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Vangelis Th. Paschos

Publications and source records attributed to Vangelis Th. Paschos.

At least 19 recordsLinked to original sources
arXiv2021

New Algorithms for Mixed Dominating Set

A mixed dominating set is a collection of vertices and edges that dominates all vertices and edges of a graph. We study the complexity of exact and parameterized algorithms for \textsc{Mixed Dominating Set}, resolving some open questions. In particular, we settle the problem's complexity parameterized by treewidth and pathwidth by giving an algorithm running in time $O^*(5^{tw})$ (improving the current best $O^*(6^{tw})$), as well as a lower bound showing that our algorithm cannot be improved under the Strong Exponential Time Hypothesis (SETH), even if parameterized by pathwidth (improving a lower bound of $O^*((2 - \varepsilon)^{pw})$). Furthermore, by using a simple but so far overlooked observation on the structure of minimal solutions, we obtain branching algorithms which improve both the best known FPT algorithm for this problem, from $O^*(4.172^k)$ to $O^*(3.510^k)$, and the best known exponential-time exact algorithm, from $O^*(2^n)$ and exponential space, to $O^*(1.912^n)$ and polynomial space.

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arXiv2021

Upper Dominating Set: Tight Algorithms for Pathwidth and Sub-Exponential Approximation

An upper dominating set is a minimal dominating set in a graph. In the \textsc{Upper Dominating Set} problem, the goal is to find an upper dominating set of maximum size. We study the complexity of parameterized algorithms for \textsc{Upper Dominating Set}, as well as its sub-exponential approximation. First, we prove that, under ETH, \textsc{$k$-Upper Dominating Set} cannot be solved in time $O(n^{o(k)})$ (improving on $O(n^{o(\sqrt{k})})$), and in the same time we show under the same complexity assumption that for any constant ratio $r$ and any $\varepsilon > 0$, there is no $r$-approximation algorithm running in time $O(n^{k^{1-\varepsilon}})$. Then, we settle the problem's complexity parameterized by pathwidth by giving an algorithm running in time $O^*(6^{pw})$ (improving the current best $O^*(7^{pw})$), and a lower bound showing that our algorithm is the best we can get under the SETH. Furthermore, we obtain a simple sub-exponential approximation algorithm for this problem: an algorithm that produces an $r$-approximation in time $n^{O(n/r)}$, for any desired approximation ratio $r < n$. We finally show that this time-approximation trade-off is tight, up to an arbitrarily small constant in the second exponent: under the randomized ETH, and for any ratio $r > 1$ and $\varepsilon > 0$, no algorithm can output an $r$-approximation in time $n^{(n/r)^{1-\varepsilon}}$. Hence, we completely characterize the approximability of the problem in sub-exponential time.

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arXiv2019

Improved (In-)Approximability Bounds for d-Scattered Set

In the $d$-Scattered Set problem we are asked to select at least $k$ vertices of a given graph, so that the distance between any pair is at least $d$. We study the problem's (in-)approximability and offer improvements and extensions of known results for Independent Set, of which the problem is a generalization. Specifically, we show: - A lower bound of $Δ^{\lfloor d/2\rfloor-ε}$ on the approximation ratio of any polynomial-time algorithm for graphs of maximum degree $Δ$ and an improved upper bound of $O(Δ^{\lfloor d/2\rfloor})$ on the approximation ratio of any greedy scheme for this problem. - A polynomial-time $2\sqrt{n}$-approximation for bipartite graphs and even values of $d$, that matches the known lower bound by considering the only remaining case. - A lower bound on the complexity of any $ρ$-approximation algorithm of (roughly) $2^{\frac{n^{1-ε}}{ρd}}$ for even $d$ and $2^{\frac{n^{1-ε}}{ρ(d+ρ)}}$ for odd $d$ (under the randomized ETH), complemented by $ρ$-approximation algorithms of running times that (almost) match these bounds.

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arXiv2019

Average-case complexity of a branch-and-bound algorithm for min dominating set

The average-case complexity of a branch-and-bound algorithms for Minimum Dominating Set problem in random graphs in the G(n,p) model is studied. We identify phase transitions between subexponential and exponential average-case complexities, depending on the growth of the probability p with respect to the number n of nodes.

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arXiv2018

Structural Parameters, Tight Bounds, and Approximation for (k,r)-Center

In $(k,r)$-Center we are given a (possibly edge-weighted) graph and are asked to select at most $k$ vertices (centers), so that all other vertices are at distance at most $r$ from a center. In this paper we provide a number of tight fine-grained bounds on the complexity of this problem with respect to various standard graph parameters. Specifically: - For any $r\ge 1$, we show an algorithm that solves the problem in $O^*((3r+1)^{\textrm{cw}})$ time, where $\textrm{cw}$ is the clique-width of the input graph, as well as a tight SETH lower bound matching this algorithm's performance. As a corollary, for $r=1$, this closes the gap that previously existed on the complexity of Dominating Set parameterized by $\textrm{cw}$. - We strengthen previously known FPT lower bounds, by showing that $(k,r)$-Center is W[1]-hard parameterized by the input graph's vertex cover (if edge weights are allowed), or feedback vertex set, even if $k$ is an additional parameter. Our reductions imply tight ETH-based lower bounds. Finally, we devise an algorithm parameterized by vertex cover for unweighted graphs. - We show that the complexity of the problem parameterized by tree-depth is $2^{Θ(\textrm{td}^2)}$ by showing an algorithm of this complexity and a tight ETH-based lower bound. We complement these mostly negative results by providing FPT approximation schemes parameterized by clique-width or treewidth which work efficiently independently of the values of $k,r$. In particular, we give algorithms which, for any $ε>0$, run in time $O^*((\textrm{tw}/ε)^{O(\textrm{tw})})$, $O^*((\textrm{cw}/ε)^{O(\textrm{cw})})$ and return a $(k,(1+ε)r)$-center, if a $(k,r)$-center exists, thus circumventing the problem's W-hardness.

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arXiv2018

Structurally Parameterized d-Scattered Set

In $d$-Scattered Set we are given an (edge-weighted) graph and are asked to select at least $k$ vertices, so that the distance between any pair is at least $d$, thus generalizing Independent Set. We provide upper and lower bounds on the complexity of this problem with respect to various standard graph parameters. In particular, we show the following: - For any $d\ge2$, an $O^*(d^{\textrm{tw}})$-time algorithm, where $\textrm{tw}$ is the treewidth of the input graph. - A tight SETH-based lower bound matching this algorithm's performance. These generalize known results for Independent Set. - $d$-Scattered Set is W[1]-hard parameterized by vertex cover (for edge-weighted graphs), or feedback vertex set (for unweighted graphs), even if $k$ is an additional parameter. - A single-exponential algorithm parameterized by vertex cover for unweighted graphs, complementing the above-mentioned hardness. - A $2^{O(\textrm{td}^2)}$-time algorithm parameterized by tree-depth ($\textrm{td}$), as well as a matching ETH-based lower bound, both for unweighted graphs. We complement these mostly negative results by providing an FPT approximation scheme parameterized by treewidth. In particular, we give an algorithm which, for any error parameter $ε> 0$, runs in time $O^*((\textrm{tw}/ε)^{O(\textrm{tw})})$ and returns a $d/(1+ε)$-scattered set of size $k$, if a $d$-scattered set of the same size exists.

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arXiv2016

Parameterized Exact and Approximation Algorithms for Maximum $k$-Set Cover and Related Satisfiability Problems

Given a family of subsets $\mathcal S$ over a set of elements~$X$ and two integers~$p$ and~$k$, Max k-Set Cover consists of finding a subfamily~$\mathcal T \subseteq \mathcal S$ of cardinality at most~$k$, covering at least~$p$ elements of~$X$. This problem is W[2]-hard when parameterized by~$k$, and FPT when parameterized by $p$. We investigate the parameterized approximability of the problem with respect to parameters~$k$ and~$p$. Then, we show that Max Sat-k, a satisfiability problem generalizing Max k-Set Cover, is also FPT with respect to parameter~$p$.

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arXiv2015

Sub-exponential Approximation Schemes for CSPs: from Dense to Almost Sparse

It has long been known, since the classical work of (Arora, Karger, Karpinski, JCSS~99), that \MC\ admits a PTAS on dense graphs, and more generally, \kCSP\ admits a PTAS on "dense" instances with $Ω(n^k)$ constraints. In this paper we extend and generalize their exhaustive sampling approach, presenting a framework for $(1-\eps)$-approximating any \kCSP\ problem in \emph{sub-exponential} time while significantly relaxing the denseness requirement on the input instance. Specifically, we prove that for any constants $δ\in (0, 1]$ and $\eps > 0$, we can approximate \kCSP\ problems with $Ω(n^{k-1+δ})$ constraints within a factor of $(1-\eps)$ in time $2^{O(n^{1-δ}\ln n /\eps^3)}$. The framework is quite general and includes classical optimization problems, such as \MC, {\sc Max}-DICUT, \kSAT, and (with a slight extension) $k$-{\sc Densest Subgraph}, as special cases. For \MC\ in particular (where $k=2$), it gives an approximation scheme that runs in time sub-exponential in $n$ even for "almost-sparse" instances (graphs with $n^{1+δ}$ edges). We prove that our results are essentially best possible, assuming the ETH. First, the density requirement cannot be relaxed further: there exists a constant $r < 1$ such that for all $δ> 0$, \kSAT\ instances with $O(n^{k-1})$ clauses cannot be approximated within a ratio better than $r$ in time $2^{O(n^{1-δ})}$. Second, the running time of our algorithm is almost tight \emph{for all densities}. Even for \MC\ there exists $r<1$ such that for all $δ' > δ>0$, \MC\ instances with $n^{1+δ}$ edges cannot be approximated within a ratio better than $r$ in time $2^{n^{1-δ'}}$.

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arXiv2015

Algorithmic Aspects of Upper Domination

In this paper we study combinatorial and algorithmic resp. complexity questions of upper domination, i.e., the maximum cardinality of a minimal dominating set in a graph. We give a full classification of the related maximisation and minimisation problems, as well as the related parameterised problems, on general graphs and on graphs of bounded degree, and we also study planar graphs.

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arXiv2015

Time-Approximation Trade-offs for Inapproximable Problems

In this paper we focus on problems which do not admit a constant-factor approximation in polynomial time and explore how quickly their approximability improves as the allowed running time is gradually increased from polynomial to (sub-)exponential. We tackle a number of problems: For Min Independent Dominating Set, Max Induced Path, Forest and Tree, for any $r(n)$, a simple, known scheme gives an approximation ratio of $r$ in time roughly $r^{n/r}$. We show that, for most values of $r$, if this running time could be significantly improved the ETH would fail. For Max Minimal Vertex Cover we give a non-trivial $\sqrt{r}$-approximation in time $2^{n/r}$. We match this with a similarly tight result. We also give a $\log r$-approximation for Min ATSP in time $2^{n/r}$ and an $r$-approximation for Max Grundy Coloring in time $r^{n/r}$. Furthermore, we show that Min Set Cover exhibits a curious behavior in this super-polynomial setting: for any $δ> 0$ it admits an $m^δ$-approximation, where $m$ is the number of sets, in just quasi-polynomial time. We observe that if such ratios could be achieved in polynomial time, the ETH or the Projection Games Conjecture would fail.

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arXiv2014

Sparsification and subexponential approximation

Instance sparsification is well-known in the world of exact computation since it is very closely linked to the Exponential Time Hypothesis. In this paper, we extend the concept of sparsification in order to capture subexponential time approximation. We develop a new tool for inapproximability, called approximation preserving sparsification and use it in order to get strong inapproximability results in subexponential time for several fundamental optimization problems as Max Independent Set, Min Dominating Set, Min Feedback Vertex Set, and Min Set Cover.

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arXiv2013

Parameterized (in)approximability of subset problems

We discuss approximability and inapproximability in FPT-time for a large class of subset problems where a feasible solution $S$ is a subset of the input data and the value of $S$ is $|S|$. The class handled encompasses many well-known graph, set, or satisfiability problems such as Dominating Set, Vertex Cover, Set Cover, Independent Set, Feedback Vertex Set, etc. In a first time, we introduce the notion of intersective approximability that generalizes the one of safe approximability and show strong parameterized inapproximability results for many of the subset problems handled. Then, we study approximability of these problems with respect to the dual parameter $n-k$ where $n$ is the size of the instance and $k$ the standard parameter. More precisely, we show that under such a parameterization, many of these problems, while W[$\cdot$]-hard, admit parameterized approximation schemata.

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arXiv2013

Playing with parameters: structural parameterization in graphs

When considering a graph problem from a parameterized point of view, the parameter chosen is often the size of an optimal solution of this problem (the "standard" parameter). A natural subject for investigation is what happens when we parameterize such a problem by various other parameters, some of which may be the values of optimal solutions to different problems. Such research is known as parameterized ecology. In this paper, we investigate seven natural vertex problems, along with their respective parameters: the size of a maximum independent set, the size of a minimum vertex cover, the size of a maximum clique, the chromatic number, the size of a minimum dominating set, the size of a minimum independent dominating set and the size of a minimum feedback vertex set. We study the parameterized complexity of each of these problems with respect to the standard parameter of the others.

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arXiv2013

An exact algorithm for 1-in-3 SAT

1-in-3 SAT is an NP-complete variant of 3-SAT\ where a "clause" is satisfied iff exactly one of its three literal is satisfied. We present here an exact algorithm solving \oit\ in time $O^*(1.260^n)$.

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arXiv2013

Multi-parameter complexity analysis for constrained size graph problems: using greediness for parameterization

We study the parameterized complexity of a broad class of problems called "local graph partitioning problems" that includes the classical fixed cardinality problems as max k-vertex cover, k-densest subgraph, etc. By developing a technique "greediness-for-parameterization", we obtain fixed parameter algorithms with respect to a pair of parameters k, the size of the solution (but not its value) and Δ, the maximum degree of the input graph. In particular, greediness-for-parameterization improves asymptotic running times for these problems upon random separation (that is a special case of color coding) and is more intuitive and simple. Then, we show how these results can be easily extended for getting standard-parameterization results (i.e., with parameter the value of the optimal solution) for a well known local graph partitioning problem.

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arXiv2012

Subexponential and FPT-time Inapproximability of Independent Set and Related Problems

Fixed-parameter algorithms, approximation algorithms and moderately exponential algorithms are three major approaches to algorithms design. While each of them being very active in its own, there is an increasing attention to the connection between different approaches. In particular, whether Maximum Independent Set would be better approximable once endowed with subexponential-time or FPT-time is a central question. In this paper, we present a strong link between the linear PCP conjecture and the inapproximability, thus partially answering this question.

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