19 research outputs found
On Conflict-Free Cuts: Algorithms and Complexity
One way to define the Matching Cut problem is: Given a graph , is there an
edge-cut of such that is an independent set in the line graph of
? We propose the more general Conflict-Free Cut problem: Together with the
graph , we are given a so-called conflict graph on the edges of
, and we ask for an edge-cutset of that is independent in .
Since conflict-free settings are popular generalizations of classical
optimization problems and Conflict-Free Cut was not considered in the
literature so far, we start the study of the problem. We show that the problem
is -complete even when the maximum degree of is 5 and
is 1-regular. The same reduction implies an exponential lower bound
on the solvability based on the Exponential Time Hypothesis. We also give
parameterized complexity results: We show that the problem is fixed-parameter
tractable with the vertex cover number of as a parameter, and we show
-hardness even when has a feedback vertex set of size one,
and the clique cover number of is the parameter. Since the clique
cover number of is an upper bound on the independence number of
and thus the solution size, this implies -hardness
when parameterized by the cut size. We list polynomial-time solvable cases and
interesting open problems. At last, we draw a connection to a symmetric variant
of SAT.Comment: 13 pages, 3 figure
Knapsack: Connectedness, Path, and Shortest-Path
We study the knapsack problem with graph theoretic constraints. That is, we
assume that there exists a graph structure on the set of items of knapsack and
the solution also needs to satisfy certain graph theoretic properties on top of
knapsack constraints. In particular, we need to compute in the connected
knapsack problem a connected subset of items which has maximum value subject to
the size of knapsack constraint. We show that this problem is strongly
NP-complete even for graphs of maximum degree four and NP-complete even for
star graphs. On the other hand, we develop an algorithm running in time
where
are respectively treewidth of the graph, size, and target value of the
knapsack. We further exhibit a factor approximation algorithm
running in time for
every . We show similar results for several other graph theoretic
properties, namely path and shortest-path under the problem names path-knapsack
and shortestpath-knapsack. Our results seems to indicate that
connected-knapsack is computationally hardest followed by path-knapsack and
shortestpath-knapsack.Comment: Under revie
Parameterized Complexity of Conflict-Free Matchings and Paths
An input to a conflict-free variant of a classical problem Gamma, called Conflict-Free Gamma, consists of an instance I of Gamma coupled with a graph H, called the conflict graph. A solution to Conflict-Free Gamma in (I,H) is a solution to I in Gamma, which is also an independent set in H. In this paper, we study conflict-free variants of Maximum Matching and Shortest Path, which we call Conflict-Free Matching (CF-Matching) and Conflict-Free Shortest Path (CF-SP), respectively. We show that both CF-Matching and CF-SP are W[1]-hard, when parameterized by the solution size. Moreover, W[1]-hardness for CF-Matching holds even when the input graph where we want to find a matching is itself a matching, and W[1]-hardness for CF-SP holds for conflict graph being a unit-interval graph. Next, we study these problems with restriction on the conflict graphs. We give FPT algorithms for CF-Matching when the conflict graph is chordal. Also, we give FPT algorithms for both CF-Matching and CF-SP, when the conflict graph is d-degenerate. Finally, we design FPT algorithms for variants of CF-Matching and CF-SP, where the conflicting conditions are given by a (representable) matroid
Fair allocation of indivisible goods under conflict constraints
We consider the fair allocation of indivisible items to several agents and
add a graph theoretical perspective to this classical problem. Thereby we
introduce an incompatibility relation between pairs of items described in terms
of a conflict graph. Every subset of items assigned to one agent has to form an
independent set in this graph. Thus, the allocation of items to the agents
corresponds to a partial coloring of the conflict graph. Every agent has its
own profit valuation for every item. Aiming at a fair allocation, our goal is
the maximization of the lowest total profit of items allocated to any one of
the agents. The resulting optimization problem contains, as special cases, both
{\sc Partition} and {\sc Independent Set}. In our contribution we derive
complexity and algorithmic results depending on the properties of the given
graph. We can show that the problem is strongly NP-hard for bipartite graphs
and their line graphs, and solvable in pseudo-polynomial time for the classes
of chordal graphs, cocomparability graphs, biconvex bipartite graphs, and
graphs of bounded treewidth. Each of the pseudo-polynomial algorithms can also
be turned into a fully polynomial approximation scheme (FPTAS).Comment: A preliminary version containing some of the results presented here
appeared in the proceedings of IWOCA 2020. Version 3 contains an appendix
with a remark about biconvex bipartite graph
Shortest Path with Positive Disjunctive Constraints -- a Parameterized Perspective
We study the SHORTEST PATH problem with positive disjunctive constraints from
the perspective of parameterized complexity. For positive disjunctive
constraints, there are certain pair of edges such that any feasible solution
must contain at least one edge from every such pair. In this paper, we initiate
the study of SHORTEST PATH problem subject to some positive disjunctive
constraints the classical version is known to be NP-Complete. Formally, given
an undirected graph G = (V, E) with a forcing graph H = (E, F) such that the
vertex set of H is same as the edge set of G. The goal is to find a set S of at
most k edges from G such that S forms a vertex cover in H and there is a path
from s to t in the subgraph of G induced by the edge set S. In this paper, we
consider two natural parameterizations for this problem. One natural parameter
is the solution size, i.e. k for which we provide a kernel with O(k^5) vertices
when both G and H are general graphs. Additionally, when either G or H (but not
both) belongs to some special graph classes, we provied kernelization results
with O(k^3) vertices . The other natural parameter we consider is structural
properties of H, i.e. the size of a vertex deletion set of H to some special
graph classes. We provide some fixed-parameter tractability results for those
structural parameterizations.Comment: 14 page
On Strong NP-Completeness of Rational Problems
The computational complexity of the partition, 0-1 subset sum, unbounded
subset sum, 0-1 knapsack and unbounded knapsack problems and their multiple
variants were studied in numerous papers in the past where all the weights and
profits were assumed to be integers. We re-examine here the computational
complexity of all these problems in the setting where the weights and profits
are allowed to be any rational numbers. We show that all of these problems in
this setting become strongly NP-complete and, as a result, no pseudo-polynomial
algorithm can exist for solving them unless P=NP. Despite this result we show
that they all still admit a fully polynomial-time approximation scheme.Comment: to appear in Proc. of CSR 201