170 research outputs found
The Firefighter Problem: A Structural Analysis
We consider the complexity of the firefighter problem where b>=1 firefighters
are available at each time step. This problem is proved NP-complete even on
trees of degree at most three and budget one (Finbow et al.,2007) and on trees
of bounded degree b+3 for any fixed budget b>=2 (Bazgan et al.,2012). In this
paper, we provide further insight into the complexity landscape of the problem
by showing that the pathwidth and the maximum degree of the input graph govern
its complexity. More precisely, we first prove that the problem is NP-complete
even on trees of pathwidth at most three for any fixed budget b>=1. We then
show that the problem turns out to be fixed parameter-tractable with respect to
the combined parameter "pathwidth" and "maximum degree" of the input graph
New Integrality Gap Results for the Firefighters Problem on Trees
The firefighter problem is NP-hard and admits a approximation based
on rounding the canonical LP. In this paper, we first show a matching
integrality gap of on the canonical LP. This result relies
on a powerful combinatorial gadget that can be used to prove integrality gap
results for many problem settings. We also consider the canonical LP augmented
with simple additional constraints (as suggested by Hartke). We provide several
evidences that these constraints improve the integrality gap of the canonical
LP: (i) Extreme points of the new LP are integral for some known tractable
instances and (ii) A natural family of instances that are bad for the canonical
LP admits an improved approximation algorithm via the new LP. We conclude by
presenting a integrality gap instance for the new LP.Comment: 22 page
Applications of Bee Colony Optimization
Many computationally difficult problems are attacked using non-exact algorithms, such as approximation algorithms and heuristics. This thesis investigates an ex- ample of the latter, Bee Colony Optimization, on both an established optimization problem in the form of the Quadratic Assignment Problem and the FireFighting problem, which has not been studied before as an optimization problem. Bee Colony Optimization is a swarm intelligence algorithm, a paradigm that has increased in popularity in recent years, and many of these algorithms are based on natural pro- cesses.
We tested the Bee Colony Optimization algorithm on the QAPLIB library of Quadratic Assignment Problem instances, which have either optimal or best known solutions readily available, and enabled us to compare the quality of solutions found by the algorithm. In addition, we implemented a couple of other well known algorithms for the Quadratic Assignment Problem and consequently we could analyse the runtime of our algorithm.
We introduce the Bee Colony Optimization algorithm for the FireFighting problem. We also implement some greedy algorithms and an Ant Colony Optimization al- gorithm for the FireFighting problem, and compare the results obtained on some randomly generated instances.
We conclude that Bee Colony Optimization finds good solutions for the Quadratic Assignment Problem, however further investigation on speedup methods is needed to improve its performance to that of other algorithms. In addition, Bee Colony Optimization is effective on small instances of the FireFighting problem, however as instance size increases the results worsen in comparison to the greedy algorithms, and more work is needed to improve the decisions made on these instances
Firefighting as a game
The Firefighter Problem was proposed in 1995 [16] as a deterministic discrete-time model for the spread (and containment) of a fire. Its applications reach from real fires to the spreading of diseases and the containment of floods. Furthermore, it can be used to model the spread of computer viruses or viral marketing in communication networks.
In this work, we study the problem from a game-theoretical perspective. Such a context seems very appropriate when applied to large networks, where entities may act and make decisions based on their own interests, without global coordination.
We model the Firefighter Problem as a strategic game where there is one player for each time step who decides where to place the firefighters. We show that the Price of Anarchy is linear in the general case, but at most 2 for trees. We prove that the quality of the equilibria improves when allowing coalitional cooperation among players. In general, we have that the Price of Anarchy is in T(n/k) where k is the coalition size. Furthermore, we show that there are topologies which have a constant Price of Anarchy even when constant sized coalitions are considered.Peer ReviewedPostprint (author’s final draft
Space-Efficient Parameterized Algorithms on Graphs of Low Shrubdepth
Dynamic programming on various graph decompositions is one of the most
fundamental techniques used in parameterized complexity. Unfortunately, even if
we consider concepts as simple as path or tree decompositions, such dynamic
programming uses space that is exponential in the decomposition's width, and
there are good reasons to believe that this is necessary. However, it has been
shown that in graphs of low treedepth it is possible to design algorithms which
achieve polynomial space complexity without requiring worse time complexity
than their counterparts working on tree decompositions of bounded width. Here,
treedepth is a graph parameter that, intuitively speaking, takes into account
both the depth and the width of a tree decomposition of the graph, rather than
the width alone.
Motivated by the above, we consider graphs that admit clique expressions with
bounded depth and label count, or equivalently, graphs of low shrubdepth (sd).
Here, sd is a bounded-depth analogue of cliquewidth, in the same way as td is a
bounded-depth analogue of treewidth. We show that also in this setting,
bounding the depth of the decomposition is a deciding factor for improving the
space complexity. Precisely, we prove that on -vertex graphs equipped with a
tree-model (a decomposition notion underlying sd) of depth and using
labels, we can solve
- Independent Set in time using
space;
- Max Cut in time using space; and
- Dominating Set in time using space via
a randomized algorithm.
We also establish a lower bound, conditional on a certain assumption about
the complexity of Longest Common Subsequence, which shows that at least in the
case of IS the exponent of the parametric factor in the time complexity has to
grow with if one wishes to keep the space complexity polynomial.Comment: Conference version to appear at the European Symposium on Algorithms
(ESA 2023
Spread and defend infection in graphs
The spread of an infection, a contagion, meme, emotion, message and various
other spreadable objects have been discussed in several works. Burning and
firefighting have been discussed in particular on static graphs. Graph burning
simulates the notion of the spread of "fire" throughout a graph (plus, one
unburned node burned at each time-step); graph firefighting simulates the
defending of nodes by placing firefighters on the nodes which have not been
already burned while the fire is being spread (started by only a single fire
source).
This article studies a combination of firefighting and burning on a graph
class which is a variation (generalization) of temporal graphs. Nodes can be
infected from "outside" a network. We present a notion of both upgrading (of
unburned nodes, similar to firefighting) and repairing (of infected nodes). The
nodes which are burned, firefighted, or repaired are chosen probabilistically.
So a variable amount of nodes are allowed to be infected, upgraded and repaired
in each time step.
In the model presented in this article, both burning and firefighting proceed
concurrently, we introduce such a system to enable the community to study the
notion of spread of an infection and the notion of upgrade/repair against each
other. The graph class that we study (on which, these processes are simulated)
is a variation of temporal graph class in which at each time-step,
probabilistically, a communication takes place (iff an edge exists in that time
step). In addition, a node can be "worn out" and thus can be removed from the
network, and a new healthy node can be added to the network as well. This class
of graphs enables systems with high complexity to be able to be simulated and
studied
Improved Hardness for Cut, Interdiction, and Firefighter Problems
We study variants of the classic s-t cut problem and prove the following improved hardness results assuming the Unique Games Conjecture (UGC).
* For Length-Bounded Cut and Shortest Path Interdiction, we show that both problems are hard to approximate within any constant factor, even if we allow bicriteria approximation. If we want to cut vertices or the graph is directed, our hardness ratio for Length-Bounded Cut matches the best approximation ratio up to a constant. Previously, the best hardness ratio was 1.1377 for Length-Bounded Cut and 2 for Shortest Path Interdiction.
* For any constant k >= 2 and epsilon > 0, we show that Directed Multicut with k source-sink pairs is hard to approximate within a factor k - epsilon. This matches the trivial k-approximation algorithm. By a simple reduction, our result for k = 2 implies that Directed Multiway Cut with two terminals (also known as s-t Bicut} is hard to approximate within a factor 2 - epsilon, matching the trivial 2-approximation algorithm.
* Assuming a variant of the UGC (implied by another variant of Bansal and Khot), we prove that it is hard to approximate Resource Minimization Fire Containment within any constant factor. Previously, the best hardness ratio was 2. For directed layered graphs with b layers, our hardness ratio Omega(log b) matches the best approximation algorithm.
Our results are based on a general method of converting an integrality gap instance to a length-control dictatorship test for variants of the s-t cut problem, which may be useful for other problems
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