Showing papers by "Center for Discrete Mathematics and Theoretical Computer Science published in 2012"

Beck's Three Permutations Conjecture: A Counterexample and Some Consequences

Abstract: Given three permutations on the integers 1 through $n$, consider the set system consisting of each interval in each of the three permutations. In 1982, Beck conjectured that the discrepancy of this set system is $O(1)$. In other words, the conjecture says that each integer from 1 through $n$ can be colored either red or blue so that the number of red and blue integers in each interval of each permutations differs only by a constant. (The discrepancy of a set system based on two permutations is at most two.) Our main result is a counterexample to this conjecture: for any positive integer $n = 3^k$, we construct three permutations whose corresponding set system has discrepancy $\Omega(\log{n})$. Our counterexample is based on a simple recursive construction, and our proof of the discrepancy lower bound is by induction. This construction also disproves a generalization of Beck's conjecture due to Spencer, Srinivasan and Tetali, who conjectured that a set system corresponding to $\ell$ permutations has discrepancy $O(\sqrt{\ell})$. Our work was inspired by an intriguing paper from SODA 2011 by Eisenbrand, P{\'a}lv{\"o}lgyi and Rothvo\ss, who show a surprising connection between the discrepancy of three permutations and the bin packing problem: They show that Beck's conjecture implies a constant worst-case bound on the additive integrality gap for the Gilmore-Gomory LP relaxation for bin packing in the special case when all items have sizes strictly between $1/4$ and $1/2$, also known as the three partition problem. Our counterexample shows that this approach to bounding the additive integrality gap for bin packing will not work. We can, however, prove an interesting implication of our construction in the reverse direction: there are instances of bin packing and corresponding optimal basic feasible solutions for the Gilmore-Gomory LP relaxation such that any packing that contains only patterns from the support of these solutions requires at least ${\rm opt} + \Omega(\log{m})$ bins, where $m$ is the number of items. Finally, we discuss some implications that our construction has for other areas of discrepancy theory.

An artificial bee colony algorithm approach for routing in VLSI

Abstract: This paper presents an approach that applies the Artificial Bee Colony algorithm to the Two-Terminals-Net-Routing(TTNR) problem in VLSI physical design and compares its performance with the maze algorithm variant known as the state-of-the-art global routing algorithm. An effectively encoding method is described in this paper to solve the TTNR problem. In order to improve the convergence speed of the algorithm, some guiding solutions are employed as the initial solutions. The experimental results demonstrate that Artificial Bee Colony algorithm can find the less cost routing paths for TTNR problems than the maze algorithm.

A heuristic algorithm for the strip packing problem

Abstract: The two-dimensional strip packing problem is to pack a given set of rectangles into a strip with a given width and infinite height so as to minimize the required height of the packing. From the computational point of view, the strip packing problem is an NP-hard problem. With the B*-tree representation, this paper first presents a heuristic packing strategy which evaluates the positions used by the rectangles. Then an effective local search method is introduced to improve the results and a heuristic algorithm (HA) is further developed to find a desirable solution. Computational results on randomly generated instances and popular test instances show that the proposed method is efficient for the strip packing problem.

A discrete dynamic convexized method for the max-cut problem

Abstract: The max-cut problem is a classical NP-hard problem in graph theory. In this paper, we adopt a local search method, called MCFM, which is a simple modification of the Fiduccia-Mattheyses heuristic method in Fiduccia and Mattheyses (Proc. ACM/IEEE DAC, pp. 175–181, 1982) for the circuit partitioning problem in very large scale integration of circuits and systems. The method uses much less computational cost than general local search methods. Then, an auxiliary function is presented which has the same global maximizers as the max-cut problem. We show that maximization of the function using MCFM can escape successfully from previously converged discrete local maximizers by taking increasing values of a parameter. An algorithm is proposed for the max-cut problem, by maximizing the auxiliary function using MCFM from random initial solutions. Computational experiments were conducted on three sets of standard test instances from the literature. Experimental results show that the proposed algorithm is effective for the three sets of standard test instances.

The competition number of a generalized line graph is at most two

Abstract: In 1982, Opsut showed that the competition number of a line graph is at most two and gave a necessary and sufficient condition for the competition number of a line graph being one. In this paper, we generalize this result to the competition numbers of generalized line graphs, that is, we show that the competition number of a generalized line graph is at most two, and give necessary conditions and sufficient con ditions for the competition number of a generalized line graph being one.

A real-time capacitor placement considering systematic and random mismatches in analog IC

Abstract: Capacitor mismatches result from systematic mismatches and random mismatches. It is well-known that placement in common-centroid and symmetric structure can be used to efficiently reduce systematic mismatch. However, such structure is useless for reducing random mismatch. It is found that, based on a spatial correlation model, higher correlation coefficient (a value for measuring dispersion) results in lower random mismatch and higher chip yield. This paper proposes an algorithm which can immediately achieve placements in common-centroid, symmetric structure to reduce systematic mismatches, and high correlation coefficient to reduce random mismatches. The experiment results show that the proposed algorithm can reduce the running times from dozens of minutes to zeros, and achieve correlation coefficients, in average, up to 94.29% of the known best results which are derived from searching almost the whole solution space.

MSOL Restricted Contractibility to Planar Graphs

Abstract: We study the computational complexity of graph planarization via edge contraction. The problem CONTRACT asks whether there exists a set $S$ of at most $k$ edges that when contracted produces a planar graph. We work with a more general problem called $P$-RESTRICTEDCONTRACT in which $S$, in addition, is required to satisfy a fixed MSOL formula $P(S,G)$. We give an FPT algorithm in time $O(n^2 f(k))$ which solves $P$-RESTRICTEDCONTRACT, where $P(S,G)$ is (i) inclusion-closed and (ii) inert contraction-closed (where inert edges are the edges non-incident to any inclusion minimal solution $S$). As a specific example, we can solve the $\ell$-subgraph contractibility problem in which the edges of a set $S$ are required to form disjoint connected subgraphs of size at most $\ell$. This problem can be solved in time $O(n^2 f'(k,\ell))$ using the general algorithm. We also show that for $\ell \ge 2$ the problem is NP-complete.