Yash Raj Shrestha

Parameterized Complexity of Edge Interdiction Problems

For an optimization problem on edge-weighted graphs, the corresponding interdiction problem can be formulated as a game consisting of two players, namely, an interdictor and an evader, who compete on an objective with opposing interests. In an edge interdiction problem, every edge of the input graph is associated with an interdiction cost. The interdictor interdicts the graph by modifying the edges in the graph and the number of such modifications is bounded by the interdictor’s budget. The evader then solves the given optimization problem on the modified graph. The action of the interdictor must impede the evader as much as possible.

Kernelization of Two Path Searching Problems on Split Graphs

In the k-Vertex-Disjoint Paths problem, we are given a graph G and k terminal pairs of vertices, and are asked whether there is a set of k vertex-disjoint paths linking these terminal pairs, respectively. In the k-Path problem, we are given a graph and are asked whether there is a path of length k. It is known that both problems are NP-hard even in split graphs, which are the graphs whose vertices can be partitioned into a clique and an independent set. We study kernelization for the two problems in split graphs. In particular, we derive a 4k vertex-kernel for the k-Vertex-Disjoint Paths problem and a \(\frac{3}{2}k^2+\frac{1}{2}k\) vertex-kernel for the k-Path problem.

Complexity of Disjoint Π-Vertex Deletion for Disconnected Forbidden Subgraphs

We investigate the computational complexity of Disjoint Π-Vertex Deletion. Here, given an input graph G = (V,E) and a vertex set S ⊆ V, called a solution set, whose removal results in a graph satisfying a non-trivial, hereditary property Π, we are asked to find a solution set S′ with |S′| < |S| and S′ ∩ S = ∅. This problem is partially motivated by the “compression task” occurring in the iterative compression technique. The complexity of this problem has already been studied, with the restriction that Π is satisfied by a graph G iff Π is satisfied by each connected component of G [7]. In this work, we remove this restriction and show that, except for few cases which are polynomial-time solvable, almost all other cases of Disjoint Π-Vertex Deletion are \(\mathcal{NP}\)-hard.

Controlling two-stage voting rules

We study the computational complexity of control problems for two-stage voting rules. An example of a two-stage voting rule is the Blacks procedure. The first stage of the Blacks procedure selects the Condorcet winner if one exists; otherwise, in the second stage the Borda winner is selected. The computational complexity of the manipulation problem of two-stage voting rules has recently been studied by Narodytska and Walsh [20] and Fitzsimmons et al. [14]. Extending their work, we consider the control problems for similar scenarios, focusing on constructive control by adding or deleting votes, denoted as CCAV and CCDV, respectively. Let X be the voting rule applied in the first stage and Y the one in the second stage. As for the manipulation problem shown in [20, 14], we prove that there is basically no connection between the complexity of CCAV and CCDV for X or Y and the complexity of CCAV and CCDV for the two-stage election X THEN Y : CCAV and CCDV for X THEN Y could be NP-hard, while both problems are polynomial-time solvable for X and Y. On the other hand, combining two rules X and Y, both with NP-hard CCAV and CCDV, could lead to a two-stage election, where both CCAV and CCDV become polynomial-time solvable. Hereby, we also achieve some complexity results for the special case X THEN X. In addition, we show that, compared to the manipulation problem, the control problems for two-stage elections admit more diverse behaviors concerning their complexity. For example, there exist rules X and Y, for each of which CCAV and CCDV have the same complexity, but CCAV and CCDV behave differently for X THEN Y.

On the complexity of bribery with distance restrictions

Abstract We study the complexity of the constructive/destructive bribery problem with distance restrictions. In the constructive/destructive bribery problem, we are given an election and a distinguished candidate p, and are asked whether we can make p a winner/loser by bribing a limited number of voters to recast their votes. In the constructive/destructive bribery problem with distance restrictions, we require that the votes recast by the bribed voters are close to their original votes. In particular, we measure the closeness of two votes by the minimum number of swaps of candidates needed to transform one vote into the other. We consider both the case where swaps only take place between consecutive candidates, and the case where swaps can take place between any arbitrary candidates. We achieve a wide range of complexity results for the voting correspondences Borda, Condorcet, Copeland α for every rational number 0 ≤ α ≤ 1 , and Maximin.

On the kernelization of split graph problems

Abstract A split graph is a graph whose vertices can be partitioned into a clique and an independent set. We study numerous problems on split graphs, namely the k -Vertex-Disjoint Paths , k -Cycle , k -Path and k - l -Stable Set problems. In the k -Vertex-Disjoint Paths problem, we are given a graph and k terminal pairs of vertices, and are asked whether there is a set of k vertex-disjoint paths linking these terminal pairs, respectively. In the k -Cycle / k -Path problem, we are given a graph and are asked whether there is a path/cycle of length k . The k - l -Stable Set problem takes a graph and an integer k as input, and asks whether the graph has a subset of k vertices such that the distance between every two vertices in the subset is at least l + 1 . It is known that all the above problems are NP-complete on split graphs. We derive a 4 k -vertex kernel for the k -Vertex-Disjoint Paths problem and an O ( k 2 ) -vertex kernel for both the k -Path problem and the k -Cycle problem. Concerning the k - l -Stable Set problem, for l = 1 or l ≥ 3 , the problem is polynomial-time solvable on split graphs. For l = 2 , we prove that the k - l -Stable Set problem is W[1]-complete on split graphs, with respect to k . However, if the given split graph contains no K 1 , r as an induced subgraph, and every vertex in the independent set of the split graph has degree at most d , we derive a linear vertex kernel for the k - 2 -Stable Set problem, where both r and d are constants.

Kernelization and Parameterized Complexity of Star Editing and Union Editing

The NP-hard Star Editing problem has as input a graph G = (V,E) with edges colored red and black and two positive integers k 1 and k 2, and determines whether one can recolor at most k 1 black edges to red and at most k 2 red edges to black, such that the resulting graph has an induced subgraph whose edge set is exactly the set of black edges. A generalization of Star Editing is Union Editing, which, given a hypergraph H with the vertices colored by red and black and two positive integers k 1 and k 2, determines whether one can recolor at most k 1 black vertices to red and at most k 2 red vertices to black, such that the set of red vertices becomes exactly the union of some hyperedges. Star Editing is equivalent to Union Editing when the maximum degree of H is bounded by 2. Both problems are NP-hard and have applications in chemical analytics. Damaschke and Molokov [WADS 2011] introduced another version of Star Editing, which has only one integer k in the input and asks for a solution of totally at most k recolorings, and proposed an O(k 3)-edge kernel for this new version. We improve this bound to O(k 2) and show that the O(k 2)-bound is basically tight. Moreover, we also derive a kernel with O((k 1 + k 2)2) edges for Star Editing. Fixed-parameter intractability results are achieved for Star Editing parameterized by any one of k 1 and k 2. Finally, we extend and complete the parameterized complexity picture of Union Editing parameterized by k 1 + k 2.