the maximum size of the edge cut.
This is the first FPT algorithm for the MATCHING CUT (and -CUT) with an
explicit dependence on this parameter. We also observe a lower bound of
with same parameter for MATCHING CUT assuming ETH
Parameterized Complexity of Path Set Packing
In PATH SET PACKING, the input is an undirected graph , a collection of simple paths in , and a positive integer . The problem is to decide
whether there exist edge-disjoint paths in . We study the
parameterized complexity of PATH SET PACKING with respect to both natural and
structural parameters. We show that the problem is -hard with respect to
vertex cover plus the maximum length of a path in , and -hard
respect to pathwidth plus maximum degree plus solution size. These results
answer an open question raised in COCOON 2018. On the positive side, we show an
FPT algorithm parameterized by feedback vertex set plus maximum degree, and
also show an FPT algorithm parameterized by treewidth plus maximum degree plus
maximum length of a path in . Both the positive results complement the
hardness of PATH SET PACKING with respect to any subset of the parameters used
in the FPT algorithms
The chromatic discrepancy of graphs
For a proper vertex coloring cc of a graph GG, let φc(G)φc(G) denote the maximum, over all induced subgraphs HH of GG, the difference between the chromatic number χ(H)χ(H) and the number of colors used by cc to color HH. We define the chromatic discrepancy of a graph GG, denoted by φ(G)φ(G), to be the minimum φc(G)φc(G), over all proper colorings cc of GG. If HH is restricted to only connected induced subgraphs, we denote the corresponding parameter by View the MathML sourceφˆ(G). These parameters are aimed at studying graph colorings that use as few colors as possible in a graph and all its induced subgraphs. We study the parameters φ(G)φ(G) and View the MathML sourceφˆ(G) and obtain bounds on them. We obtain general bounds, as well as bounds for certain special classes of graphs including random graphs. We provide structural characterizations of graphs with φ(G)=0φ(G)=0 and graphs with View the MathML sourceφˆ(G)=0. We also show that computing these parameters is NP-hard
On Polynomial Kernelization of -free Edge Deletion
For a set of graphs , the \textsc{-free Edge
Deletion} problem asks to find whether there exist at most edges in the
input graph whose deletion results in a graph without any induced copy of
. In \cite{cai1996fixed}, it is shown that the problem is
fixed-parameter tractable if is of finite cardinality. However,
it is proved in \cite{cai2013incompressibility} that if is a
singleton set containing , for a large class of , there exists no
polynomial kernel unless . In this paper, we present a
polynomial kernel for this problem for any fixed finite set of
connected graphs and when the input graphs are of bounded degree. We note that
there are \textsc{-free Edge Deletion} problems which remain
NP-complete even for the bounded degree input graphs, for example
\textsc{Triangle-free Edge Deletion}\cite{brugmann2009generating} and
\textsc{Custer Edge Deletion(-free Edge
Deletion)}\cite{komusiewicz2011alternative}. When contains
, we obtain a stronger result - a polynomial kernel for -free
input graphs (for any fixed ). We note that for , there is an
incompressibility result for \textsc{-free Edge Deletion} for general
graphs \cite{cai2012polynomial}. Our result provides first polynomial kernels
for \textsc{Claw-free Edge Deletion} and \textsc{Line Edge Deletion} for
-free input graphs which are NP-complete even for -free
graphs\cite{yannakakis1981edge} and were raised as open problems in
\cite{cai2013incompressibility,open2013worker}.Comment: 12 pages. IPEC 2014 accepted pape
Chess is hard even for a single player
We introduce a generalization of "Solo Chess", a single-player variant of the
game that can be played on chess.com. The standard version of the game is
played on a regular 8 x 8 chessboard by a single player, with only white
pieces, using the following rules: every move must capture a piece, no piece
may capture more than 2 times, and if there is a King on the board, it must be
the final piece. The goal is to clear the board, i.e, make a sequence of
captures after which only one piece is left.
We generalize this game to unbounded boards with pieces, each of which
have a given number of captures that they are permitted to make. We show that
Generalized Solo Chess is NP-complete, even when it is played by only rooks
that have at most two captures remaining. It also turns out to be NP-complete
even when every piece is a queen with exactly two captures remaining in the
initial configuration. In contrast, we show that solvable instances of
Generalized Solo Chess can be completely characterized when the game is: a)
played by rooks on a one-dimensional board, and b) played by pawns with two
captures left on a 2D board.
Inspired by Generalized Solo Chess, we also introduce the Graph Capture Game,
which involves clearing a graph of tokens via captures along edges. This game
subsumes Generalized Solo Chess played by knights. We show that the Graph
Capture Game is NP-complete for undirected graphs and DAGs.Comment: 22 pages, a slightly shorter version to appear in FUN 202
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