
Subexponentialtime Algorithms for Maximum Independent Set in P_tfree and Broomfree Graphs
In algorithmic graph theory, a classic open question is to determine the...
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Disconnected Cuts in Clawfree Graphs
A disconnected cut of a connected graph is a vertex cut that itself also...
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The Computational Complexity of Finding Temporal Paths under Waiting Time Constraints
Computing a (shortest) path between two vertices in a graph is one of th...
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On rSimple kPath and Related Problems Parameterized by k/r
Abasi et al. (2014) and Gabizon et al. (2015) studied the following prob...
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Compilability of Abduction
Abduction is one of the most important forms of reasoning; it has been s...
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Detecting Feedback Vertex Sets of Size k in O^(2.7^k) Time
In the Feedback Vertex Set problem, one is given an undirected graph G a...
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The Parameterized Complexity of Guarding Almost Convex Polygons
Art Gallery is a fundamental visibility problem in Computational Geometr...
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Gerrymandering on graphs: Computational complexity and parameterized algorithms
Partitioning a region into districts to favor a particular candidate or a party is commonly known as gerrymandering. In this paper, we investigate the gerrymandering problem in graph theoretic setting as proposed by CohenZemach et al. [AAMAS 2018]. Our contributions in this article are twofold, conceptual and computational. We first resolve the open question posed by Ito et al. [AAMAS 2019] about the computational complexity of the problem when the input graph is a path. Next, we propose a generalization of their model, where the input consists of a graph on n vertices representing the set of voters, a set of m candidates š, a weight function w_v: šāā¤^+ for each voter vā V(G) representing the preference of the voter over the candidates, a distinguished candidate pāš, and a positive integer k. The objective is to decide if one can partition the vertex set into k pairwise disjoint connected sets (districts) s.t p wins more districts than any other candidate. The problem is known to be NPC even if k=2, m=2, and G is either a complete bipartite graph (in fact K_2,n) or a complete graph. This means that in search for FPT algorithms we need to either focus on the parameter n, or subclasses of forest. Circumventing these intractable results, we give a deterministic and a randomized algorithms for the problem on paths running in times 2.619^k(n+m)^O(1) and 2^k(n+m)^O(1), respectively. Additionally, we prove that the problem on general graphs is solvable in time 2^n (n+m)^O(1). Our algorithmic results use sophisticated technical tools such as representative set family and Fast Fourier transform based polynomial multiplication, and their (possibly first) application to problems arising in social choice theory and/or game theory may be of independent interest to the community.
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