
On graphs of bounded degree that are far from being Hamiltonian
Hamiltonian cycles in graphs were first studied in the 1850s. Since then...
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Parameterizing the Permanent: Hardness for K_8minorfree graphs
In the 1960s, statistical physicists discovered a fascinating algorithm ...
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A complexity dichotomy for hitting connected minors on bounded treewidth graphs: the chair and the banner draw the boundary
For a fixed connected graph H, the {H}MDELETION problem asks, given a ...
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Hamiltonicity: Variants and Generalization in P_5free Chordal Bipartite graphs
A bipartite graph is chordal bipartite if every cycle of length at least...
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Approximation in (Poly) Logarithmic Space
We develop new approximation algorithms for classical graph and set prob...
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NC Algorithms for Perfect Matching and Maximum Flow in OneCrossingMinorFree Graphs
In 1988, Vazirani gave an NC algorithm for computing the number of perfe...
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Isolation schemes for problems on decomposable graphs
The Isolation Lemma of Mulmuley, Vazirani and Vazirani [Combinatorica'87] provides a selfreduction scheme that allows one to assume that a given instance of a problem has a unique solution, provided a solution exists at all. Since its introduction, much effort has been dedicated towards derandomization of the Isolation Lemma for specific classes of problems. So far, the focus was mainly on problems solvable in polynomial time. In this paper, we study a setting that is more typical for ππ―complete problems, and obtain partial derandomizations in the form of significantly decreasing the number of required random bits. In particular, motivated by the advances in parameterized algorithms, we focus on problems on decomposable graphs. For example, for the problem of detecting a Hamiltonian cycle, we build upon the rankbased approach from [Bodlaender et al., Inf. Comput.'15] and design isolation schemes that use  O(tlog n + log^2n) random bits on graphs of treewidth at most t;  O(β(n)) random bits on planar or Hminor free graphs; and  O(n)random bits on general graphs. In all these schemes, the weights are bounded exponentially in the number of random bits used. As a corollary, for every fixed H we obtain an algorithm for detecting a Hamiltonian cycle in an Hminorfree graph that runs in deterministic time 2^O(β(n)) and uses polynomial space; this is the first algorithm to achieve such complexity guarantees. For problems of more local nature, such as finding an independent set of maximum size, we obtain isolation schemes on graphs of treedepth at most d that use O(d) random bits and assign polynomiallybounded weights. We also complement our findings with several unconditional and conditional lower bounds, which show that many of the results cannot be significantly improved.
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