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### Maximum Induced Matching Algorithms via Vertex Ordering Characterizations [article]

2017 arXiv   pre-print
Furthermore, these orderings on L2(g) can be exploited algorithmically to compute a maximum induced matching on G faster.  ...  We study the maximum induced matching problem on a graph g. Induced matchings correspond to independent sets in L2(g), the square of the line graph of g.  ...  Algorithm 2 2 Cocomparability Weighted Maximum Induced Matching (CCWMIM) Input: G = (V, E, w) an edge weighted cocomparability graph where w : E ae R >0 Output: A maximum weight induced matching of G 1  ...

### Maximum Induced Matching Algorithms via Vertex Ordering Characterizations * †

unpublished
Furthermore, these orderings on L 2 (G) can be exploited algorithmically to compute a maximum induced matching on G faster.  ...  We study the maximum induced matching problem on a graph G. Induced matchings correspond to independent sets in L 2 (G), the square of the line graph of G.  ...  In Section 4, we present the maximum weight induced matching algorithm and its analysis on cocomparability graphs.  ...

### Vertex-Weighted Matching in Two-Directional Orthogonal Ray Graphs [chapter]

C. Gregory Plaxton
2013 Lecture Notes in Computer Science
Given such a compact representation of G, and a (possibly negative) weight for each vertex, we show how to compute a maximum weight matching of G in O(n log 2 n) time.  ...  As an application of our more general result, we obtain an O(n log 2 n)-time algorithm for computing the VCG outcome of a sealed-bid unit-demand auction in which each item has two associated numerical  ...  Assume that we are given a bicolored 2D-graph representation of an n-vertex, vertex-weighted 2DORG G such that each x-value, y-value, or weight is an O(1)-word integer.  ...

### Mapping Matchings to Minimum Vertex Covers: Kőnig's Theorem Revisited [article]

Jacob Turner
2020 arXiv   pre-print
Kőnig's proof of this fact gave an algorithm for finding a minimum vertex cover from a maximum matching.  ...  We find that all minimum vertex covers can be found by applying this algorithm to some matching and then classify which matchings give minimum vertex covers when this algorithm is applied to them.  ...  This algorithm is still used in most state of the art algorithms today in order to find a minimum vertex cover after finding a maximum matching (which may be accomplished by one of several different algorithms  ...

### The Sparse Awakens: Streaming Algorithms for Matching Size Estimation in Sparse Graphs

Graham Cormode, Hossein Jowhari, Morteza Monemizadeh, S. Muthukrishnan, Marc Herbstritt
2017 European Symposium on Algorithms
In contrast to prior work, our results take more advantage of the streaming access to the input and characterize the matching size based on the ordering of the edges in the stream in addition to the degree  ...  We present improved streaming algorithms for approximating the size of maximum matching with sparse (bounded arboricity) graphs.  ...  We prove this characterization gives an O(α) approximation of the maximum matching size.  ...

### The Sparse Awakens: Streaming Algorithms for Matching Size Estimation in Sparse Graphs [article]

Graham Cormode and Hossein Jowhari and Morteza Monemizadeh and S. Muthukrishnan
2016 arXiv   pre-print
In contrast to the previous works, our results take more advantage of the streaming access to the input and characterize the matching size based on the ordering of the edges in the stream in addition to  ...  arboricity graph, we present a one-pass algorithm that uses space O (c^10/3n^2/3) and returns an O(c)-estimator for the size of the maximum matching.  ...  In the following we let M µ denote the size of maximum matching in G L . Let G L = (V ′ , E ′ ) be an induced subgraph of G where V ′ = {v| deg G (v) ≤ µ} Characterization lemmas Lemma 3 ([9] ).  ...

### Optimal Lower Bounds for Matching and Vertex Cover in Dynamic Graph Streams [article]

2020 arXiv   pre-print
[SODA 2016] gave an optimal space lower bound for insertion-deletion streaming algorithms for Maximum Matching via the simultaneous model of communication.  ...  Our results imply optimal space lower bounds for insertion-deletion streaming algorithms for Maximum Matching and Minimum Vertex Cover. Previously, Assadi et al.  ...  induced matchings.  ...

### Optimal Lower Bounds for Matching and Vertex Cover in Dynamic Graph Streams

Jacques Dark, Christian Konrad, Shubhangi Saraf
2020 Computational Complexity Conference
[SODA 2016] gave an optimal space lower bound for insertion-deletion streaming algorithms for Maximum Matching via the simultaneous model of communication.  ...  Our results imply optimal space lower bounds for insertion-deletion streaming algorithms for Maximum Matching and Minimum Vertex Cover. Previously, Assadi et al.  ...  induced matchings.  ...

### On the strong p-Helly property

Mitre C. Dourado, Fábio Protti, Jayme L. Szwarcfiter
2008 Discrete Applied Mathematics
We describe a characterization for this class and obtain an algorithm for recognizing such graphs.  ...  In this paper, we characterize strong p-Helly hypergraphs. This characterization leads to an algorithm for recognizing such hypergraphs, which terminates within polynomial time whenever p is fixed.  ...  This means that V int ⊆ U and the vertices of V int and V ext form an induced k + -matching in G. The rank of H is the maximum size of a hyperdge of H.  ...

### A Network Game with Attackers and a Defender

Marios Mavronicolas, Vicky Papadopoulou, Anna Philippou, Paul Spirakis
2007 Algorithmica
So, using a polynomial time algorithm to compute a Maximum Matching for a bipartite graph, we obtain, as our main result, a deterministic, polynomial time algorithm to compute a Matching Nash equilibrium  ...  The characterization enables a non-deterministic, polynomial time algorithm to compute a Matching Nash equilibrium for any such graph. -Bipartite graphs are shown to satisfy the characterization.  ...  a Minimum Vertex Cover V C of a bipartite graph (via reduction to computing a Maximum Matching), followed by setting IS to include all vertices in the complementary vertex set of V C.  ...

### The Price of Defense [chapter]

Marios Mavronicolas, Loizos Michael, Vicky Papadopoulou, Anna Philippou, Paul Spirakis
2006 Lecture Notes in Computer Science
In particular, we prove that in a Matching Nash equilibrium, the support of the vertex players is a Maximum Independent Set of the graph (Proposition 5.1) and the support of the edge player is a Minimum  ...  Each attacker (called a vertex player) targets a node of the network chosen via its own probability distribution on nodes; the defender (called the edge player) chooses a single link via its own probability  ...  In turn, the graph-theoretic algorithm relies on computing a Minimum Edge Cover (via computing a Maximum Matching) and a subsequent reduction to 2SAT. • We prove that the Price of Defense for a Matching  ...

### Stabilizing Weighted Graphs

Zhuan Khye Koh, Laura Sanità, Michael Wagner
2018 International Colloquium on Automata, Languages and Programming
Motivated by this, in the last few years many researchers have investigated the algorithmic problem of turning a given graph into a stable one, via edge-and vertex-removal operations.  ...  An edge-weighted graph G = (V, E) is called stable if the value of a maximum-weight matching equals the value of a maximum-weight fractional matching.  ...  Our first step is to characterize basic maximum-weight fractional matchings which have more than γ(G) odd cycles.  ...

### A parallel algorithm for computing the critical independence number and related sets

Ermelinda DeLaViña, Craig Eric Larson
2012 Ars Mathematica Contemporanea
The graph induced on this set is a König-Egerváry graph whose components are either isolated vertices or which have perfect matchings.  ...  It is demonstrated here that there is a parallel algorithm using n processors that runs in O(n 1.5 m/ log n) time. The new algorithm returns the union of all maximum critical independent sets.  ...  If M is a matching in a graph G and w is a vertex incident to an edge in M , let w be the vertex matched to w under M . The new algorithm can now be stated. The parallelism occurs in step 1.  ...

### Computing cutwidth and pathwidth of semi-complete digraphs via degree orderings [article]

Michał Pilipczuk
2012 arXiv   pre-print
In this work we introduce a new approach to computing these width measures on semi-complete digraphs, via degree orderings.  ...  First, we present polynomial-time approximation algorithms for both cutwidth and pathwidth, faster and simpler than the previously known ones; the most significant improvement is in case of pathwidth,  ...  Moreover, we have that every vertex of N (A) must be in fact matched to a vertex of A, as the vertices matched to N (A) are also reachable via alternating paths from A 0 .  ...

### An exact characterization of tractable demand patterns for maximum disjoint path problems [article]

Dániel Marx, Paul Wollan
2014 arXiv   pre-print
of large induced matchings and large induced skew bicliques in the demand graph H (a skew biclique is a bipartite graph on vertices a_1, ..., a_n, b_1, ..., b_n with a_i and b_j being adjacent if and  ...  If H does not contain every matching and does not contain every skew biclique, then H-Maximum Disjoint Paths is FPT. 2.  ...  Excluding induced matchings and skew bicliques: the exact FPT algorithm The goal of this section is to prove the algorithmic part of Theorem 1.4: an FPT algorithm for Maximum Disjoint H-Paths if H does  ...
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