The framework of Bodlaender et al. (ICALP 2008) and Fortnow and Santhanam (STOC 2008) allows us to exclude the existence of polynomial kernels for a range of problems under reasonable complexity-theoretical assumptions. However, there are also some issues that are not addressed by this framework, including the existence of Turing kernels such as the “kernelization” of Leaf Out Branching(k) into...

This paper formulates a necessary and sufficient condition for a generic graph matching problem to be equivalent to the maximum vertex and edge weight clique problem in a derived association graph. The consequences of this results are threefold: first, the condition is general enough to cover a broad range of practical graph matching problems; second, a proof to establish equivalence between gr...

Let G be a simple graph on vert ices having a maximum matching M. The deficiency def(G) of G is the number of M-unsaturated vertices in G. The vertex clique covering number vcc(G) of G is the smallest number of cliques (complete subgraphs) needed to cover the vertex set of G. In this determine def(G) and vcc(G) for the case when G is a tree with each vertex having degree 1 or k.

K. F. Fraughnaugh et al. proved that a graph G is the competition graph of a hamiltonian digraph possibly having loops if and only if G has an edge clique cover C = {C1, . . . , Cn} that has a system of distinct representatives. [SIAM J. Discrete Math., 8 (1995), pp. 179–185]. We settle a question left open by their work, by showing that the words “possibly having loops” may be removed.

For each m ≥ 1, we construct a graph G = (V, E) with ω(G) = m such that max 1≤i≤k ω(G[Vi]) = m for arbitrary partition V = V1 ∪ · · · ∪ Vk, where ω(G) is the clique number of G and G[Vi] is the induced graph of Vi. Using this result, we show that for each m ≥ 2 there exists an exact m-cover of Z which is not the union of two 1-covers.

Quantum walks have received a great deal of attention recently because they can be used to develop new quantum algorithms and to simulate interesting quantum systems. In this work, we focus on a model called staggered quantum walk, which employs advanced ideas of graph theory and has the advantage of including the most important instances of other discrete-time models. The evolution operator of...

We present a heuristic scheme for nding a clique of maximum size or weight. Our heuristic scheme uses approximation techniques for the weighted vertex cover problem, due to R. Bar-Yehuda and S. Even BE85]. The approach is based on the notion of making incremental progress towards nding a clique. At each step of our algorithm, one or more local optimization techniques are attempted, and when the...

We study the min-max problem in factor graphs, which seeks the assignment that minimizes the maximum value over all factors. We reduce this problem to both min-sum and sum-product inference, and focus on the later. This approach reduces the min-max inference problem to a sequence of constraint satisfaction problems (CSPs) which allows us to sample from a uniform distribution over the set of sol...

This paper formulates a necessary and sufficient condition for a generic graph matching problem to be equivalent to the maximum vertex and edge weight clique problem in a derived association graph. The consequences of this results are threefold: first, the condition is general enough to cover a broad range of practical graph matching problems; second, a proof to establish equivalence between gr...