Ke Xu

Many hard examples in exact phase transitions

This paper analyzes the resolution complexity of two random constraint satisfaction problem (CSP) models (i.e. Model RB/RD) for which we can establish the existence of phase transitions and identify the threshold points exactly. By encoding CSPs into CNF formulas, it is proved that almost all instances of Model RB/RD have no tree-like resolution proofs of less than exponential size. Thus, we not only introduce new families of CSPs and CNF formulas hard to solve, which can be useful in the experimental evaluation of CSP and SAT algorithms, but also propose models with both many hard instances and exact phase transitions. Finally, conclusions are presented, as well as a detailed comparison of Model RB/RD with the Hamiltonian cycle problem and random 3-SAT, which, respectively, exhibit three different kinds of phase transition behavior in NP-complete problems.

The SAT phase transition

Phase transition is an important feature of SAT problem. For randomk-SAT model, it is proved that asr (ratio of clauses to variables) increases, the structure of solutions will undergo a sudden change like satisfiability phase transition whenr reaches a threshold point (r=rσ). This phenomenon shows that the satisfying truth assignments suddenly shift from being relatively different from each other to being very similar to each other.

Two hardness results on feedback vertex sets

A feedback vertex set is a subset of vertices whose deletion makes the remaining graph a forest. We show that the minimum FVS (MFVS) in star convex bipartite graphs is NP-hard to find, and give a tighter lower bound on the size of MFVS in sparse random graphs, to provide further evidence on the hardness of random CSPs.

Feedback vertex sets on restricted bipartite graphs

A feedback vertex set (FVS) in a graph is a subset of vertices whose complement induces a forest. Finding a minimum FVS is NP-complete on bipartite graphs, but tractable on convex bipartite graphs and on chordal bipartite graphs. A bipartite graph is called tree convex, if a tree is defined on one part of the vertices, such that for every vertex in the other part, its neighborhood induces a subtree. When the tree is a path, a triad or a star, the bipartite graph is called convex bipartite, triad convex bipartite or star convex bipartite, respectively. We show that: (1) FVS is tractable on triad convex bipartite graphs; (2) FVS is NP-complete on star convex bipartite graphs and on tree convex bipartite graphs where the maximum degree of vertices on the tree is at most three.

Independent Domination: Reductions from Circular- and Triad-Convex Bipartite Graphs to Convex Bipartite Graphs

An independent dominating set in a graph is a subset of vertices, such that no edge has both ends in the subset, and each vertex either itself is in the subset or has a neighbor in the subset. In a convex bipartite (circular convex bipartite, triad convex bipartite, respectively) graph, there is a linear ordering (a circular ordering, a triad, respectively) defined on one class of vertices, such that for every vertex in the other class, the neighborhood of this vertex is an interval (a circular arc, a subtree, respectively), where a triad is three paths with a common end. The problem of finding a minimum independent dominating set, called independent domination, is known (mathcal{NP})-complete for bipartite graphs and tractable for convex bipartite graphs. In this paper, we make polynomial time reductions for independent domination from triad- and circular-convex bipartite graphs to convex bipartite graphs.

Circular convex bipartite graphs

A feedback vertex set is a subset of vertices, such that the removal of this subset renders the remaining graph cycle-free. The weight of a feedback vertex set is the sum of weights of its vertices. Finding a minimum weighted feedback vertex set is tractable for convex bipartite graphs, but NP -complete even for unweighted bipartite graphs. In a circular convex (convex, respectively) bipartite graph, there is a circular (linear, respectively) ordering defined on one class of vertices, such that for every vertex in another class, the neighborhood of this vertex is a circular arc (an interval, respectively). The minimum weighted feedback vertex set problem is shown tractable for circular convex bipartite graphs in this paper, by making a Cook reduction (i.e. polynomial time Turing reduction) for this problem from circular convex bipartite graphs to convex bipartite graphs.

Large hypertree width for sparse random hypergraphs

Hypertree width is a graph-theoretic parameter similar to treewidth. It has many equivalent characterizations and many applications. If the hypertree width of the constraint graphs of the instances of a constraint satisfaction problem is bounded by a constant, then the CSP is tractable In this paper, we show that with high probability, hypertree width is large on sparse random

Tractable connected domination for restricted bipartite graphs

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