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WinWin Kernelization for Degree Sequence Completion Problems
, 2014
"... We study the kernelizability of a class of NPhard graph modification problems based on vertex degree properties. Our main positive results refer to NPhard graph completion (that is, edge addition) cases while we show that there is no hope to achieve analogous results for the corresponding vertex ..."
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We study the kernelizability of a class of NPhard graph modification problems based on vertex degree properties. Our main positive results refer to NPhard graph completion (that is, edge addition) cases while we show that there is no hope to achieve analogous results for the corresponding vertex or edge deletion versions. Our algorithms are based on a method that transforms graph completion problems into efficiently solvable number problems and exploits ffactor computations for translating the results back into the graph setting. Indeed, our core observation is that we encounter a winwin situation in the sense that either the number of edge additions is small (and thus faster to find) or the problem is polynomialtime solvable. This approach helps in answering an open question by Mathieson and Szeider [JCSS 2012].
Tree Deletion Set has a Polynomial Kernel (but no OPT O(1) approximation
 CoRR
, 2013
"... In the Tree Deletion Set problem the input is a graph G together with an integer k. The objective is to determine whether there exists a set S of at most k vertices such that G \ S is a tree. The problem is NPcomplete and even NPhard to approximate within any factor of OPTc for any constant c. In ..."
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In the Tree Deletion Set problem the input is a graph G together with an integer k. The objective is to determine whether there exists a set S of at most k vertices such that G \ S is a tree. The problem is NPcomplete and even NPhard to approximate within any factor of OPTc for any constant c. In this paper we give an O(k5) size kernel for the Tree Deletion Set problem. An appealing feature of our kernelization algorithm is a new reduction rule, based on system of linear equations, that we use to handle the instances on which Tree Deletion Set is hard to approximate.
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"... Abstract This paper analyzes to what extent it is possible to efficiently reduce the number of clauses in NPhard satisfiability problems, without changing the answer. Upper and lower bounds are established using the concept of kernelization. Existing results show that if NP ⊆ coNP/poly, no efficie ..."
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Abstract This paper analyzes to what extent it is possible to efficiently reduce the number of clauses in NPhard satisfiability problems, without changing the answer. Upper and lower bounds are established using the concept of kernelization. Existing results show that if NP ⊆ coNP/poly, no efficient preprocessing algorithm can reduce nvariable instances of cnfsat with d literals per clause, to equivalent instances with O(n d−ε ) bits for any ε > 0. For the NotAllEqual sat problem, a compression to size O(n d−1 ) exists. We put these results in a common framework by analyzing the compressibility of binary CSPs. We characterize constraint types based on the minimum degree of multivariate polynomials whose roots correspond to the satisfying assignments, obtaining (nearly) matching upper and lower bounds in several settings. Our lower bounds show that not just the number of constraints, but also the encoding size of individual constraints plays an important role. For example, for Exact Satisfiability with unbounded clause length it is possible to efficiently reduce the number of constraints to n+1, yet no polynomialtime algorithm can reduce to an equivalent instance with O(n 2−ε ) bits for any ε > 0, unless NP ⊆ coNP/poly. ACM Subject Classification Introduction The goal of sparsification is to make an object such as a graph or logical structure less dense, without changing the outcome of a computational task of interest. Sparsification can be used to speed up the solution of NPhard problems, by sparsifying a problem instance before solving it. The notion of kernelization, originating in the field of parameterized complexity Optimal Sparsification for Some Binary CSPs Using Lowdegree Polynomials The vast majority of the currently known results in this direction are negative Before presenting our results, we give an example to illustrate our methods. Consider the NPcomplete Exact dCNFSatisfiability (Exact dsat) problem, which asks whether there is a truth assignment that satisfies exactly one literal in each clause; the clauses have size at most d. While there are Θ(n d ) different clauses that can occur in an instance with n variables, the exact nature of the problem makes it possible to reduce any instance to an equivalent one with n + 1 clauses. A clause such as x 1 ∨ x 3 ∨ ¬x 5 naturally corresponds to an equality constraint of the form x 1 + x 3 + (1 − x 5 ) = 1, since a 0/1assignment to the variables satisfies exactly one literal of the clause if and only if it satisfies the equality. To find redundant clauses, transform each of the m clauses into an equality to obtain a system of equalities Ax = b where A is an m × n matrix, x is the column vector (x 1 , . . . , x n ), and b is an integer column vector. Using Gaussian elimination, one can efficiently compute a basis B for the row space of the extended matrix (Ab): a set of equalities such that every equality can be written as a linear combination of equalities in B. Since (Ab) has n + 1 columns, its rank is at most n + 1 and the basis B contains at most n + 1 equalities. To perform data reduction, remove all clauses from the Exact dsat instance whose corresponding equalities do not occur in B. If an assignment satisfies f 1 (x) = b 1 and f 2 (x) = b 2 , then it also satisfies their sum f 1 (x) + f 2 (x) = b 1 + b 2 , and any linear combination of the satisfied equalities. Since any equality not in B can be written as a linear combination of equalities in B, a truth assignment satisfying all clauses from B must necessarily also satisfy the remaining clauses, which shows the correctness of the data reduction procedure. The resulting instance can be encoded in O(n log n) bits, as each of the remaining n + 1 clauses has d ∈ O(1) literals. Our results Our positive results are generalizations of the linearalgebraic data reduction tool for binary CSPs presented above. They reveal that the O(n)bit compression for
Using Patterns to Form Homogeneous Teams
 ALGORITHMICA
"... Homogeneous team formation is the task of grouping individuals into teams, each of which consists of members who fulfill the same set of prespecified properties. In this theoretical work, we propose, motivate, and analyze a combinatorial model where, given a matrix over a finite alphabet whose rows ..."
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Homogeneous team formation is the task of grouping individuals into teams, each of which consists of members who fulfill the same set of prespecified properties. In this theoretical work, we propose, motivate, and analyze a combinatorial model where, given a matrix over a finite alphabet whose rows correspond to individuals and columns correspond to attributes of individuals, the user specifies lower and upper bounds on team sizes as well as combinations of attributes that have to be homogeneous (that is, identical) for all members of the corresponding teams. Furthermore, the user can define a cost for assigning any individual to a certain team. We show that some special cases of our new model lead to NPhard problems while others allow for (fixedparameter) tractability results. For example, the problem is already NPhard even