∈ S ′ : i ≥ 0}; T = {i ∈ T ′ : i ≥ 0}. Thus Min UnCut is equivalent to the following problem: Definition 2 Given a graph G =(V,E), where V = {−n,...,−1} ∪{−1,...,−n} findacut (S, T = −S) that minimizes the number of cut edges i.e. the number of edges going from the part S to T. 1.1 SDP relaxation Write

We show that the MULTICUT, SPARSEST-CUT, and MIN-2CNF ≡ DELETION problems are NP-hard to approximate within every constant factor, assuming the Unique Games Conjecture of Khot [STOC, 2002]. A quantitatively stronger version of the conjecture implies inapproximability factor of Ω(log log n). 1.

We show that the MULTICUT, SPARSEST-CUT and MIN-2CNF £ DELETION problems are hard to approximate, assuming the Unique Games Conjecture of Khot [Kho02]. In particular, we obtain an arbitrarily large constant factor hardness for these problems, and show that a quantitatively stronger version

’, ’2-CNF deletion’, ’2-SAT deletion’. The status of fixed-parameter tractability of this problem is a long-standing open question in the area of Parameterized Complexity. We resolve this open question by proposing an algorithm which solves this problem in O(15 k ∗ k ∗ m 3) and thus we show

, and constraint satisfaction problems such as Min UnCut and Min 2CNF Deletion. 2. An ~O(n3) time derandomization of the Alon-Roichman construction of expanders using Cayley graphs. The algorithm yields a set of O(log n) elements which generates an expanding Cayley graph in any group of n elements. 3. An ~O(n3

Various technologies are based on the capability to find small unsatisfiable cores given an unsatisfiable CNF formula, i.e., a subset of the clauses that are unsatisfiable regardless of the rest of the formula. If that subset is irreducible, it is called a Minimal Unsatisfiable Core (MUC). In many

d ≥ 2. The case d = 2 implies that no NP-hard vertex deletion problem based on a graph property that is inherited by subgraphs can have kernels consisting of O(k 2−ǫ) edges unless coNP is in NP/poly, where k denotes the size of the deletion set. Kernels consisting of O(k 2) edges are known

We consider the minimal unsatisfiablity problem MU (k) for propositional formulas in conjunctive normal form (CNF) over n variables and n + k clauses, where k is fixed. It will be shown that MU (k) is in NP. Based on the non--deterministic algorithm we prove for MU(2) that after a simplification

-of-the-art constraint solvers. In particular, it was recently shown that the problem of identifying a strong Horn- or 2CNF-backdoor can be solved by exploiting equivalence with deletion backdoors, and is NPcomplete. We prove that strong backdoor identification becomes harder than NP (unless NP=coNP) as soon

f, and before applying any other rule, a simplification rule deletes all its other occurrences (at any level of nesting) and perform some suitable boolean reduction. 2 Theoretical Results A number of techniques can be shown to be restricted forms of a local simplification rule (LSR): ffl