arXiv subjects

G. Gutin

Publications and source records attributed to G. Gutin.

32 records · Page 2Linked to original sources

arXiv2009

Fixed-Parameter Algorithms in Analysis of Heuristics for Extracting Networks in Linear Programs

We consider the problem of extracting a maximum-size reflected network in a linear program. This problem has been studied before and a state-of-the-art SGA heuristic with two variations have been proposed. In this paper we apply a new approach to evaluate the quality of SGA\@. In particular, we solve majority of the instances in the testbed to optimality using a new fixed-parameter algorithm, i.e., an algorithm whose runtime is polynomial in the input size but exponential in terms of an additional parameter associated with the given problem. This analysis allows us to conclude that the the existing SGA heuristic, in fact, produces solutions of a very high quality and often reaches the optimal objective values. However, SGA contain two components which leave some space for improvement: building of a spanning tree and searching for an independent set in a graph. In the hope of obtaining even better heuristic, we tried to replace both of these components with some equivalent algorithms. We tried to use a fixed-parameter algorithm instead of a greedy one for searching of an independent set. But even the exact solution of this subproblem improved the whole heuristic insignificantly. Hence, the crucial part of SGA is building of a spanning tree. We tried three different algorithms, and it appears that the Depth-First search is clearly superior to the other ones in building of the spanning tree for SGA. Thereby, by application of fixed-parameter algorithms, we managed to check that the existing SGA heuristic is of a high quality and selected the component which required an improvement. This allowed us to intensify the research in a proper direction which yielded a superior variation of SGA.

cs.DS↗

arXiv2009

A Probabilistic Approach to Problems Parameterized Above or Below Tight Bounds

We introduce a new approach for establishing fixed-parameter tractability of problems parameterized above tight lower bounds. To illustrate the approach we consider three problems of this type of unknown complexity that were introduced by Mahajan, Raman and Sikdar (J. Comput. Syst. Sci. 75, 2009). We show that a generalization of one of the problems and non-trivial special cases of the other two are fixed-parameter tractable.

cs.DS↗

arXiv2008

Minimum Leaf Out-branching and Related Problems

Given a digraph $D$, the Minimum Leaf Out-Branching problem (MinLOB) is the problem of finding in $D$ an out-branching with the minimum possible number of leaves, i.e., vertices of out-degree 0. We prove that MinLOB is polynomial-time solvable for acyclic digraphs. In general, MinLOB is NP-hard and we consider three parameterizations of MinLOB. We prove that two of them are NP-complete for every value of the parameter, but the third one is fixed-parameter tractable (FPT). The FPT parametrization is as follows: given a digraph $D$ of order $n$ and a positive integral parameter $k$, check whether $D$ contains an out-branching with at most $n-k$ leaves (and find such an out-branching if it exists). We find a problem kernel of order $O(k^2)$ and construct an algorithm of running time $O(2^{O(k\log k)}+n^6),$ which is an `additive' FPT algorithm. We also consider transformations from two related problems, the minimum path covering and the maximum internal out-tree problems into MinLOB, which imply that some parameterizations of the two problems are FPT as well.

cs.DS↗

arXiv2008

Spanning directed trees with many leaves

The {\sc Directed Maximum Leaf Out-Branching} problem is to find an out-branching (i.e. a rooted oriented spanning tree) in a given digraph with the maximum number of leaves. In this paper, we obtain two combinatorial results on the number of leaves in out-branchings. We show that - every strongly connected $n$-vertex digraph $D$ with minimum in-degree at least 3 has an out-branching with at least $(n/4)^{1/3}-1$ leaves; - if a strongly connected digraph $D$ does not contain an out-branching with $k$ leaves, then the pathwidth of its underlying graph UG($D$) is $O(k\log k)$. Moreover, if the digraph is acyclic, the pathwidth is at most $4k$. The last result implies that it can be decided in time $2^{O(k\log^2 k)}\cdot n^{O(1)}$ whether a strongly connected digraph on $n$ vertices has an out-branching with at least $k$ leaves. On acyclic digraphs the running time of our algorithm is $2^{O(k\log k)}\cdot n^{O(1)}$.

cs.DS↗

arXiv2007

Algorithms for Generating Convex Sets in Acyclic Digraphs

A set $X$ of vertices of an acyclic digraph $D$ is convex if $X\neq \emptyset$ and there is no directed path between vertices of $X$ which contains a vertex not in $X$. A set $X$ is connected if $X\neq \emptyset$ and the underlying undirected graph of the subgraph of $D$ induced by $X$ is connected. Connected convex sets and convex sets of acyclic digraphs are of interest in the area of modern embedded processor technology. We construct an algorithm $\cal A$ for enumeration of all connected convex sets of an acyclic digraph $D$ of order $n$. The time complexity of $\cal A$ is $O(n\cdot cc(D))$, where $cc(D)$ is the number of connected convex sets in $D$. We also give an optimal algorithm for enumeration of all (not just connected) convex sets of an acyclic digraph $D$ of order $n$. In computational experiments we demonstrate that our algorithms outperform the best algorithms in the literature. Using the same approach as for $\cal A$, we design an algorithm for generating all connected sets of a connected undirected graph $G$. The complexity of the algorithm is $O(n\cdot c(G)),$ where $n$ is the order of $G$ and $c(G)$ is the number of connected sets of $G.$ The previously reported algorithm for connected set enumeration is of running time $O(mn\cdot c(G))$, where $m$ is the number of edges in $G.$

cs.DM↗

arXiv2007

Convex sets in acyclic digraphs

A non-empty set $X$ of vertices of an acyclic digraph is called connected if the underlying undirected graph induced by $X$ is connected and it is called convex if no two vertices of $X$ are connected by a directed path in which some vertices are not in $X$. The set of convex sets (connected convex sets) of an acyclic digraph $D$ is denoted by $\sco(D)$ ($\scc(D)$) and its size by $\co(D)$ ($\cc(D)$). Gutin, Johnstone, Reddington, Scott, Soleimanfallah, and Yeo (Proc. ACiD'07) conjectured that the sum of the sizes of all (connected) convex sets in $D$ equals $Θ(n \cdot \co(D))$ ($Θ(n \cdot \cc(D))$) where $n$ is the order of $D$. In this paper we exhibit a family of connected acyclic digraphs with $\sum_{C\in \sco(D)}|C| = o(n\cdot \co(D))$ and $\sum_{C\in \scc(D)}|C| = o(n\cdot \cc(D))$. We also show that the number of connected convex sets of order $k$ in any connected acyclic digraph of order $n$ is at least $n-k+1$. This is a strengthening of a theorem by Gutin and Yeo.

cs.DM↗

arXiv2007

Minimum Cost Homomorphisms to Locally Semicomplete and Quasi-Transitive Digraphs

For digraphs $G$ and $H$, a homomorphism of $G$ to $H$ is a mapping $f:\ V(G)\dom V(H)$ such that $uv\in A(G)$ implies $f(u)f(v)\in A(H)$. If, moreover, each vertex $u \in V(G)$ is associated with costs $c_i(u), i \in V(H)$, then the cost of a homomorphism $f$ is $\sum_{u\in V(G)}c_{f(u)}(u)$. For each fixed digraph $H$, the minimum cost homomorphism problem for $H$, denoted MinHOM($H$), can be formulated as follows: Given an input digraph $G$, together with costs $c_i(u)$, $u\in V(G)$, $i\in V(H)$, decide whether there exists a homomorphism of $G$ to $H$ and, if one exists, to find one of minimum cost. Minimum cost homomorphism problems encompass (or are related to) many well studied optimization problems such as the minimum cost chromatic partition and repair analysis problems. We focus on the minimum cost homomorphism problem for locally semicomplete digraphs and quasi-transitive digraphs which are two well-known generalizations of tournaments. Using graph-theoretic characterization results for the two digraph classes, we obtain a full dichotomy classification of the complexity of minimum cost homomorphism problems for both classes.

cs.DM↗

arXiv2007

On the Complexity of the Minimum Cost Homomorphism Problem for Reflexive Multipartite Tournaments

For digraphs $D$ and $H$, a mapping $f: V(D)\dom V(H)$ is a homomorphism of $D$ to $H$ if $uv\in A(D)$ implies $f(u)f(v)\in A(H).$ For a fixed digraph $H$, the homomorphism problem is to decide whether an input digraph $D$ admits a homomorphism to $H$ or not, and is denoted as HOMP($H$). Digraphs are allowed to have loops, but not allowed to have parallel arcs. A natural optimization version of the homomorphism problem is defined as follows. If each vertex $u \in V(D)$ is associated with costs $c_i(u), i \in V(H)$, then the cost of the homomorphism $f$ is $\sum_{u\in V(D)}c_{f(u)}(u)$. For each fixed digraph $H$, we have the {\em minimum cost homomorphism problem for} $H$ and denote it as MinHOMP($H$). The problem is to decide, for an input graph $D$ with costs $c_i(u),$ $u \in V(D), i\in V(H)$, whether there exists a homomorphism of $D$ to $H$ and, if one exists, to find one of minimum cost. In a recent paper, we posed a problem of characterizing polynomial time solvable and NP-hard cases of the minimum cost homomorphism problem for acyclic multipartite tournaments with possible loops (w.p.l.). In this paper, we solve the problem for reflexive multipartite tournaments and demonstrate a considerate difficulty of the problem for the whole class of multipartite tournaments w.p.l. using, as an example, acyclic 3-partite tournaments of order 4 w.p.l.\footnote{This paper was submitted to Discrete Mathematics on April 6, 2007}

cs.DM↗

arXiv2007

Complexity of the Minimum Cost Homomorphism Problem for Semicomplete Digraphs with Possible Loops

For digraphs $D$ and $H$, a mapping $f: V(D)\dom V(H)$ is a homomorphism of $D$ to $H$ if $uv\in A(D)$ implies $f(u)f(v)\in A(H).$ For a fixed digraph $H$, the homomorphism problem is to decide whether an input digraph $D$ admits a homomorphism to $H$ or not, and is denoted as HOM($H$). An optimization version of the homomorphism problem was motivated by a real-world problem in defence logistics and was introduced in \cite{gutinDAM154a}. If each vertex $u \in V(D)$ is associated with costs $c_i(u), i \in V(H)$, then the cost of the homomorphism $f$ is $\sum_{u\in V(D)}c_{f(u)}(u)$. For each fixed digraph $H$, we have the {\em minimum cost homomorphism problem for} $H$ and denote it as MinHOM($H$). The problem is to decide, for an input graph $D$ with costs $c_i(u),$ $u \in V(D), i\in V(H)$, whether there exists a homomorphism of $D$ to $H$ and, if one exists, to find one of minimum cost. Although a complete dichotomy classification of the complexity of MinHOM($H$) for a digraph $H$ remains an unsolved problem, complete dichotomy classifications for MinHOM($H$) were proved when $H$ is a semicomplete digraph \cite{gutinDAM154b}, and a semicomplete multipartite digraph \cite{gutinDAM}. In these studies, it is assumed that the digraph $H$ is loopless. In this paper, we present a full dichotomy classification for semicomplete digraphs with possible loops, which solves a problem in \cite{gutinRMS}.\footnote{This paper was submitted to SIAM J. Discrete Math. on October 27, 2006}

cs.DM↗

arXiv2006

Minimum Cost Homomorphisms to Semicomplete Bipartite Digraphs

For digraphs $D$ and $H$, a mapping $f: V(D)\dom V(H)$ is a homomorphism of $D$ to $H$ if $uv\in A(D)$ implies $f(u)f(v)\in A(H).$ If, moreover, each vertex $u \in V(D)$ is associated with costs $c_i(u), i \in V(H)$, then the cost of the homomorphism $f$ is $\sum_{u\in V(D)}c_{f(u)}(u)$. For each fixed digraph $H$, we have the {\em minimum cost homomorphism problem for} $H$. The problem is to decide, for an input graph $D$ with costs $c_i(u),$ $u \in V(D), i\in V(H)$, whether there exists a homomorphism of $D$ to $H$ and, if one exists, to find one of minimum cost. Minimum cost homomorphism problems encompass (or are related to) many well studied optimization problems. We describe a dichotomy of the minimum cost homomorphism problem for semicomplete multipartite digraphs $H$. This solves an open problem from an earlier paper. To obtain the dichotomy of this paper, we introduce and study a new notion, a $k$-Min-Max ordering of digraphs.

cs.DM↗

arXiv2006

The Linear Arrangement Problem Parameterized Above Guaranteed Value

A linear arrangement (LA) is an assignment of distinct integers to the vertices of a graph. The cost of an LA is the sum of lengths of the edges of the graph, where the length of an edge is defined as the absolute value of the difference of the integers assigned to its ends. For many application one hopes to find an LA with small cost. However, it is a classical NP-complete problem to decide whether a given graph $G$ admits an LA of cost bounded by a given integer. Since every edge of $G$ contributes at least one to the cost of any LA, the problem becomes trivially fixed-parameter tractable (FPT) if parameterized by the upper bound of the cost. Fernau asked whether the problem remains FPT if parameterized by the upper bound of the cost minus the number of edges of the given graph; thus whether the problem is FPT ``parameterized above guaranteed value.'' We answer this question positively by deriving an algorithm which decides in time $O(m+n+5.88^k)$ whether a given graph with $m$ edges and $n$ vertices admits an LA of cost at most $m+k$ (the algorithm computes such an LA if it exists). Our algorithm is based on a procedure which generates a problem kernel of linear size in linear time for a connected graph $G$. We also prove that more general parameterized LA problems stated by Serna and Thilikos are not FPT, unless P=NP.

cs.DS↗

arXiv2006

Minimum Cost Homomorphisms to Proper Interval Graphs and Bigraphs

For graphs $G$ and $H$, a mapping $f: V(G)\dom V(H)$ is a homomorphism of $G$ to $H$ if $uv\in E(G)$ implies $f(u)f(v)\in E(H).$ If, moreover, each vertex $u \in V(G)$ is associated with costs $c_i(u), i \in V(H)$, then the cost of the homomorphism $f$ is $\sum_{u\in V(G)}c_{f(u)}(u)$. For each fixed graph $H$, we have the {\em minimum cost homomorphism problem}, written as MinHOM($H)$. The problem is to decide, for an input graph $G$ with costs $c_i(u),$ $u \in V(G), i\in V(H)$, whether there exists a homomorphism of $G$ to $H$ and, if one exists, to find one of minimum cost. Minimum cost homomorphism problems encompass (or are related to) many well studied optimization problems. We describe a dichotomy of the minimum cost homomorphism problems for graphs $H$, with loops allowed. When each connected component of $H$ is either a reflexive proper interval graph or an irreflexive proper interval bigraph, the problem MinHOM($H)$ is polynomial time solvable. In all other cases the problem MinHOM($H)$ is NP-hard. This solves an open problem from an earlier paper. Along the way, we prove a new characterization of the class of proper interval bigraphs.

cs.DM↗

arXiv2005

Minimum Cost Homomorphisms to Semicomplete Multipartite Digraphs

For digraphs $D$ and $H$, a mapping $f: V(D)\dom V(H)$ is a {\em homomorphism of $D$ to $H$} if $uv\in A(D)$ implies $f(u)f(v)\in A(H).$ For a fixed directed or undirected graph $H$ and an input graph $D$, the problem of verifying whether there exists a homomorphism of $D$ to $H$ has been studied in a large number of papers. We study an optimization version of this decision problem. Our optimization problem is motivated by a real-world problem in defence logistics and was introduced very recently by the authors and M. Tso. Suppose we are given a pair of digraphs $D,H$ and a positive integral cost $c_i(u)$ for each $u\in V(D)$ and $i\in V(H)$. The cost of a homomorphism $f$ of $D$ to $H$ is $\sum_{u\in V(D)}c_{f(u)}(u)$. Let $H$ be a fixed digraph. The minimum cost homomorphism problem for $H$, MinHOMP($H$), is stated as follows: For input digraph $D$ and costs $c_i(u)$ for each $u\in V(D)$ and $i\in V(H)$, verify whether there is a homomorphism of $D$ to $H$ and, if it does exist, find such a homomorphism of minimum cost. In our previous paper we obtained a dichotomy classification of the time complexity of \MiP for $H$ being a semicomplete digraph. In this paper we extend the classification to semicomplete $k$-partite digraphs, $k\ge 3$, and obtain such a classification for bipartite tournaments.

cs.DM↗

arXiv2005

Minimum Cost and List Homomorphisms to Semicomplete Digraphs

The following optimization problem was introduced in \cite{gutinDAM}, where it was motivated by a real-world problem in defence logistics. Suppose we are given a pair of digraphs $D,H$ and a positive cost $c_i(u)$ for each $u\in V(D)$ and $i\in V(H)$. The cost of a homomorphism $f$ of $D$ to $H$ is $\sum_{u\in V(D)}c_{f(u)}(u)$. For a fixed digraph $H$, the minimum cost homomorphism problem for $H$, MinHOMP($H$), is stated as follows: For an input digraph $D$ and costs $c_i(u)$ for each $u\in V(D)$ and $i\in V(H)$, verify whether there is a homomorphism of $D$ to $H$ and, if it exists, find such a homomorphism of minimum cost. We obtain dichotomy classifications of the computational complexity of the list homomorphism problem and MinHOMP($H$), when $H$ is a semicomplete digraph (a digraph in which every two vertices have at least one arc between them). Our dichotomy for the list homomorphism problem coincides with the one obtained by Bang-Jensen, Hell and MacGillivray in 1988 for the homomorphism problem when $H$ is a semicomplete digraph: both problems are polynomial solvable if $H$ has at most one cycle; otherwise, both problems are NP-complete. The dichotomy for \MiP is different: the problem is polynomial time solvable if $H$ is acyclic or $H$ is a cycle of length 2 or 3; otherwise, the problem is NP-hard.

cs.DM↗