arXiv subjects
Publications and source records attributed to Gregory Gutin.
A spanning subgraph $F$ of a graph $G$ is called perfect if $F$ is a forest, the degree $d_F(x)$ of each vertex $x$ in $F$ is odd, and each tree of $F$ is an induced subgraph of $G$. We provide a short proof of the following theorem of A.D. Scott (Graphs & Combin., 2001): a connected graph $G$ contains a perfect forest if and only if $G$ has an even number of vertices.
The Workflow Satisfiability Problem (WSP) is a problem of practical interest that arises whenever tasks need to be performed by authorized users, subject to constraints defined by business rules. We are required to decide whether there exists a plan -- an assignment of tasks to authorized users -- such that all constraints are satisfied. The WSP is, in fact, the conservative Constraint Satisfaction Problem (i.e., for each variable, here called task, we have a unary authorization constraint) and is, thus, NP-complete. It was observed by Wang and Li (2010) that the number k of tasks is often quite small and so can be used as a parameter, and several subsequent works have studied the parameterized complexity of WSP regarding parameter k. We take a more detailed look at the kernelization complexity of WSP(Γ) when Γ denotes a finite or infinite set of allowed constraints. Our main result is a dichotomy for the case that all constraints in Γ are regular: (1) We are able to reduce the number n of users to n' <= k. This entails a kernelization to size poly(k) for finite Γ, and, under mild technical conditions, to size poly(k+m) for infinite Γ, where m denotes the number of constraints. (2) Already WSP(R) for some R \in Γ allows no polynomial kernelization in k+m unless the polynomial hierarchy collapses.
In the Travelling Salesman Problem (TSP), we are given a complete graph $K_n$ together with an integer weighting $w$ on the edges of $K_n$, and we are asked to find a Hamilton cycle of $K_n$ of minimum weight. Let $h(w)$ denote the average weight of a Hamilton cycle of $K_n$ for the weighting $w$. Vizing (1973) asked whether there is a polynomial-time algorithm which always finds a Hamilton cycle of weight at most $h(w)$. He answered this question in the affirmative and subsequently Rublineckii (1973) and others described several other TSP heuristics satisfying this property. In this paper, we prove a considerable generalisation of Vizing's result: for each fixed $k$, we give an algorithm that decides whether, for any input edge weighting $w$ of $K_n$, there is a Hamilton cycle of $K_n$ of weight at most $h(w)-k$ (and constructs such a cycle if it exists). For $k$ fixed, the running time of the algorithm is polynomial in $n$, where the degree of the polynomial does not depend on $k$ (i.e., the generalised Vizing problem is fixed-parameter tractable with respect to the parameter $k$).
The \emph{Workflow Satisfiability Problem (WSP)} is a problem of practical interest that arises whenever tasks need to be performed by authorized users, subject to constraints defined by business rules. We are required to decide whether there exists a \emph{plan} -- an assignment of tasks to authorized users -- such that all constraints are satisfied. Several bespoke algorithms have been constructed for solving the WSP, optimised to deal with constraints (business rules) of particular types. It is natural to see the WSP as a subclass of the {\em Constraint Satisfaction Problem (CSP)} in which the variables are tasks and the domain is the set of users. What makes the WSP distinctive as a CSP is that we can assume that the number of tasks is very small compared to the number of users. This is in sharp contrast with traditional CSP models where the domain is small and the number of variables is very large. As such, it is appropriate to ask for which constraint languages the WSP is fixed-parameter tractable (FPT), parameterized by the number of tasks. We have identified a new FPT constraint language, user-independent constraint, that includes many of the constraints of interest in business processing systems. We are also able to prove that the union of FPT languages remains FPT if they satisfy a simple compatibility condition. In this paper we present our generic algorithm, in which plans are grouped into equivalence classes, each class being associated with a \emph{pattern}. We demonstrate that our generic algorithm has running time $O^*(2^{k\log k})$, where $k$ is the number of tasks, for the language of user-independent constraints. We also show that there is no algorithm of running time $O^*(2^{o(k\log k)})$ for user-independent constraints unless the Exponential Time Hypothesis fails.
One way to state the Load Coloring Problem (LCP) is as follows. Let $G=(V,E)$ be graph and let $f:V\rightarrow \{{\rm red}, {\rm blue}\}$ be a 2-coloring. An edge $e\in E$ is called red (blue) if both end-vertices of $e$ are red (blue). For a 2-coloring $f$, let $r'_f$ and $b'_f$ be the number of red and blue edges and let $μ_f(G)=\min\{r'_f,b'_f\}$. Let $μ(G)$ be the maximum of $μ_f(G)$ over all 2-colorings. We introduce the parameterized problem $k$-LCP of deciding whether $μ(G)\ge k$, where $k$ is the parameter. We prove that this problem admits a kernel with at most $7k$. Ahuja et al. (2007) proved that one can find an optimal 2-coloring on trees in polynomial time. We generalize this by showing that an optimal 2-coloring on graphs with tree decomposition of width $t$ can be found in time $O^*(2^t)$. We also show that either $G$ is a Yes-instance of $k$-LCP or the treewidth of $G$ is at most $2k$. Thus, $k$-LCP can be solved in time $O^*(4^k).$
The Directed Rural Postman Problem (DRPP) can be formulated as follows: given a strongly connected directed multigraph $D=(V,A)$ with nonnegative integral weights on the arcs, a subset $R$ of $A$ and a nonnegative integer $\ell$, decide whether $D$ has a closed directed walk containing every arc of $R$ and of total weight at most $\ell$. Let $k$ be the number of weakly connected components in the the subgraph of $D$ induced by $R$. Sorge et al. (2012) ask whether the DRPP is fixed-parameter tractable (FPT) when parameterized by $k$, i.e., whether there is an algorithm of running time $O^*(f(k))$ where $f$ is a function of $k$ only and the $O^*$ notation suppresses polynomial factors. Sorge et al. (2012) note that this question is of significant practical relevance and has been open for more than thirty years. Using an algebraic approach, we prove that DRPP has a randomized algorithm of running time $O^*(2^k)$ when $\ell$ is bounded by a polynomial in the number of vertices in $D$. We also show that the same result holds for the undirected version of DRPP, where $D$ is a connected undirected multigraph.
In the Directed $k$-Chinese Postman Problem ($k$-DCPP), we are given a connected weighted digraph $G$ and asked to find $k$ non-empty closed directed walks covering all arcs of $G$ such that the total weight of the walks is minimum. Gutin, Muciaccia and Yeo (Theor. Comput. Sci. 513 (2013) 124--128) asked for the parameterized complexity of $k$-DCPP when $k$ is the parameter. We prove that the $k$-DCPP is fixed-parameter tractable. We also consider a related problem of finding $k$ arc-disjoint directed cycles in an Euler digraph, parameterized by $k$. Slivkins (ESA 2003) showed that this problem is W[1]-hard for general digraphs. Generalizing another result by Slivkins, we prove that the problem is fixed-parameter tractable for Euler digraphs. The corresponding problem on vertex-disjoint cycles in Euler digraphs remains W[1]-hard even for Euler digraphs.
We consider the following problem called the $k$-Chinese Postman Problem ($k$-CPP): given a connected edge-weighted graph $G$ and integers $p$ and $k$, decide whether there are at least $k$ closed walks such that every edge of $G$ is contained in at least one of them and the total weight of the edges in the walks is at most $p$? The problem $k$-CPP is NP-complete, and van Bevern et al. (to appear) and Sorge (2013) asked whether the $k$-CPP is fixed-parameter tractable when parameterized by $k$. We prove that the $k$-CPP is indeed fixed-parameter tractable. In fact, we prove a stronger result: the problem admits a kernel with $O(k^2\log k)$ vertices. We prove that the directed version of $k$-CPP is NP-complete and ask whether the directed version is fixed-parameter tractable when parameterized by $k$.
We study ordinal embedding relaxations in the realm of parameterized complexity. We prove the existence of a quadratic kernel for the {\sc Betweenness} problem parameterized above its tight lower bound, which is stated as follows. For a set $V$ of variables and set $\mathcal C$ of constraints "$v_i$ \mbox{is between} $v_j$ \mbox{and} $v_k$", decide whether there is a bijection from $V$ to the set $\{1,\ldots,|V|\}$ satisfying at least $|\mathcal C|/3 + κ$ of the constraints in $\mathcal C$. Our result solves an open problem attributed to Benny Chor in Niedermeier's monograph "Invitation to Fixed-Parameter Algorithms." The betweenness problem is of interest in molecular biology. An approach developed in this paper can be used to determine parameterized complexity of a number of other optimization problems on permutations parameterized above or below tight bounds.
A workflow specification defines a set of steps and the order in which those steps must be executed. Security requirements and business rules may impose constraints on which users are permitted to perform those steps. A workflow specification is said to be satisfiable if there exists an assignment of authorized users to workflow steps that satisfies all the constraints. An algorithm for determining whether such an assignment exists is important, both as a static analysis tool for workflow specifications, and for the construction of run-time reference monitors for workflow management systems. We develop new methods for determining workflow satisfiability based on the concept of constraint expressions, which were introduced recently by Khan and Fong. These methods are surprising versatile, enabling us to develop algorithms for, and determine the complexity of, a number of different problems related to workflow satisfiability.
In the {\sc Test Cover} problem we are given a hypergraph $H=(V, \mathcal{E})$ with $|V|=n, |\mathcal{E}|=m$, and we assume that $\mathcal{E}$ is a test cover, i.e. for every pair of vertices $x_i, x_j$, there exists an edge $e \in \mathcal{E}$ such that $|{x_i,x_j}\cap e|=1$. The objective is to find a minimum subset of $\mathcal{E}$ which is a test cover. The problem is used for identification across many areas, and is NP-complete. From a parameterized complexity standpoint, many natural parameterizations of {\sc Test Cover} are either $W[1]$-complete or have no polynomial kernel unless $coNP\subseteq NP/poly$, and thus are unlikely to be solveable efficiently. However, in practice the size of the edges is often bounded. In this paper we study the parameterized complexity of {\sc Test-$r$-Cover}, the restriction of {\sc Test Cover} in which each edge contains at most $r \ge 2$ vertices. In contrast to the unbounded case, we show that the following below-bound parameterizations of {\sc Test-$r$-Cover} are fixed-parameter tractable with a polynomial kernel: (1) Decide whether there exists a test cover of size $n-k$, and (2) decide whether there exists a test cover of size $m-k$, where $k$ is the parameter. In addition, we prove a new lower bound $\lceil \frac{2(n-1)}{r+1} \rceil$ on the minimum size of a test cover when the size of each edge is bounded by $r$. {\sc Test-$r$-Cover} parameterized above this bound is unlikely to be fixed-parameter tractable; in fact, we show that it is para-NP-complete, as it is NP-hard to decide whether an instance of {\sc Test-$r$-Cover} has a test cover of size exactly $\frac{2(n-1)}{r+1}$.
A workflow specification defines a set of steps and the order in which those steps must be executed. Security requirements may impose constraints on which groups of users are permitted to perform subsets of those steps. A workflow specification is said to be satisfiable if there exists an assignment of users to workflow steps that satisfies all the constraints. An algorithm for determining whether such an assignment exists is important, both as a static analysis tool for workflow specifications, and for the construction of run-time reference monitors for workflow management systems. Finding such an assignment is a hard problem in general, but work by Wang and Li in 2010 using the theory of parameterized complexity suggests that efficient algorithms exist under reasonable assumptions about workflow specifications. In this paper, we improve the complexity bounds for the workflow satisfiability problem. We also generalize and extend the types of constraints that may be defined in a workflow specification and prove that the satisfiability problem remains fixed-parameter tractable for such constraints. Finally, we consider preprocessing for the problem and prove that in an important special case, in polynomial time, we can reduce the given input into an equivalent one, where the number of users is at most the number of steps. We also show that no such reduction exists for two natural extensions of this case, which bounds the number of users by a polynomial in the number of steps, provided a widely-accepted complexity-theoretical assumption holds.
A function $f:\ \{-1,1\}^n\rightarrow \mathbb{R}$ is called pseudo-Boolean. It is well-known that each pseudo-Boolean function $f$ can be written as $f(x)=\sum_{I\in {\cal F}}\hat{f}(I)χ_I(x),$ where ${\cal F}\subseteq \{I:\ I\subseteq [n]\}$, $[n]=\{1,2,...,n\}$, and $χ_I(x)=\prod_{i\in I}x_i$ and $\hat{f}(I)$ are non-zero reals. The degree of $f$ is $\max \{|I|:\ I\in {\cal F}\}$ and the width of $f$ is the minimum integer $ρ$ such that every $i\in [n]$ appears in at most $ρ$ sets in $\cal F$. For $i\in [n]$, let $\mathbf{x}_i$ be a random variable taking values 1 or -1 uniformly and independently from all other variables $\mathbf{x}_j$, $j\neq i.$ Let $\mathbf{x}=(\mathbf{x}_1,...,\mathbf{x}_n)$. The $p$-norm of $f$ is $||f||_p=(\mathbb E[|f(\mathbf{x})|^p])^{1/p}$ for any $p\ge 1$. It is well-known that $||f||_q\ge ||f||_p$ whenever $q> p\ge 1$. However, the higher norm can be bounded by the lower norm times a coefficient not directly depending on $f$: if $f$ is of degree $d$ and $q> p>1$ then $ ||f||_q\le (\frac{q-1}{p-1})^{d/2}||f||_p.$ This inequality is called the Hypercontractive Inequality. We show that one can replace $d$ by $ρ$ in the Hypercontractive Inequality for each $q> p\ge 2$ as follows: $ ||f||_q\le ((2r)!ρ^{r-1})^{1/(2r)}||f||_p,$ where $r=\lceil q/2\rceil$. For the case $q=4$ and $p=2$, which is important in many applications, we prove a stronger inequality: $ ||f||_4\le (2ρ+1)^{1/4}||f||_2.$
We consider vectors from $\{0,1\}^n$. The weight of such a vector $v$ is the sum of the coordinates of $v$. The distance ratio of a set $L$ of vectors is ${\rm dr}(L):=\max \{ρ(x,y):\ x,y \in L\}/ \min \{ρ(x,y):\ x,y \in L,\ x\neq y\},$ where $ρ(x,y)$ is the Hamming distance between $x$ and $y$. We prove that (a) for every constant $λ>1$ there are no positive constants $α$ and $C$ such that every set $K$ of at least $λ^p$ vectors with weight $p$ contains a subset $K'$ with $|K'|\ge |K|^α$ and ${\rm dr}(K')\le C$, % even when $|K|\ge λ$, (b) For a set $K$ of vectors with weight $p$, and a constant $C>2$, there exists $K'\subseteq K$ such that ${\rm dr}(K')\le C$ and $|K'| \ge |K|^α$, where $α= 1/ \lceil \log(p/2)/\log(C/2) \rceil$.
Let F be a CNF formula with n variables and m clauses. F is 3-satisfiable if for any 3 clauses in F, there is a truth assignment which satisfies all of them. Lieberherr and Specker (1982) and, later, Yannakakis (1994) proved that in each 3-satisfiable CNF formula at least 2/3 of its clauses can be satisfied by a truth assignment. We improve this result by showing that every 3-satisfiable CNF formula F contains a subset of variables U, such that some truth assignment $τ$ will satisfy at least $2m/3+ m_U/3+ρn'$ clauses, where m is the number of clauses of F, m_U is the number of clauses of F containing a variable from U, n' is the total number of variables in clauses not containing a variable in U, and ρis a positive absolute constant. Both U and $τ$ can be found in polynomial time. We use our result to show that the following parameterized problem is fixed-parameter tractable and, moreover, has a kernel with a linear number of variables. In 3-S-MAXSAT-AE, we are given a 3-satisfiable CNF formula F with m clauses and asked to determine whether there is an assignment which satisfies at least 2m/3 + k clauses, where k is the parameter.
An oriented graph is a directed graph without directed 2-cycles. Poljak and Turzík (1986) proved that every connected oriented graph $G$ on $n$ vertices and $m$ arcs contains an acyclic subgraph with at least $\frac{m}{2}+\frac{n-1}{4}$ arcs. Raman and Saurabh (2006) gave another proof of this result and left it as an open question to establish the parameterized complexity of the following problem: does $G$ have an acyclic subgraph with least $\frac{m}{2}+\frac{n-1}{4}+k$ arcs, where $k$ is the parameter? We answer this question by showing that the problem can be solved by an algorithm of runtime $(12k)!n^{O(1)}$. Thus, the problem is fixed-parameter tractable. We also prove that there is a polynomial time algorithm that either establishes that the input instance of the problem is a Yes-instance or reduces the input instance to an equivalent one of size $O(k^2)$.
The Generalized Traveling Salesman Problem (GTSP) is a well-known combinatorial optimization problem with a host of applications. It is an extension of the Traveling Salesman Problem (TSP) where the set of cities is partitioned into so-called clusters, and the salesman has to visit every cluster exactly once. While the GTSP is a very important combinatorial optimization problem and is well studied in many aspects, the local search algorithms used in the literature are mostly basic adaptations of simple TSP heuristics. Hence, a thorough and deep research of the neighborhoods and local search algorithms specific to the GTSP is required. We formalize the procedure of adaptation of a TSP neighborhood for the GTSP and classify all other existing and some new GTSP neighborhoods. For every neighborhood, we provide efficient exploration algorithms that are often significantly faster than the ones known from the literature. Finally, we compare different local search implementations empirically.
In MaxSat, we are given a CNF formula $F$ with $n$ variables and $m$ clauses and asked to find a truth assignment satisfying the maximum number of clauses. Let $r_1,..., r_m$ be the number of literals in the clauses of $F$. Then $asat(F)=\sum_{i=1}^m (1-2^{-r_i})$ is the expected number of clauses satisfied by a random truth assignment (the truth values to the variables are distributed uniformly and independently). It is well-known that, in polynomial time, one can find a truth assignment satisfying at least $asat(F)$ clauses. In the parameterized problem MaxSat-AA, we are to decide whether there is a truth assignment satisfying at least $asat(F)+k$ clauses, where $k$ is the parameter. We prove that MaxSat-AA is para-NP-complete and, thus, MaxSat-AA is not fixed-parameter tractable unless P$=$NP. This is in sharp contrast to MaxLin2-AA which was recently proved to be fixed-parameter tractable by Crowston et al. (arXiv:1104.1135v3). In fact, we consider a more refined version of {\sc MaxSat-AA}, {\sc Max-$r(n)$-Sat-AA}, where $r_j\le r(n)$ for each $j$. Alon {\em et al.} (SODA 2010) proved that if $r=r(n)$ is a constant, then {\sc Max-$r$-Sat-AA} is fixed-parameter tractable. We prove that {\sc Max-$r(n)$-Sat-AA} is para-NP-complete for $r(n)=\lceil \log n\rceil.$ We also prove that assuming the exponential time hypothesis, {\sc Max-$r(n)$-Sat-AA} is not in XP already for any $r(n)\ge \log \log n +ϕ(n)$, where $ϕ(n)$ is any unbounded strictly increasing function. This lower bound on $r(n)$ cannot be decreased much further as we prove that {\sc Max-$r(n)$-Sat-AA} is (i) in XP for any $r(n)\le \log \log n - \log \log \log n$ and (ii) fixed-parameter tractable for any $r(n)\le \log \log n - \log \log \log n - ϕ(n)$, where $ϕ(n)$ is any unbounded strictly increasing function. The proof uses some results on {\sc MaxLin2-AA}.