arXiv subjects

Viktor Zamaraev

Publications and source records attributed to Viktor Zamaraev.

48 records · Page 3Linked to original sources

arXiv2019

Sliding Window Temporal Graph Coloring

Graph coloring is one of the most famous computational problems with applications in a wide range of areas such as planning and scheduling, resource allocation, and pattern matching. So far coloring problems are mostly studied on static graphs, which often stand in stark contrast to practice where data is inherently dynamic and subject to discrete changes over time. A temporal graph is a graph whose edges are assigned a set of integer time labels, indicating at which discrete time steps the edge is active. In this paper we present a natural temporal extension of the classical graph coloring problem. Given a temporal graph and a natural number $Δ$, we ask for a coloring sequence for each vertex such that (i) in every sliding time window of $Δ$ consecutive time steps, in which an edge is active, this edge is properly colored (i.e. its endpoints are assigned two different colors) at least once during that time window, and (ii) the total number of different colors is minimized. This sliding window temporal coloring problem abstractly captures many realistic graph coloring scenarios in which the underlying network changes over time, such as dynamically assigning communication channels to moving agents. We present a thorough investigation of the computational complexity of this temporal coloring problem. More specifically, we prove strong computational hardness results, complemented by efficient exact and approximation algorithms. Some of our algorithms are linear-time fixed-parameter tractable with respect to appropriate parameters, while others are asymptotically almost optimal under the Exponential Time Hypothesis (ETH).

cs.DM↗

arXiv2018

Linear read-once and related Boolean functions

It is known that a positive Boolean function f depending on n variables has at least n + 1 extremal points, i.e. minimal ones and maximal zeros. We show that f has exactly n + 1 extremal points if and only if it is linear read-once. The class of linear read-once functions is known to be the intersection of the classes of read-once and threshold functions. Generalizing this result we show that the class of linear read-once functions is the intersection of read-once and Chow functions. We also find the set of minimal read-once functions which are not linear read-once and the set of minimal threshold functions which are not linear read-once. In other words, we characterize the class of linear read-once functions by means of minimal forbidden subfunctions within the universe of read-once and the universe of threshold functions. Within the universe of threshold functions the importance of linear read-once func- tions is due to the fact that they attain the minimum value of the specification number, which is n + 1 for functions depending on n variables. In 1995 Anthony et al. conjec- tured that for all other threshold functions the specification number is strictly greater than n + 1. We disprove this conjecture by exhibiting a threshold non-linear read-once function depending on n variables whose specification number is n + 1.

math.CO↗

arXiv2018

Distributed Minimum Vertex Coloring and Maximum Independent Set in Chordal Graphs

We give deterministic distributed $(1+ε)$-approximation algorithms for Minimum Vertex Coloring and Maximum Independent Set on chordal graphs in the LOCAL model. Our coloring algorithm runs in $O(\frac{1}ε \log n)$ rounds, and our independent set algorithm has a runtime of $O(\frac{1}ε\log(\frac{1}ε)\log^* n)$ rounds. For coloring, existing lower bounds imply that the dependencies on $\frac{1}ε$ and $\log n$ are best possible. For independent set, we prove that $O(\frac{1}ε)$ rounds are necessary. Both our algorithms make use of a tree decomposition of the input chordal graph. They iteratively peel off interval subgraphs, which are identified via the tree decomposition of the input graph, thereby partitioning the vertex set into $O(\log n)$ layers. For coloring, each interval graph is colored independently, which results in various coloring conflicts between the layers. These conflicts are then resolved in a separate phase, using the particular structure of our partitioning. For independent set, only the first $O( \log \frac{1}ε)$ layers are required as they already contain a large enough independent set. We develop a $(1+ε)$-approximation maximum independent set algorithm for interval graphs, which we then apply to those layers. This work raises the question as to how useful tree decompositions are for distributed computing.

cs.DS↗

arXiv2018

Letter graphs and geometric grid classes of permutations: characterization and recognition

In this paper, we reveal an intriguing relationship between two seemingly unrelated notions: letter graphs and geometric grid classes of permutations. An important property common for both of them is well-quasi-orderability, implying, in a non-constructive way, a polynomial-time recognition of geometric grid classes of permutations and $k$-letter graphs for a fixed $k$. However, constructive algorithms are available only for $k=2$. In this paper, we present the first constructive polynomial-time algorithm for the recognition of $3$-letter graphs. It is based on a structural characterization of graphs in this class.

math.CO↗

arXiv2017

Clique-Width for Graph Classes Closed under Complementation

Clique-width is an important graph parameter due to its algorithmic and structural properties. A graph class is hereditary if it can be characterized by a (not necessarily finite) set ${\cal H}$ of forbidden induced subgraphs. We initiate a systematic study into the boundedness of clique-width of hereditary graph classes closed under complementation. First, we extend the known classification for the $|{\cal H}|=1$ case by classifying the boundedness of clique-width for every set ${\cal H}$ of self-complementary graphs. We then completely settle the $|{\cal H}|=2$ case. In particular, we determine one new class of $(H,\overline{H})$-free graphs of bounded clique-width (as a side effect, this leaves only six classes of $(H_1,H_2)$-free graphs, for which it is not known whether their clique-width is bounded). Once we have obtained the classification of the $|{\cal H}|=2$ case, we research the effect of forbidding self-complementary graphs on the boundedness of clique-width. Surprisingly, we show that for a set ${\cal F}$ of self-complementary graphs on at least five vertices, the classification of the boundedness of clique-width for $(\{H,\overline{H}\}\cup {\cal F})$-free graphs coincides with the one for the $|{\cal H}|=2$ case if and only if ${\cal F}$ does not include the bull (the only non-empty self-complementary graphs on fewer than five vertices are $P_1$ and $P_4$, and $P_4$-free graphs have clique-width at most $2$). Finally, we discuss the consequences of our results for the Colouring problem.

cs.DM↗

arXiv2017

Specifying a positive threshold function via extremal points

An extremal point of a positive threshold Boolean function $f$ is either a maximal zero or a minimal one. It is known that if $f$ depends on all its variables, then the set of its extremal points completely specifies $f$ within the universe of threshold functions. However, in some cases, $f$ can be specified by a smaller set. The minimum number of points in such a set is the specification number of $f$. It was shown in [S.-T. Hu. Threshold Logic, 1965] that the specification number of a threshold function of $n$ variables is at least $n+1$. In [M. Anthony, G. Brightwell, and J. Shawe-Taylor. On specifying Boolean functions by labelled examples. Discrete Applied Mathematics, 1995] it was proved that this bound is attained for nested functions and conjectured that for all other threshold functions the specification number is strictly greater than $n+1$. In the present paper, we resolve this conjecture negatively by exhibiting threshold Boolean functions of $n$ variables, which are non-nested and for which the specification number is $n+1$. On the other hand, we show that the set of extremal points satisfies the statement of the conjecture, i.e., a positive threshold Boolean function depending on all its $n$ variables has $n+1$ extremal points if and only if it is nested. To prove this, we reveal an underlying structure of the set of extremal points.

math.CO↗

arXiv2017

More results on weighted independent domination

Weighted independent domination is an NP-hard graph problem, which remains computationally intractable in many restricted graph classes. In particular, the problem is NP-hard in the classes of sat-graphs and chordal graphs. We strengthen these results by showing that the problem is NP-hard in a proper subclass of the intersection of sat-graphs and chordal graphs. On the other hand, we identify two new classes of graphs where the problem admits polynomial-time solutions.

cs.DM↗

arXiv2016

Upper Domination: towards a dichotomy through boundary properties

An upper dominating set in a graph is a minimal (with respect to set inclusion) dominating set of maximum cardinality. The problem of finding an upper dominating set is generally NP-hard. We study the complexity of this problem in classes of graphs defined by finitely many forbidden induced subgraphs and conjecture that the problem admits a dichotomy in this family, i.e. it is either NP-hard or polynomial-time solvable for each class in the family. A helpful tool to study the complexity of an algorithmic problem on finitely defined classes of graphs is the notion of boundary classes. However, none of such classes has been identified so far for the upper dominating set problem. In the present paper, we discover the first boundary class for this problem and prove the dichotomy for classes defined by a single forbidden induced subgraph.

cs.DM↗

arXiv2016

The structure and the number of $P_7$-free bipartite graphs

We show that the number of labelled $P_7$-free bipartite graphs with $n$ vertices grows as $n^{Θ(n)}$. This resolves an open problem posed by Allen [P. Allen, Forbidden induced bipartite graphs. J. Graph Theory 60 (2009), no. 3, 219--241.], and completes the description of speeds of monogenic classes of bipartite graphs. Our solution is based on a new decomposition scheme of bipartite graphs, which is of independent interest.

math.CO↗

arXiv2016

On forbidden induced subgraphs for unit disk graphs

A unit disk graph is the intersection graph of disks of equal radii in the plane. The class of unit disk graphs is hereditary, and therefore admits a characterization in terms of minimal forbidden induced subgraphs. In spite of quite active study of unit disk graphs very little is known about minimal forbidden induced subgraphs for this class. We found only finitely many minimal non unit disk graphs in the literature. In this paper we study in a systematic way forbidden induced subgraphs for the class of unit disk graphs. We develop several structural and geometrical tools, and use them to reveal infinitely many new minimal non unit disk graphs. Further we use these results to investigate structure of co-bipartite unit disk graphs. In particular, we give structural characterization of those co-bipartite unit disk graphs whose edges between parts form a C4-free bipartite graph, and show that bipartite complements of these graphs are also unit disk graphs. Our results lead us to propose a conjecture that the class of co-bipartite unit disk graphs is closed under bipartite complementation.

math.CO↗

arXiv2015

Well-quasi-ordering does not imply bounded clique-width

We present a hereditary class of graphs of unbounded clique-width which is well-quasi-ordered by the induced subgraph relation. This result provides a negative answer to the question asked by Daligault, Rao and Thomassé in ("Well-quasi-order of relabel functions", Order, 27(3):301--315, 2010).

cs.DM↗

arXiv2014

Combinatorics and algorithms for augmenting graphs

The notion of augmenting graphs generalizes Berge's idea of augmenting chains, which was used by Edmonds in his celebrated solution of the maximum matching problem. This problem is a special case of the more general maximum independent set (MIS) problem. Recently, the augmenting graph approach has been successfully applied to solve MIS in various other special cases. However, our knowledge of augmenting graphs is still very limited, and we do not even know what the minimal infinite classes of augmenting graphs are. In the present paper, we find an answer to this question and apply it to extend the area of polynomial-time solvability of the maximum independent set problem.

cs.DM↗