12 research outputs found
Exact Matching and the Top-k Perfect Matching Problem
The aim of this note is to provide a reduction of the Exact Matching problem
to the Top- Perfect Matching Problem. Together with earlier work by El
Maalouly, this shows that the two problems are polynomial-time equivalent.
The Exact Matching Problem is a well-known 40 years old problem for which a
randomized, but no deterministic poly-time algorithm has been discovered. The
Top- Perfect Matching Problem is the problem of finding a perfect matching
which maximizes the total weight of the heaviest edges contained in it
An Approximation Algorithm for the Exact Matching Problem in Bipartite Graphs
In 1982 Papadimitriou and Yannakakis introduced the Exact Matching problem, in which given a red and blue edge-colored graph G and an integer k one has to decide whether there exists a perfect matching in G with exactly k red edges. Even though a randomized polynomial-time algorithm for this problem was quickly found a few years later, it is still unknown today whether a deterministic polynomial-time algorithm exists. This makes the Exact Matching problem an important candidate to test the RP=P hypothesis.
In this paper we focus on approximating Exact Matching. While there exists a simple algorithm that computes in deterministic polynomial-time an almost perfect matching with exactly k red edges, not a lot of work focuses on computing perfect matchings with almost k red edges. In fact such an algorithm for bipartite graphs running in deterministic polynomial-time was published only recently (STACS\u2723). It outputs a perfect matching with k\u27 red edges with the guarantee that 0.5k ? k\u27 ? 1.5k. In the present paper we aim at approximating the number of red edges without exceeding the limit of k red edges. We construct a deterministic polynomial-time algorithm, which on bipartite graphs computes a perfect matching with k\u27 red edges such that k/3 ? k\u27 ? k
Counting Perfect Matchings in Dense Graphs Is Hard
We show that the problem of counting perfect matchings remains #P-complete
even if we restrict the input to very dense graphs, proving the conjecture in
[5]. Here "dense graphs" refer to bipartite graphs of bipartite independence
number , or general graphs of independence number . Our proof
is by reduction from counting perfect matchings in bipartite graphs, via
elementary linear algebra tricks and graph constructions
Exact Matching: Correct Parity and FPT Parameterized by Independence Number
Given an integer and a graph where every edge is colored either red or
blue, the goal of the exact matching problem is to find a perfect matching with
the property that exactly of its edges are red. Soon after Papadimitriou
and Yannakakis (JACM 1982) introduced the problem, a randomized polynomial-time
algorithm solving the problem was described by Mulmuley et al. (Combinatorica
1987). Despite a lot of effort, it is still not known today whether a
deterministic polynomial-time algorithm exists. This makes the exact matching
problem an important candidate to test the popular conjecture that the
complexity classes P and RP are equal. In a recent article (MFCS 2022),
progress was made towards this goal by showing that for bipartite graphs of
bounded bipartite independence number, a polynomial time algorithm exists. In
terms of parameterized complexity, this algorithm was an XP-algorithm
parameterized by the bipartite independence number. In this article, we
introduce novel algorithmic techniques that allow us to obtain an
FPT-algorithm. If the input is a general graph we show that one can at least
compute a perfect matching which has the correct number of red edges modulo
2, in polynomial time. This is motivated by our last result, in which we prove
that an FPT algorithm for general graphs, parameterized by the independence
number, reduces to the problem of finding in polynomial time a perfect matching
with at most red edges and the correct number of red edges modulo 2
The Complexity of Recognizing Geometric Hypergraphs
As set systems, hypergraphs are omnipresent and have various representations
ranging from Euler and Venn diagrams to contact representations. In a geometric
representation of a hypergraph , each vertex is associated
with a point and each hyperedge is associated
with a connected set such that for all . We say that a given
hypergraph is representable by some (infinite) family of sets in
, if there exist and such
that is a geometric representation of . For a family F, we define
RECOGNITION(F) as the problem to determine if a given hypergraph is
representable by F. It is known that the RECOGNITION problem is
-hard for halfspaces in . We study the
families of translates of balls and ellipsoids in , as well as of
other convex sets, and show that their RECOGNITION problems are also
-complete. This means that these recognition problems are
equivalent to deciding whether a multivariate system of polynomial equations
with integer coefficients has a real solution.Comment: Appears in the Proceedings of the 31st International Symposium on
Graph Drawing and Network Visualization (GD 2023) 17 pages, 11 figure
The Complexity of Recognizing Geometric Hypergraphs
As set systems, hypergraphs are omnipresent and have various representations. In a geometric representation of a hypergraph H=(V,E), each vertex v∈V is a associated with a point pv∈Rd and each hyperedge e∈E is associated with a connected set se⊂Rd such that {pv∣v∈V}∩se={pv∣v∈e} for all e∈E. We say that a given hypergraph H is representable by some (infinite) family F of sets in Rd, if there exist P⊂Rd and S⊆F such that (P,S) is a geometric representation of H. For a family F, we define RECOGNITION(F) as the problem to determine if a given hypergraph is representable by F. It is known that the RECOGNITION problem is ER-hard for halfspaces in Rd. We study the families of balls and ellipsoids in Rd, as well as other convex sets, and show that their RECOGNITION problems are also ER-complete. This means that these recognition problems are equivalent to deciding whether a multivariate system of polynomial equations with integer coefficients has a real solution
Exact Matching in Graphs of Bounded Independence Number
In the Exact Matching Problem (EM), we are given a graph equipped with a fixed coloring of its edges with two colors (red and blue), as well as a positive integer k. The task is then to decide whether the given graph contains a perfect matching exactly k of whose edges have color red. EM generalizes several important algorithmic problems such as perfect matching and restricted minimum weight spanning tree problems.
When introducing the problem in 1982, Papadimitriou and Yannakakis conjectured EM to be NP-complete. Later however, Mulmuley et al. presented a randomized polynomial time algorithm for EM, which puts EM in RP. Given that to decide whether or not RP=P represents a big open challenge in complexity theory, this makes it unlikely for EM to be NP-complete, and in fact indicates the possibility of a deterministic polynomial time algorithm. EM remains one of the few natural combinatorial problems in RP which are not known to be contained in P, making it an interesting instance for testing the hypothesis RP=P.
Despite EM being quite well-known, attempts to devise deterministic polynomial algorithms have remained illusive during the last 40 years and progress has been lacking even for very restrictive classes of input graphs. In this paper we push the frontier of positive results forward by proving that EM can be solved in deterministic polynomial time for input graphs of bounded independence number, and for bipartite input graphs of bounded bipartite independence number. This generalizes previous positive results for complete (bipartite) graphs which were the only known results for EM on dense graphs
Exact Matching: Algorithms and Related Problems
In 1982, Papadimitriou and Yannakakis introduced the Exact Matching (EM)
problem where given an edge colored graph, with colors red and blue, and an
integer , the goal is to decided whether or not the graph contains a perfect
matching with exactly red edges. Although they conjectured it to be
-complete, soon after it was shown to be solvable in randomized
polynomial time in the seminal work of Mulmuley et al. placing it in the
complexity class . Since then, all attempts at finding a
deterministic algorithm for EM have failed, thus leaving it as one of the few
natural combinatorial problems in but not known to be contained
in , and making it an interesting instance for testing the
hypothesis . Progress has been lacking even on very
restrictive classes of graphs despite the problem being quite well known as
evidenced by the number of works citing it. In this paper we aim to gain more
insight into the problem by considering two directions of study. In the first
direction, we study EM on bipartite graphs with a relaxation of the color
constraint and provide an algorithm where the output is required to be a
perfect matching with a number of red edges differing from by at most
. We also introduce an optimisation problem we call Top-k Perfect Matching
(TkPM) that shares many similarities with EM. By virtue of being an
optimization problem, it is more natural to approximate so we provide
approximation algorithms for it. In the second direction, we look at the
parameterized algorithms. Here we introduce new tools and FPT algorithms for
the study of EM and TkPM
An Approximation Algorithm for the Exact Matching Problem in Bipartite Graph
In 1982 Papadimitriou and Yannakakis introduced the Exact Matching problem, in which given a red and blue edge-colored graph G and an integer k one has to decide whether there exists a perfect matching in G with exactly k red edges. Even though a randomized polynomial-time algorithm for this problem was quickly found a few years later, it is still unknown today whether a deterministic polynomial-time algorithm exists. This makes the Exact Matching problem an important candidate to test the RP=P hypothesis. In this paper we focus on approximating Exact Matching. While there exists a simple algorithm that computes in deterministic polynomial-time an almost perfect matching with exactly k red edges, not a lot of work focuses on computing perfect matchings with almost k red edges. In fact such an algorithm for bipartite graphs running in deterministic polynomial-time was published only recently (STACS’23). It outputs a perfect matching with k′ red edges with the guarantee that 0.5k ≤ k′ ≤ 1.5k. In the present paper we aim at approximating the number of red edges without exceeding the limit of k red edges. We construct a deterministic polynomial-time algorithm, which on bipartite graphs computes a perfect matching with k′ red edges such that k3 ≤ k′ ≤ k.ISSN:1868-896
Topological Art in Simple Galleries
Let P be a simple polygon, then the art gallery problem is looking for a minimum set of points (guards) that can see every point in P. We say two points a,b∈P can see each other if the line segment seg(a,b) is contained in P. We denote by V(P) the family of all minimum guard placements. The Hausdorff distance makes V(P) a metric space and thus a topological space. We show homotopy-universality, that is for every semi-algebraic set S there is a polygon P such that V(P) is homotopy equivalent to S. Furthermore, for various concrete topological spaces T, we describe instances I of the art gallery problem such that V(I) is homeomorphic to T