On Feedback Vertex Set: New Measure and New Structures

We present a new parameterized algorithm for the {feedback vertex set} problem ({\sc fvs}) on undirected graphs. We approach the problem by considering a variation of it, the {disjoint feedback vertex set} problem ({\sc disjoint-fvs}), which finds a feedback vertex set of size $k$ that has no overlap with a given feedback vertex set $F$ of the graph $G$. We develop an improved kernelization algorithm for {\sc disjoint-fvs} and show that {\sc disjoint-fvs} can be solved in polynomial time when all vertices in $G \setminus F$ have degrees upper bounded by three. We then propose a new branch-and-search process on {\sc disjoint-fvs}, and introduce a new branch-and-search measure. The process effectively reduces a given graph to a graph on which {\sc disjoint-fvs} becomes polynomial-time solvable, and the new measure more accurately evaluates the efficiency of the process. These algorithmic and combinatorial studies enable us to develop an $O^*(3.83^k)$-time parameterized algorithm for the general {\sc fvs} problem, improving all previous algorithms for the problem.Comment: Final version, to appear in Algorithmic

Clique complexes and graph powers

We study the behaviour of clique complexes of graphs under the operation of taking graph powers. As an example we compute the clique complexes of powers of cycles, or, in other words, the independence complexes of circular complete graphs.Comment: V3: final versio

On Locally Decodable Index Codes

Index coding achieves bandwidth savings by jointly encoding the messages demanded by all the clients in a broadcast channel. The encoding is performed in such a way that each client can retrieve its demanded message from its side information and the broadcast codeword. In general, in order to decode its demanded message symbol, a receiver may have to observe the entire transmitted codeword. Querying or downloading the codeword symbols might involve costs to a client -- such as network utilization costs and storage requirements for the queried symbols to perform decoding. In traditional index coding solutions, this 'client aware' perspective is not considered during code design. As a result, for these codes, the number of codeword symbols queried by a client per decoded message symbol, which we refer to as 'locality', could be large. In this paper, considering locality as a cost parameter, we view index coding as a trade-off between the achievable broadcast rate (codeword length normalized by the message length) and locality, where the objective is to minimize the broadcast rate for a given value of locality and vice versa. We show that the smallest possible locality for any index coding problem is 1, and that the optimal index coding solution with locality 1 is the coding scheme based on fractional coloring of the interference graph. We propose index coding schemes with small locality by covering the side information graph using acyclic subgraphs and subgraphs with small minrank. We also show how locality can be accounted for in conventional partition multicast and cycle covering solutions to index coding. Finally, applying these new techniques, we characterize the locality-broadcast rate trade-off of the index coding problem whose side information graph is the directed 3-cycle.Comment: 10 pages, 1 figur

Hitting Weighted Even Cycles in Planar Graphs

A classical branch of graph algorithms is graph transversals, where one seeks a minimum-weight subset of nodes in a node-weighted graph G which intersects all copies of subgraphs F from a fixed family F. Many such graph transversal problems have been shown to admit polynomial-time approximation schemes (PTAS) for planar input graphs G, using a variety of techniques like the shifting technique (Baker, J. ACM 1994), bidimensionality (Fomin et al., SODA 2011), or connectivity domination (Cohen-Addad et al., STOC 2016). These techniques do not seem to apply to graph transversals with parity constraints, which have recently received significant attention, but for which no PTASs are known. In the even-cycle transversal (ECT) problem, the goal is to find a minimum-weight hitting set for the set of even cycles in an undirected graph. For ECT, Fiorini et al. (IPCO 2010) showed that the integrality gap of the standard covering LP relaxation is ?(log n), and that adding sparsity inequalities reduces the integrality gap to 10. Our main result is a primal-dual algorithm that yields a 47/7 ? 6.71-approximation for ECT on node-weighted planar graphs, and an integrality gap of the same value for the standard LP relaxation on node-weighted planar graphs

Fixed-parameter tractability for the subset feedback set problem and the S-cycle packing problem

AbstractWe investigate generalizations of the following well-known problems in the framework of parameterized complexity: the feedback set problem and the cycle packing problem. Our problem setting is that we are given a graph and a vertex set S called “terminals”. Our purpose here is to consider the following problems:1.The feedback set problem with respect to the terminals S. We call it the subset feedback set problem.2.The cycle packing problem with respect to the terminals S, i.e., each cycle has to contain a vertex in S (such a cycle is called an S-cycle). We call it the S-cycle packing problem. We give the first fixed parameter algorithms for the two problems. Namely;1.For fixed k, we can either find a vertex set X of size k such that G−X has no S-cycle, or conclude that such a vertex set does not exist in O(n2m) time, where n is the number of vertices of the input graph and m is the number of edges of the input graph.2.For fixed k, we can either find k vertex-disjoint S-cycles or conclude that such k disjoint cycles do not exist in O(n3) time

One-Way Communication Complexity of Partial XOR Functions

Boolean function $F(x,y)$ for $x,y \in \{0,1\}^n$ is an XOR function if $F(x,y)=f(x\oplus y)$ for some function $f$ on $n$ input bits, where $\oplus$ is a bit-wise XOR. XOR functions are relevant in communication complexity, partially for allowing Fourier analytic technique. For total XOR functions it is known that deterministic communication complexity of $F$ is closely related to parity decision tree complexity of $f$. Montanaro and Osbourne (2009) observed that one-sided communication complexity $D_{cc}^{\rightarrow}(F)$ of $F$ is exactly equal to nonadaptive parity decision tree complexity $NADT^{\oplus}(f)$ of $f$. Hatami et al. (2018) showed that unrestricted communication complexity of $F$ is polynomially related to parity decision tree complexity of $f$. We initiate the studies of a similar connection for partial functions. We show that in case of one-sided communication complexity whether these measures are equal, depends on the number of undefined inputs of $f$. On the one hand, if $D_{cc}^{\rightarrow}(F)=t$ and $f$ is undefined on at most $O(\frac{2^{n-t}}{\sqrt{n-t}})$, then $NADT^{\oplus}(f)=t$. On the other hand, for a wide range of values of $D_{cc}^{\rightarrow}(F)$ and $NADT^{\oplus}(f)$ (from constant to $n-2$) we provide partial functions for which $D_{cc}^{\rightarrow}(F) < NADT^{\oplus}(f)$. In particular, we provide a function with an exponential gap between the two measures. Our separation results translate to the case of two-sided communication complexity as well, in particular showing that the result of Hatami et al. (2018) cannot be generalized to partial functions. Previous results for total functions heavily rely on Boolean Fourier analysis and the technique does not translate to partial functions. For the proofs of our results we build a linear algebraic framework instead. Separation results are proved through the reduction to covering codes