Centdian Computation in Cactus Graphs

Ben‐Moshe, Boaz; Dvir, Amit; Segal, Michael; Tamir, Arie

doi:10.7155/jgaa.00255

Cited by 13 publications

(4 citation statements)

References 43 publications

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“…Cactus graphs were first defined by Harary and Uhlenbeck [15] who attributed them to the physicist Husimi and therefore called them Husimi Trees. Cactus graphs arise, for example, in the modelling of wireless sensor networks [2] and in the comparison of genomes [21]. Some NP-hard graph problems can be solved in polynomial time on cactus graphs [1].…”

Section: Introductionmentioning

confidence: 99%

The complexity of counting homomorphisms to cactus graphs modulo 2

Göbel

Goldberg

Richerby

2014

ACM Trans. Comput. Theory

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A homomorphism from a graph G to a graph H is a function from V (G) to V (H) that preserves edges. Many combinatorial structures that arise in mathematics and in computer science can be represented naturally as graph homomorphisms and as weighted sums of graph homomorphisms. In this paper, we study the complexity of counting homomorphisms modulo 2. The complexity of modular counting was introduced by Papadimitriou and Zachos and it has been pioneered by Valiant who famously introduced a problem for which counting modulo 7 is easy where counting modulo 2 is intractable. Modular counting provides a rich setting in which to study the structure of homomorphism problems. In this case, the structure of the graph H has a big influence on the complexity of the problem. Thus, our approach is graph-theoretic. We give a complete solution for the class of cactus graphs, which are connected graphs in which every edge belongs to at most one cycle. Cactus graphs arise in many applications such as the modelling of wireless sensor networks and the comparison of genomes. We show that, for some cactus graphs H, counting homomorphisms to H modulo 2 can be done in polynomial time. For every other fixed cactus graph H, the problem is complete in the complexity class ⊕P which is a wide complexity class to which every problem in the polynomial hierarchy can be reduced (using randomised reductions). Determining which H lead to tractable problems can be done in polynomial time. Our result builds upon the work of Faben and Jerrum, who gave a dichotomy for the case in which H is a tree.

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Section: Introductionmentioning

confidence: 99%

The complexity of counting homomorphisms to cactus graphs modulo 2

Göbel

Goldberg

Richerby

2014

ACM Trans. Comput. Theory

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“…Note that since α ≤ 1/π and = α/m, it follows that π 2 ≤ 1/m for all m ≥ 1, so the conditions of Lemma 15 are satisfied. Moreover, as shown in the proof of Theorem 12, the probability that two segments of length intersect is at least p5π 16 2 , where p 5 is the probability that two segments of length 1 intersect, given that their centers are distance 0.5 apart. By the same argument, if the center of s t is within distance /2 of the center of some other segment s i , 1 ≤ i ≤ t − 1, the conditional probability that s t intersects s i is at least p 5 .…”

Section: Buffon Segmentsmentioning

confidence: 84%

“…Properties of cactus graphs have been studied with some applications in mind; for example, cactus graphs arise in the design of telecommunication systems, material handling networks, and local area networks (cf. [2,7,23,28,31,33] and the bibliographies therein). Proposition 7.…”

Section: Segment Cactus Graphsmentioning

confidence: 99%

Intersections, circuits, and colorability of line segments

Brimkov¹,

Geneson²,

Jensen³

et al. 2018

Preprint

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We derive sharp upper and lower bounds on the number of intersection points and closed regions that can occur in sets of line segments with certain structure, in terms of the number of segments. We consider sets of segments whose underlying planar graphs are Halin graphs, cactus graphs, maximal planar graphs, and triangle-free planar graphs, as well as randomly produced segment sets. We also apply these results to a variant of the Erdős-Faber-Lovász (EFL) Conjecture stating that the intersection points of m segments can be colored with m colors so that no segment contains points with the same color. We investigate an optimization problem related to the EFL Conjecture for line segments, determine its complexity, and provide some computational approaches.

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“…Tamir et al [50] presented a polynomial time exact algorithm for the pDP on trees. Ben-Moshe et al [6] gave O(|V |log|V |) time exact algorithms for the 1DP on cycle graphs and cactus graphs, respectively. If the induced subgraph by the centdian set is connected, Nguyen et al [39] proposed a linear time algorithm for the pDP on unweighted block graphs and proved the problem is NP-Complete on weighted block graphs.…”

Section: Introductionmentioning

confidence: 99%

Approximability results for the $p$-centdian and the converse centdian problems

Chen¹

2022

Discrete Mathematics &Amp; Theoretical Computer Science

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Given an undirected graph $G=(V,E)$ with a nonnegative edge length function and an integer $p$, $0 < p < |V|$, the $p$-centdian problem is to find $p$ vertices (called the {\it centdian set}) of $V$ such that the {\it eccentricity} plus {\it median-distance} is minimized, in which the {\it eccentricity} is the maximum (length) distance of all vertices to their nearest {\it centdian set} and the {\it median-distance} is the total (length) distance of all vertices to their nearest {\it centdian set}. The {\it eccentricity} plus {\it median-distance} is called the {\it centdian-distance}. The purpose of the $p$-centdian problem is to find $p$ open facilities (servers) which satisfy the quality-of-service of the minimum total distance ({\it median-distance}) and the maximum distance ({\it eccentricity}) to their service customers, simultaneously. If we converse the two criteria, that is given the bound of the {\it centdian-distance} and the objective function is to minimize the cardinality of the {\it centdian set}, this problem is called the converse centdian problem. In this paper, we prove the $p$-centdian problem is NP-Complete. Then we design the first non-trivial brute force exact algorithms for the $p$-centdian problem and the converse centdian problem, respectively. Finally, we design two approximation algorithms for both problems.

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Centdian Computation in Cactus Graphs

Cited by 13 publications

References 43 publications

The complexity of counting homomorphisms to cactus graphs modulo 2

The complexity of counting homomorphisms to cactus graphs modulo 2

Intersections, circuits, and colorability of line segments

Approximability results for the $p$-centdian and the converse centdian problems

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