
An optimal algorithm for Bisection for boundedtreewidth graphs
The maximum/minimum bisection problems are, given an edgeweighted graph...
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Finegrained complexity of graph homomorphism problem for boundedtreewidth graphs
For graphs G and H, a homomorphism from G to H is an edgepreserving map...
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Tight bounds for counting colorings and connected edge sets parameterized by cutwidth
We study the finegrained complexity of counting the number of colorings...
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Approximate counting CSP seen from the other side
In this paper we study the complexity of counting Constraint Satisfactio...
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Testing the complexity of a valued CSP language
A Valued Constraint Satisfaction Problem (VCSP) provides a common framew...
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On Treewidth and Stable Marriage
Stable Marriage is a fundamental problem to both computer science and ec...
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Technical Report: Antiunification of Unordered Goals
Antiunification in logic programming refers to the process of capturing...
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Degrees and Gaps: Tight Complexity Results of General Factor Problems Parameterized by Treewidth and Cutwidth
For the General Factor problem we are given an undirected graph G and for each vertex v∈ V(G) a finite set B_v of nonnegative integers. The task is to decide if there is a subset S⊆ E(G) such that deg_S(v)∈ B_v for all vertices v of G. The maxgap of a finite integer set B is the largest d≥ 0 such that there is an a≥ 0 with [a,a+d+1]∩ B={a,a+d+1}. Cornuéjols (1988) showed that if the maxgap of all sets B_v is at most 1, then the decision version of General Factor is polytime solvable. Dudycz and Paluch (2018) extended this result for the minimization and maximization versions. Using convolution techniques from van Rooij (2020), we improve upon the previous algorithm by Arulselvan et al. (2018) and present an algorithm counting the number of solutions of a certain size in time O^*((M+1)^k), given a tree decomposition of width k, where M=max_v max B_v. We prove that this algorithm is essentially optimal for all cases that are not polynomial time solvable for the decision, minimization or maximization versions. We prove that such improvements are not possible even for BFactor, which is General Factor on graphs where all sets B_v agree with the fixed set B. We show that for every fixed B where the problem is NPhard, our new algorithm cannot be significantly improved: assuming the Strong Exponential Time Hypothesis (SETH), no algorithm can solve BFactor in time O^*((max B+1ϵ)^k) for any ϵ>0. We extend this bound to the counting version of BFactor for arbitrary, nontrivial sets B, assuming #SETH. We also investigate the parameterization of the problem by cutwidth. Unlike for treewidth, a larger set B does not make the problem harder: Given a linear layout of width k we give a O^*(2^k) algorithm for any B and provide a matching lower bound that this is optimal for the NPhard cases.
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